CONSTRAINED KARTREE-FOCK WAVE FUNCTIONS FOR ATOMS by H i l a l Ahmed Qureshi A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag r ee tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t hou t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada ABSTRACT This thesis deals with the calculation of Hartree-Fock wave functions satisfying an off-diagonal hypervirial relation as a constraint. The constraint in this case implies that the dipole length form and the dipole velocity form of the transition probability give identical values. Mathematically, this is equivalent to forcing the approximate eigenfunctions of the Hamiltonian of the system to satisfy a relation which is true for exact eigenfunctions. The method of constrained variation is used to solve this problem. The constrained Hartree-Fock system of equations is solved numerically. The Z-expansions of radial wave functions, the diagonal and the off-diagonal energy parameters and the parameter of constraint are carried out. The effect of the constraint on the total energy E of the system, defined as the change in the Hartree-Fock total energy due to the constraint, i s estimated. The method of constrained variation is then applied to a few two, three and four electron systems to calculate the constrained total energy E of the system and also the oscillator strengths of a few of the transitions of the system. The results indicate that the oscillator strengths can be calculated more accurately, at practically no cost of the total energy E, with the aid of the constrained Hartree-Fock functions than with the standard Hartree-Fock functions in a l l those cases where the correlation effects are not too strong to invalidate the single configuration approximation. i ACKNOWLEDGEMENTS The author would like to express his sincere thanks to his thesis supervisor, Professor Charlotte Froese Fischer, for her constant help and advice throughout the progress of the present work. The author appreciates the efforts of the Head, Department of Mathematics, University of British Columbia, in connection with the continuation of the program of studies after the termination of the author's Commonwealth Scholarship. Thanks are also due to the Head, Department of Applied Analysis and Computer Science, University of Waterloo, for allowing the author the use of the computing f a c i l i t i e s and an office space for about two years. For financial assistance, thanks are due to the Canadian Commonwealth Scholarship Committee and the National Research Council of Canada. Finally, for the patience shown in typing this thesis, the author would like to thank Mrs. Erna Unrau and Miss Vickie MacKinnon. TABLE OF CONTENTS CHAPTER I Page 1.1 The Schrbdinger equation for an atomic system 1 1.2 The Hartree-Fock method for multi-electron atoms 4 1.3 The hypervirial relations as constraints 7 1.4 Derivation of the constraint in terms of radial 11 functions 1.5 Derivation of the radial equation for the 13 constrained system CHAPTER II 2.1 The nuclear charge as a parameter 17 2.2 Z-expansion of radial wave functions and various 19 atomic parameters 2.3 Z-expansion of the constraint 25 2.4 Z-expansion of the parameter of the constraint 27 2.5 The effect of the constraint on the total energy 31 E of the system CHAPTER III 3.1 Computational procedure 35 3.2 Results for three-electron systems 40 3.3 Results for four-electron systems 48 CHAPTER IV 4.1 Comparison of Oscillator strengths 55 CONCLUSIONS 73 BIBLIOGRAPHY 75 i i LIST OF TABLES Page 2 2 2 2 TABLE I: Results for the transition Is 2s Sx - Is 2p P, . 41 'I *S 2 2 2 2 TABLE II. Results for the transition Is 2s St - Is 3p P, . 42 -2 % 2 2 2 2 TABLE III: Results for the transition Is 2s Sx - Is 4p P, . 43 TABLE IV: Results for the transition I s 2 3s 2S1 - I s 2 2p 2P X . 44 TABLE V: Results for the transition I s 2 3s 2S1 - I s 2 3p 2Pj . 45 TABLE VI: Results for the transition I s 2 2s 2 "'"S - I s 2 2s 2p 1P 1« 49 TABLE VII: Results for the transition I s 2 2s 2 1SQ - I s 2 2s 3p "4^. 50 TABLE VIII: Results for the transition I s 2 2s 2 1 S Q - I s 2 2s 4p . 51 TABLE IX: Orthogonality integrals for the constrained ground 53 state functions. TABLE X: Comparisons of oscillator strengths for He transitions. 60 2 2 TABLE XI: Comparisons of oscillator strengths for Is 2s - Is 2p. 62 2 2 TABLE XII: Comparisons of oscillator strengths for Is 2s - Is 3p. 64 2 2 TABLE XIII: Comparisons of oscillator strengths for Is 3s - Is 3p. 65 2 2 TABLE XIV: Comparisons of oscillator strengths for Is 3s - Is 2p. 67 2 2 2 TABLE XV: Comparisons of oscillator strengths for Is 2s - Is 2s 2p. 68 2 2 2 TABLE XVI: Comparisons of oscillator strengths for Is 2s - Is 2s 3p. 70 i i i TO CHAPTER I Sec. 1.1. The Schrcidinger Equation for an Atomic System The n o n - r e l a t i v i s t i c Schrcidinger equation f o r an atomic system c o n s i s t i n g of a point nucleus, of i n f i n i t e mass, and N-electrons, of unit mass each, i s HT1 = EWT, (1) where the Hamiltonian H i s given by (with Z denoting the nuclear charge) ? 1 2 Z ? l . L T 2 I r. r. . i = l i i>j i j The subscripts 1, 2, N d i s t i n g u i s h d i f f e r e n t e lectrons, r^ i s the distance of the i - t h electron from the nucleus, r . . i s the mutual distance between the i - t h and j - t h electrons and V\ i s the gradient operator with respect to the coordinates of i - t h e l e c t r o n . In the present study we s h a l l be concerned only with the bound states of the system corresponding to the d i s c r e t e part of the spectrum of H. We postulate that each electron of the system possesses an i n t r i n s i c angular momentum c a l l e d the spin whose component i n any d i r e c t i o n i n space can take only two values. Thus each e l e c t r o n i s s p e c i f i e d by three space coordinates r , 0, ct and a spin coordinate s. Since the Hamiltonian given by (2) does not depend on the spin coordinates, T the t o t a l wave function ¥ (r-^,8^, <j>^, s^, r ^ , 8^ , ct^, s^) of the system factors into a coordinate wave function ^ ( r ^ , 8^ , <j>^, r ^ , 8^ , ct and a spin wave function x(s2» •*•» • Since each one of the spin coordinates s^, s^, ..•, s^ only takes two v a l u e s , . d i f f e r e n t values of the spin function x may be considered as a s u f f i x to the coordinate wave - 2 -T function In order that the total wave function ¥ be capable of physical interpretation we postulate that: T (i) The total wave function ¥ of an N-electron system i s antisymmetrical with respect to the permutation of any two electrons. ( i i ) The coordinate wave functions corresponding to each suffix determined by the spin function X» belong to a Hilbert space , where % - L 2 ( R 3 N ) = {u: / |u|2 dV < =0}. " R 3 N The scalar product of two wave functions u and v, denoted by (u | v), is defined as (u | v) = J u vdT r and the matrix element of an operator A between two wave functions u and v, denoted by (u | A | v), i s defined as (u | A | v) = / u*Avdt . r In the above, t denotes the complex conjugate and / dT is meant to r include summation over the spin variables as well as integration over 3N the whole coordinate space R . 2 2 2 The total angular momentum operators L , S , J and (Slater (1960); Chapters 11 and 13) for an atomic system commute with the Hamiltonian H of the system. It i s , therefore, possible to choose 2 2 2 a set of common eigenfunctions of H, L , S , J and J ^ ; each eigenfunction of H can then be characterized by the L, S, J and Mj values that determine - 3 -2 2 2 the eigenvalues of L , S , J and respectively. This leads to a classification of energy states into sets called terms and levels. A level consists of a l l the energy states having the same set of L, S and J values, whereas, a term consists of a l l the levels having the same pair of L and S values. The Hamiltonian H as given by' (2) neglects the spin-orbit interaction. In this case the z-components L and S of z z the total orbital- and the total spin-angular momentum L and S also 2 2 2 cummute with H, L , S , J and J . The eigenfunctions of H can then z be characterized by L, S, and Mg values that determine the eigenvalues 2 2 of L , S , L , and S respectively, z z Only the SchrBdinger equation for a single electron system can be solved exactly. In this case, because of the spherical symmetry of the Hamiltonian, the equation i s separable. The solution has the form u = ^ P(n£, r)Y™(6,cf>) . (3) Here -^P(n£, r) denotes the radial part of the solution and Y™ denotes the spherical harmonic. For two-electron systems, due to the presence of the inter-electron distance i - n t n e Hamiltonian, exact solution of the SchrBdinger equation is not possible. However, because of the practicability of introducing coordinate systems which include the inter-electron distance r ^ exp l i c i t l y , approximate solutions of the SchrBdinger equation have been developed to a stage of high accuracy using expansion techniques. Some highly accurate solutions for two-electron systems can be found in the works of Stewart (1963), Pekeris (1958, 1959) and Schiff et. a l . (1965). - 4 -, Sec. 1.2. The Hartree-Fock Method for Multi-electron Atoms For multi-electron atoms the eigenvalue problem, HY = EY ? c ? C = L 2 ( R 3 N ) , ( 4 ) is so complex that exact solutions are not feasible and one has to use some approximate method. Before introducing our approximations we give the following variational characterization of the eigenvalues and eigenfunctions belonging to the discrete part of the spectrum of H. Let W(u) be a functional defined on by „/ N (u I H|u) f<r / C N W ( U ) = (u|u) 5 U e % ' ( 5 ) In terms of this functional W(u), the eigenvalues ( i f there are any) and the corresponding eigenfunctions, belonging to the discrete part of the spectrum of H, can be characterized as follows (Messiah (1962); pp. 762-766): Any function u e j ( for which W(u), considered as a functional of *J£ , is stationary, is an eigenfunction of the discrete part of the spectrum of H, and conversely. The corresponding eigenvalue is the stationary value of the functional W(u). For the least eigenvalue Eg of H, one can further state that: For a l l functions u e "JC > the value of the functional W at u is equal to or greater than the least eigenvalue of H, i.e. W(u) £ E^. The Hartree-Fock method (Hartree (1957), Chapter 3.) is a variational method of finding the stationary values of the functional W(u) and the corresponding functions u, where u now belongs to a subspace - 5 -of?£ . This subspace of !KL is determined by the physical considerations and can be described as follows,. To each stationary state of the system is assigned a definite configuration which i s a l i s t i n g of the principal and the azimuthal quantum numbers of the electrons of the system (only single configuration approximation is being considered in the present work). The wave function u is taken to be a linear combination of antisymmetric products of one-electron functions * K r ) , belonging to the corresponding configuration, characterized by quantum numbers n, SL, m^ and m , with s i|; = ^ P(n£, r) Yj*(8, 4>) 3C(s, mg) /m£/-A A\ W e ™ »^ (6) Here $c is a spin function and the same radial function P(nJ£, r) i s used for a l l the electrons in one subshell, i.e. with the same n,£ quantum numbers. The boundary conditions for P(n£, r) are P(n£, 0) = P(n£, °°) - 0 • (7) The wave function u for the system can then be represented as a linear combination $, which diagonalizes a l l the four commuting operators 2 2 L , S , L z and Sz, of Slater's determinants I(Slater (1960), Chapter 12) of the type I/NT ^ v ( l ) if»v(2) ^ (N) ^g(N) * V ( N ) (8) where a, 3> V are the occupied-one-electron spin-orbitals and 1, 2, N ,are the labels of N electrons. - 6 -With $ taken as mentioned above and ^ a ( j ) as i n (6), the int e g r a t i o n s , i n the expression for W($), over the s p h e r i c a l polar angles can be carried out formally, leaving i n t e g r a l s over the r a d i a l coordinates only. Thus, W($) consists of i n t e g r a l s of the type: P(n£, r)[-*5- + ^ - *&f±\Hnl, r ) d r dr r F*(n£,n'£') = P2(n£, r)-^ ^ (n'A', nV, r)dr (9) G*(n£, n'£') - P(n£, r)P(n'£', r).--Yk(nj>, nV, r ) d r where Y^(n5,, n'£', r) i s given by Y k(n£, n ' £ \ r) -s=0 + (^-)k P(n£, s)P(n'£', s)ds (J) k + 1 P(n£, s)P(n'£\ s)ds. (10) s=r (The actual form of !'7($) i n terms of these r a d i a l i n t e g r a l s depends on the configuration being considered.) The Hartree-Fock method obtains the best possible r a d i a l functions, P(n£, r ) , i n the sense of the v a r i a t i o n p r i n c i p l e that 6w($) = 0 for a l l 6P(n£, r ) . The v a r i a t i o n , 6w($), for the v a r i a t i o n 6P(n&, r) can be derived. The condition that 6 W ( $ ) vanishes independently of 6P(n&, r) y i e l d s a non-linear i n t e g r o d i f f e r e n t i a l equation f o r P(n&, r) . The boundary conditions f o r P(n£, r) have already been determined to meet the p h y s i c a l requirements. One such equation i s obtained f o r each P(n£, r) The coupled system of equations so obtained i s solved i t e r a t i v e l y u n t i l the solutions are s e l f - c o n s i s t e n t to a c e r t a i n l e v e l . - 7 -Sec. 1.3. The H y p e r v i r i a l R e l a t i o n s as C o n s t r a i n t s The c a l c u l a t i o n of t r a n s i t i o n p r o b a b i l i t i e s , f o r t r a n s i t i o n s between s t a t i o n a r y s t a t e s of an atomic system, reduces to the e v a l u a t i o n of m a t r i x elements of c e r t a i n operators between the i n i t i a l and f i n a l s t a t e s . The exact e v a l u a t i o n of these matrix elements r e q u i r e s a knowledge of the exact s o l u t i o n s of the SchrBdinger equation f o r the i n i t i a l and f i n a l s t a t e s . For m u l t i - e l e c t r o n atoms, only approximate s o l u t i o n s are known. The use of these approximate s o l u t i o n s i n the c a l c u l a t i o n of t r a n s i t i o n p r o b a b i l i t i e s may introduce c e r t a i n e r r o r s . Our o b j e c t i v e i n the present work i s to see i f these e r r o r s can be minimized by s u b j e c t i n g the Hartree-Fock wave f u n c t i o n s to a c o n s t r a i n t , namely, that they s a t i s f y an o f f - d i a g o n a l h y p e r v i r i a l r e l a t i o n . A time-independent l i n e a r operator F w i t h a r b i t r a r y f u n c t i o n a l s t r u c t u r e expressed i n terms of the dynamical v a r i a b l e s of the system i s c a l l e d a h y p e r v i r i a l operator. Let H be the time-independent Hamiltonian of the system, F be a h y p e r v i r i a l operator and l e t F ^ denote the matrix element (u ^ ( t ) | F | u ^ ( t ) ) of the operator F i n the energy r e p r e s e n t a t i o n . The Heisenberg equation of motion i n the energy r e p r e s e n t a t i o n i s ( S c h i f f , L; (1955) pp. 139-140) dT FMN * in" ( [ F ' H ] )MN " k ( EM " V FMN (11) (provided the Hermitian p r o p e r t i e s of H are r e t a i n e d f o r F u ^ ( t ) ) - 8 -Equation (11) can be integrated to give h ( EM " V' FMN ( t ) = V 0 ) e , (12) From equation (12), we obtain 1. The diagonal hypervirial relation: In the energy representation the diagonal matrix elements of a hypervirial operator F are constant. From equation (11), we get (t^ | [F, H] | y = 0. (13) 2. The off-diagonal hypervirial relation: In the energy representation the off-diagonal matrix elements of a hypervirial operator F oscillate with frequencies related to the energy difference between the two stationary states. From equation (11), we get ( y t H] I u N) = (E N - y O i y | F ! u N). (14) The hypervirial relations (13) and (14) are the necessary conditions for the functions u^ and u^ to be exact solutions of the SchrBdinger equation corresponding to the eigenvalues E^ and E^ respectively. Coulson (1965) has derived a large number of hypervirial relations, including (13) and (14), and has also studied the conditions under which classes of hypervirial relations are sufficient for the determination of exact solutions of the SchrBdinger equation. The diagonal hypervirial relations have been used as constraints in the calculation of diagonal matrix elements of certain operators by Hirschfelder et. a l . (I960, 1962) and also by Epstein et. a l . (1961). - 9 -Their results indicate that the effects of wave function errors in the calculation of diagonal matrix elements of certain operators can partially be eliminated by the use of diagonal relations as constraints. Chen (1964) and Coulson (1965) independently have suggested the use of off-diagonal hypervirial relations as a constraint to improve approximate wave functions. They have arrived at a number of very general relations. One of their relations of special importance in the calculation of transition probabilities is N N A E ( n I I r, | Y) = (II | I V | Y) (15) j-1 2 j=l J where r^ and are the position and gradient operators of the j-.th electron, ¥ and II denote two stationary states of energy E(VI') and E(IT) respectively, and AE is the transition energy E(Y) - E(JI). Vetchinkin (1964) has derived the hypervirial relation (15) as a necessary condition for the matrix element of an operator F to be independent of wave packet dispersion. In the theory of radiation, relation (15) implies that the numerical values of the transition probability obtained by the use of dipole length and dipole velocity formulae be identical. Recently Chong et. a l . (1968) and Yue et. a l . (1969) have used off-diagonal hypervirial relations as constraints in the calculation of transition moments for a few He-like transitions. They have applied the constraint only to the lower state whose wave functions already involve several configurations. Furthermore, the one-electron radial functions are restricted to a particular form. Our aim in the present work i s : - 10 -(1) To solve the SchrBdinger equation for an atomic system approximately under the. single-configuration Hartree-Fock scheme. (2) To apply the constraint symmetrically on both the upper and the lower states. (3) To solve the complete variational problem for the determination of radial wave functions without restricting the form of radial functions. (4) To study the Z-expansions of radial wave functions and of various atomic parameters. (5) To calculate the oscillator strengths for various transitions of two-, three- and four-electron systems and to compare our values with those reported by others. The values of oscillator strengths reported by several other authors include both experimental values as well as theoretical values obtained by a large variety of approximations, particularly the very accurate theoretical values of Schiff and Pekeris (1964). We shall refer to the wave function obtained by our method of constrained variation as Constrained Hartree-Fock wave functions and abbreviate them as C.H.F. functions compared to the standard Hartree-Fock wave functions written as S.H.F. functions. - 11 -Sec. 1.4. Derivation of the Constraint in Terms of Radial Functions Let IT and ¥ denote two states of an N-electron atomic system, whose energies are E(II) and E(¥) respectively. Let the principal and azimuthal quantum numbers of the active electron be denoted by n and -it in the state ¥ and by n* and I* in the state II. We further assume that: (1) The states denoted by II and f a r e represented by a single configuration approximation. (2) The two configurations of II and ¥ differ from each other in one electron only, namely, the active electron and that \Z - Sl*\ = 1. The constraint that we are imposing on the system i s , with AE = E O 0 - E(n), N N A E ( n | I r. | Y) = (II | I V. | ¥). (16) j=l J j=l 3 This constraint consists of the following matrix elements II and J l , N N of the operators £ r. and £ V., between the angular momentum states: j=l 3 j=l 3 N N II = (II | I r | ¥); J l = (II | I V | V. j=l 3 j - l 3 According to the Wigner-Eckart theorem (Edmonds (1957)) each of II and J l is the product of two factors, the angular part, depending on the choice of axes orientation, and the physical part called the reduced matrix element. Through repeated use of Racah's algebra (Racah (1942)), these reduced matrix elements can be reduced further to matrix elements I and J depending only on the radial functions of the active electon i n the two states. - 12 -The constraint expressed by (16) reduces to where and AE x I (nil, n*£*) - J(n£, n*£*) = 0 I (nil, n*£*) = P(n*£*, r)rP(n£, r)dr 4 0 J(n£, h*£*) = (17) (18) P(n£, r) y-P(n*£*, r ) d r (19) ar [£*(£*+!)-£(£+!)] P(n£, r)±T?(n*l*, r ) d r . Let {£*, £} denote &(&+!)] . t h e n t h e c o n s t r a i n t (17) may be written as P(n£, r)[AEr - (4~ + { £ * ' £ } ) ] p( n*£*, r ) d r = 0. (20) dr r It i s e a s i l y seen that the interchange of quantum numbers (n, £) and (n*,£*) leaves the constraint (20) unchanged. This shows that the constraint involves the two states II and Y symmetrically. I t must be noted that the Racah's method assumes that: the passive electrons are represented by the same wave functions i n both the states. This assumption w i l l not be true i n our approximation. However, the two wave functions representing the same passive electron in the two states w i l l be so nearly equal that the a p p l i c a t i o n of Racah's method s h a l l be regarded as v a l i d . - 13 -Sec. 1.5. Derivation of the Radial Equations for the Constrained System We now use the procedure of constrained v a r i a t i o n to derive the system of equations f o r the r a d i a l wave functions. We suppose that we have an N-electron system s a t i s f y i n g the assumptions given i n Sec. 1.4. The r a d i a l wave functions which make WCF) and W(II) (defined by (5)) stationary and also s a t i s f y the constraint (16) are sought f o r the states ¥ and II. Consider the quantity, Wtn, ¥ ) , given by Wtn, Y) = W(n) + W(T) + y(IT | AEr - V | V) (21) where y i s a Lagrange m u l t i p l i e r (hereafter r e f e r r e d to as the parameter of constraint) to be determined with the help of the constraint, and AE = E(¥) - E(II). Let 6Wf.II, ¥) denote the v a r i a t i o n of Wtn, ¥) corresponding to a v a r i a t i o n 6P(n£, r) of P(n£, r) . Since W(II) i s independent of P(n£, r ) , we have SWtll, ¥) = 6[W(¥) + y(n | AEr - V | ¥)]• (22) The terms of WOO that depend on P(n£, r) are numerical multiples of r a d i a l i n t e g r a l s of the type I(n£), F k(n£, n£), F k(n£, n'£') and G k(n£, n f£') defined i n Sec. 1.2. Following Hartree (1957), we can c a l c u l a t e the f i r s t order v a r i a t i o n of each one of the terms on the r i g h t hand side of equation (22). These i n d i v i d u a l v a r i a t i o n s are 61 (n£) - -2 6P(n£, r) + - ] P(n£, r ) d r dr r 6F k(n£, n'£') = 2 ( 1 + 6 ,) nn 6G k(n£, n'£') = 2 6P(n£, r ) [ i P(n£, r)Y k(n'£', n'£', r ) ] d r 6P(n£, r)[-P(n'£', r)Y (n£, n'£', r ) ] d r IT rC (23) 0 and 6[AEI(n£, n*£*) - J(n£, n*£*)] = | 6P(n£, r)[AEr - ( ^ + { £ * ^ } ) ] P( n*£*. r ) d r . - 14 -With the help of these individual variations we obtain, for 6wtn, ¥) = 0, 6P (n£, r)[Q 1(r) + Q 2(r) + J n . £ p ( n ' ^ r)]dr = 0. (24) Here n,^ are Lagrange's parameters introduced to ensure the orthonormality of the radial functions with the same £. The standard Hartree-Fock contribution to 6W(TI, Y) is denoted by Q^(r) and the contribution from the constraint by Q^(r). The requirement that 6 W(II, Y) = 0, up to f i r s t order of 6P(n£, r ) , independently of SP(n£, r ) , yields the equation for the radial function P(n£, r ) . From (24), this equation is Q l ( r ) + Q 2(r) + I X P(n«£, r) = 0. n Writing Q^(r) and (^(r) in terms of basic Hartree-Fock functions we obtain the following equation for P(n£, r), 2 £(£+1) ITT + 7 ( Z " Y ( n £ ' r ) " £n£,n£ " T ~ J r> dr t - - \ X(n£, r) + I e n £ ^ P(n'£, r) (25) n' ' Here (1) Y(n£, r) = £ q(n£)Y0(n£, ni, r) - la^Y^nl, nl, r ) . n£ k The function Y(n£, r) is called the potential function. (2) X(n£, r) = I 3» „ , v Y. (nil, n'£', r)P(n'£', r) . n'£'k The function X(n£, r) is called the exchange function. ( 3 ) en£, n ' i " ' ( 2 6 ) (4) £ denotes sununation over a l l n' except n' = n. nV (5) q(n£) denotes the number of electrons in the (n£) subshell. (6) Various constants and B^t^ take values depending on the particular configuration being considered. In the same manner as above, the variation 6W(II, ¥)' corresponding to a variation SP(n*£*, r ) , equated to zero for arbitrary SP(n*£*, r) yields the following equation for P(n*£*, r ) , I72 + 7 ( Z " Y(n*£*> r ) ) " W,n*£* " -P(n*Jl*, r) dr r -. - f X ( n*£*. r) + l' ( V £ * , n " £ * P ( n " £ * ' r ) ( 2 7 ) n ^ [AEr - 4 + % ^ ] P H r). q(n*£*) 1 Mr Here various functions and symbols have the same meaning as the corresponding functions and symbols in (26). Equations for other radial wave functions are the same as for standard Hartree-Fock cases. In this manner, a system of non-linear integrodifferential equations, one equation for each one of the subshells belonging to either - 16 -the lower or upper state, i s obtained. This system of equations i s solved by Hartree's self-consistent f i e l d method and the parameter y i s determined i n such a way that the solutions of the system, s a t i s f y the constraint (20). The solution of the above system of equations and the determination of the parameter u w i l l be taken up i n Chapter I I I . In the next chapter we s h a l l be concerned with the Z-expansion of r a d i a l wave functions and various atomic parameters. CHAPTER II Sec. 2.1. The Nuclear Charge as a Parameter The Z-expansion approach to atomic structure calculations can be based upon a technique, f i r s t used by Hylleraas for He-like systems (Bethe and Salpeter (1957), pp. 151-153), that- can be described as follows. The Hamiltonian (2) of the N-electron system is transformed, by scaling a l l lengths to so that P - Zr, (28) 2 ? - v i ( p ) i i v i H = z [ I (~V- - ~) + \ l ~ \ (29) 1-1 P i ±>j P i j = Z 2[H Q + \ H 1], (30) where H^ = \ H , H being a one-electron Hamiltonian. i-1 The Z-expansion method divides the Hamiltonian (30) into a sum HQ of one-electron Hamiltonians and a sum H^ of two-electron interactions.• The two electron interaction terms are considered as a perturbation with ^ as perturbation parameter. The variation perturbation theory (Bethe and Salpeter (1957), pp. 122-123) is used to find the eigenvalues and eigenfunctions of H. An eigenvalue E and the corresponding eigenfunction ¥ are considered expanded in powers of Z _ 1 i.e. OO 00 E = I Z" n + 2E : 1 = I Z"n (31) n=0 n n=0 n and a system of coupled equations for various iL» and E is derived. n n - 18 -The zero-order approximations to the states of the system are taken to be hydrogenic states. The energy of hydrogenic states is degenerate; therefore, according to the perturbation theory of degenerate states, the zero-order states must be chosen so that the matrix of electrostatic interaction is diagonal. This degeneracy of zero-order states introduces a particular kind of mixing of degenerate states, namely, among the configurations with the same parity and the same principal quantum numbers. The set of configurations with the same parity and same principal quantum number i s called a completex (Layzer, 1959, 1963). Linderberg (1961) has performed perturbation theoretic calculations for two-, three-, and four-electron systems. For larger atoms, the calculation of the mixing coefficients for the zero-order wave functions and the f i r s t order terms of the total energy have been made by Godfredsen (1966). Calculations to higher order, even for small systems, are quite d i f f i c u l t . A review of perturbation theoretic calculations of higher order can be found in Crossley (1969). The Z-expansions of the Kartree-Fock radial functions and various atomic parameters evaluated from them have been studied by LHwden (1954), Hartree (1958), Froese (1957, 1958) and Edlen (1964). In the present chapter we intend to incorporate the Z-expansion analysis into our method of constrained variation. - 19 -Sec. 2.2. Z-expanslon of Radial Wave Functions and Various Atomic Parameters For constiained Hartree-Fock radial functions, we expect an expansion similar to that of the S.H.F. function, namely, P(n£, r) = Z^[PH(n£,p) + |q(nil,p) + ^ R(n£,p) + ...],p = Zr (32) L zz H where P (nil, p) represents the hydrogen wave function and Q(nil, p) and R(n£, p) are the departures, of order \ and - ^ r - , of P(n£, r) from Z ZZ Coulomb f i e l d wave functions. Our object is to find the dif f e r e n t i a l equation satisfied by Q(n£, p) and also the expansionsof u and the diagonal and off-diagonal energy parameters. Changing the independent variable r to p, by (28), we see that the equation (25) for P(n£, r) is transformed into: rd2_ 2 2Y(n£,p/Z) Cn£,n£ , f l. 2 p " Zp ,2 2 1 P / Z ; dp K K Z p = - | - X ( n i l , p/Z) + I - H ^ A p ( n . A > p / z ) ( 3 3 ) + _^L_ jA|p_ _ z ( d ^ + I M J ^ J P ( N N S P / Z ) . Z 2q(nil) Z dp p We expand a l l the radial functions, appearing in (33), in powers of 1/Z. This results in the following expansions of Y(nil, p/Z) and X(n£, p/Z) Y(n£, p/Z) = Y H(n£, p) + 0(7) Z (34) X(n£, p/Z) = X H(n£, p) + 0(|) where the superscript H indicates calculations performed using hydrogen wave functions P (n£, p). - 20 -Also, from (31) we have 2 A E 1 ^2 AE = Z^[AE0 + -f-. + — | + . . . ] , (35) where AE Q = ± - | ( \ ^j) • n n* From (33), (34) and (35) i t follows that the solution of (33) w i l l satisfy the requirement P(n£, r) -»• ZT> (n£, Zr) as Z -*• °°, i f e ( D £(2) n£,n£ _ 1 n£,n£ n£,n£ Z 2 = n 2 2 Z 2 (1) (2) (36) n£,n'£ _ n£,n'£ n£,n'£ z2 " 2 z2 <1) p < 2 > and V = u + + ... z> With these expansions (34), (35) and (36) substituted into (33), the terms, independent of of equation (33) equated to zero give the la H differential equation for P (n£, p) , namely, t^L + | . i _ . w ± i i ] P H ( n 4 p ) . 0 . ( 3 7 ) dp M n p The terms of f i r s t order in \._ of equation (33) equated to zero yield the differential equation satisfied by Q(n£, p), namely, 2 \r~2 + f " \ " — f ~ 3 Q ( n £ , P) - |[YH(n£,p)PH(n£,p) - XH(n£,p)] dp n p ( 3 8 ) + «£i»/< n £'P> + L p H< n'£,p) + ^ [ A E 0 P " ( ^ •+ ± ^ ) ]PH(n*£*, p) n fn n The boundary conditions for Q(n£,p) are. Q(n£, 0) = Q(n£, ») - 0. (39) - 2 1 -The normalization condition up to order — gives Q(n£,p)PH(n£,p) dp = 0 , ( 4 0 ) and f i n a l l y , the orthogonality condition (between P(n£,p.) and P(n '£,p)) up to order ^ yields [P H(n '£,p)Q(n£,p) + Q(n'£,p)PH(n£,p)]dp = 0 . ( 4 1 ) 0 At this stage, let us introduce a certain notation in order to avoid writing the operator of the constraint repeatedly. Let DQ (p) denote the operator (AE Q p- (|- + 1 ^ J ) ) ( 4 2 ) ££* d -f£* Al-and D (p) denote the operator (AEp- (j- + 1 y J ) ) . ( 4 3 ) In this notation, the following equations are identically true (P(n£, r) | D (r) j P(n*£*, r)) = 0 ( 4 4 ) (PH(n£, r) | D^ £*(r) | PH(n*£*, r)) = 0 . ( 4 5 ) Equation ( 4 4 ) represents our constraint. Equation ( 4 5 ) is the statement that the constraint is always satisfied by hydrogen functions. In order to study atomic transitions, we have to consider configurations having one or more incomplete subshells. In such cases the off-diagonal energy parameters must be introduced into the Hartree-Fock scheme to ensure orthogonality of the corresponding radial functions. This may lead to an inconsistency, as has been shown by Sharma and Coulson ( 1 9 6 2 ) , Froese ( 1 9 6 5 ) has dealt with this problem and has been able to predict the - 22 -cases where this, inconsistency may arise. As we are subjecting our wave functions to a constraint, we should study whether or not we can introduce off-diagonal energy parameters to ensure corresponding orthogonality. Thus, our object now is to determine the expansion coefficients e^ «.^ 0, l g and The determination of w i l l have to be deferred un t i l after the expansion of the constraint. To obtain an equation for e^£^ n^> w e multiply (38) by H P (n£, p) and (37) by Q(n£, p), subtract and integrate to get =-2 n£,n£ i PH(n£,p)tYH(n£,p)PH(n£,p) - XH(n£,p)]dp - (PH(n£,p) | (p) I P*(n*£*, p)) (using (45)) = - 2 ^P H(n£,p) [YH(n£,p)PH(n£,p) - X H(n£, p)]dp. (46) Equation (46) shows that the constraint does not affect the f i r s t order diagonal energy parameters. To obtain the equation for e^ P , 0, we shall rewrite the equation for Q(n£,p) in terms of A g , 0, the symmetric Lagrange multiplier. We w i l l further relabel the two radial functions with the same values of Si and different values of n as the i t h and j t h functions, where i refers to the constrained function. In this new labeling scheme H A „ w i l l be denoted by A . . . The equation for P.(p) i s - 23 -dp K nj p J and the equation for Q^(p) is ^ + 2 _ Y_ , 2 [ Y H ( p ) p H ( p ) _ x H ( p ) J dp ^ n. p K + ea) P H(p) + J 4 4 pJJ(p) + 4 4 P H(p) 1,1 i K / k ^ ± q(i) k V K / q(i) j V K / 11 CO* u + DQ (P) P H(n*£*,p). (48) H Multiply (47) by Q ±(p), (48) by P (p), subtract and integrate to get Q.(p)p"(p)dp -oo (|)P*(P)[YJJ(P)PJ(P) " xj(p)]dp which is an equation of the type a . = g . + - 1 4 + / 1 > v i . <49> i i q(i) Similarly, combining the equations for Q.(p) and P.(p) to obtain an (1) J 1 equation for . . , we obtain an equation of the form q ( j ) a. = 8. + 4 4 - <5°) The orthogonality condition (41) implies that = = a. Thus equations (49) and (50) yield two equations in three unknowns a, and U ^ . From equations (49) and (50), eliminating a, one obtains - 24 -< Bi - V + (qciy" iTir' x i " + / 1 > v i - °- ( 5 1 ) an equation in two unknowns xf"^ and Similarly, corresponding to each of the off-diagonal parameters in either state, an equation of the type (51) is obtained. These equations form a system of m^ + linear algebraic equations in + m2 + 1 unknowns, namely, and m2 f i r s t order terms of the off-diagonal parameters in the upper and the lower states respectively and . Another equation in these m^ + + 1 unknowns w i l l be derived and the existence and the uniqueness of the solution of the system of m^ + m^ + 1 equations in m^ + + 1 unknowns, so obtained, shall be discussed in Sec. 2.4. - 25 -Sec. 2.3. Z-expansion of the Constraint In t h i s section we define the pf-values of a t r a n s i t i o n and study t h e i r asymptotic behaviour, that i s , the l i m i t i n g behaviour of these quantities as the nuclear charge Z increases without l i m i t . Having defined ( i n Chapter IV) the o s c i l l a t o r strengths gf, we s h a l l f i n d that these pf-values, as defined below, are p r o p o r t i o n a l to the o s c i l l a t o r strengths. We define the pf-values corresponding to the t r a n s i t i o n n -> ¥ as follows, pf = AEI 2(n£, n*£*) r a d - „ (52) and pf . = J (n£, n*£*)/AE, v e l . ' where AE, I(n£, n*£*) and J(n£, n*£*) are as defined i n Sec. 1.4. From the d e f i n i t i o n of pf-values, expansion (35) of AE and the Z-expansions of various r a d i a l functions, i t i s easy to see that the asymptotic behaviour (as Z •*• °°) of the pf-values i s given by f ^ constant i f n r n* p t r a d . Z-*» -1 ' . Z i f n = n* f ^ constant i f n ^ n* P v e l . Z-*» -1 • . Z i f n = n* Let us further note that the constraint (20) i s equivalent to P ^ r a d . P ^ v e l . We now obtain the expansion of the constraint (20) i n powers of -?r. The constraint (20) i s (P(n£, r) | D M * ( r ) | P(n*£*, r ) ) = 0 - 2 6 -Changing the independent variable r to p = Zr and using the expansions (32) and (35), we see that the above matrix element reduces to the following matrix element (r | A | r * ) , (54) where r S P H(n£, p) + jQ(n!L, p) + ... T* = PH(n*£*, p) + ^ Q(n*£*, p) + ... Z vdp p ' The Z-expansion of the matrix element (T | A | T*) begins with a term of order Z. This term equated to zero gives the constraint to zero-order, namely, (PH(n£, p) | D££*(p) | PH(n*£*, p)) = 0. (55) The next term in the expansion of (r | A | T*) equated to zero gives the constraint up to f i r s t order, namely, AE 1(P H(n£, p) | p | PH(n*£*,p)) + (Q(n£,p) | D££*(p) | PH(n*£*, + (PH(n£,p) j DQ £*( P) I Q(n*£*,p)) = 0. ( 5 6 ) As can be easily seen, (55) and (56) remain valid even when ££* the operator (p) degenerates. This happens when n = n* or AE„ = 0. - 27 -Sec. 2.4. Z-expansion of the Parameter of the Constraint From the formal expansion of the constraint we find that the terms of highest order in Z of the constraint vanish identically. Our object in this section is to see i f the parameter can be chosen so as to satisfy the constraint up to the next lower order of Z, namely, equation (56). To achieve this, each term of the constraint (56) shall be expressed in terms of various hydrogenic functions, the f i r s t order terms in the Z-expansions of various energy parameters and (1) H u with the aid of the differential equations satisfied by P (n£, p), H Q(n£, p), P (n*£*, p) and Q(n*£*, p). This w i l l provide an equation i n £n£,n'£' n*£*,n"£* a n d y ' Let L ^ be the operator defined by _ d 2 2 1 £(£+1) ni ' 2 p 2 2 dp r n p From the equation for Q(n£, p), namely (38), we have pL „Q(n£,p) = 2[YH(n£,p)PH(n£,p) - XH(n£,p)] +e i^ )„pP H(n£,p) n£ n£,n£ (57) n f n The equation for P (n*£*,p) gives P L n A £ A PH(n*£*,p) =0- (58) Multiply (57) by P (n*£*,p), (58) by Q(n£, p), subtract and integrate to get - 28 -r 0 0 2 2 [P H(n*£*,p)p^ Q(n£,p) - Q(n£,p)p^ T PH(n*£*,p)]dp ( 5 9 ) dp dp = (\ - ^ )(P H(n*£*,p)|p|Q(n£,p)) + 2{£,£*}(PH(n*£*,p)|p"1|Q(n£,p)) h n* + 2(PH(n*£*,p)|YH(n£,p)PH(n£,p) - XH(n£,p)) + e ( 1 ) (PH(n*£*,p)|p|PH(n£,p)) n£,n£ + I e^ n, A(P%*£*,p)|p|p H(n'£,p)) + y ( 1 ) (PH(n*£*,p) | PD J £*( P) |PH(n*£*,p)). n'^n ' q(n£) From the boundary conditions for P (n*£*,p) and Q(n£,p) we have L.H.S. of equation (59) = 2(Q(n£,p) ||^| P H(n*£*,p)). (60) 1 1 1 From (59), (60) and (42) we have, using A EQ = — (—j ^)» n n* (Q(n£,p)lDQ£*(p)|PH(n*£*,p)) (61) = -(PH(n*£*,p)|YH(n£,p)PH(n£,p) - XH(n£,p)) - (PH(n*£*. p) | p | P H(n£, p) ) , e a ) , (1) _ I nY , ; L,(P H(n*£*,p)lplP H(n'£,p)) - (PH(n*£*, p) | pD £ £*(p) | P H(n*£* ,p) ) n' . Similarly* with the help of the differential equations for Q(n*£*,p) and H P (n£,p), we obtain another term of the constraint (56), as follows: (PH(n£,p)|D5£*(p) | Q(n*£*,p)) = (PH(n£,p)|YH(n*£*,p)PH(n*£*,p) - XH(n*£*,p)) + 1 e n n ^ n * £ * ( p H ( n £ ' p ) l p l p H ( n * £ * ' p ) ) ( 6 2 ) + l[ 6n*£*,n"£* (P H(^,P)ip|P H(n"^,p)) •+ 2 Q ( ^ * ) (P H<n*.P) | P D £ % > J P V A . P ) ) -n ' 2 From (61) and (62), using (56), we get - 29 -(PH(n£,p)|YH(n*£*,p)PH(n*A*,p) - XH(n*£*,p)) J ^ 1 - - (PK(n*£*,p) |pD££*(p) |pH(n*£*,p)) 2q(n£) V i v" »^'i^o - (PH(n*£*,p)|YH(n£,p)PH(n£,p) - X H(n£, p)) 2q(n*L) (P H(^,P)|pD5 n(p)|p H(n£,p)) l ' P Y ' A (PH(n*£*,p)|p|PH(nH,p)) + I V n*£*n"** ( ? H ( n , > p ) | p | p H ( n , ^ > p ) ) n" l + V*, +TSl*,nn* - ^SU](PH<^.p)|P|PH(nn*,p)) = 0, (63) Equation (63) is a linear equation in m^ + ' + 1 unknowns, namely, (1) (1) * (1) c. An£,n'£ . . . 6n£,n'£' €n*£*,n"£* a n d y ' S i n C e £n£,n'£ = " ^ 0 " ( S E C ' 1 " 5 ' ) » equation (63) may be transformed into an equation in m^ + m^ + 1 unknowns and y ^ (of Sec. 2.2). This transformed equation together with m^ + m^ linear equations of the type (51) obtained in Sec. 2.2 forms a system of m^ + m^ + 1 equations in m^ + + 1 unknowns. This system of equations, written in the matrix notation, is - 30 -x X x x X X X X X X A i , l J ,1 .(1) / 31 " 6 i m2 J (64) where the possibly non-zero elements of the matrix of the coefficients are denoted by crosses. The last row and the last column of the matrix of coefficients as-well as the vector on the R.H.S. of (64), being hydrogenic, are known. The diagonal elements of the matrix are constant. This system of equationscan be solved, in general, to give a unique solution i f the number of vanishing diagonal elements is less than two. If p, p > 2, diagonal elements vanish, there i s either a (p - 1) - parameter family of solutions (in this case, following Hartree, we can define the standard solution as the one obtained by taking the value zero for each parameter) or no solution. - 31 -Sec. 2.5. The Effect of the Constraint on the Total Energy E of the System The effect 6E of the constraint on the total energy E of the system is defined to be the change, due to the constraint, in the total energy E of the system, i.e. 6E » E(C.H.F.) - E(S.H.F.). In the absence of any explicit expressions for Q(n£, r) and R(n£, r) of the expansion (32) of P(n£, r ) , we calculate the effect of the constraint on the total energy E from the changes, due to the constraint, in the diagonal energy parameters e . 0. We f i r s t prove a theorem connecting the coefficients in the Z-expansions of E and those of the €n£,nr -k+2 Theorem. The coefficients of Z in the expansions of the total energy E and the diagonal energy parameters e Q 0 are related by 1 (k) 2 n£,n£ „ I - i e 2 ! n £ = ( k + ^ k - ^5) a l l electrons Proof. (Ch. Layzer (1968)): With the Hamiltonian H of (30) and Y as the normalized wave function for the state>we have E = (Y | H | ¥) = Z 2(^ | H Q | + Z(¥ | H j V) = Z 2 I q(n£)I(n£) + Z(f | H | . n£ Substituting for the I(n£) from the differential equations of the corresponding P(n£, r) [Hartree (1957) Sec. 8.2], we get " = Z 2 [ " 7? L ^ ^ ^ . n ^ - ^ I H l « or " H q(n£)en£ ^ = E + Z(¥ | ^ | V). (66) n£ - 32 -1 -2 Let t = -, then H = t (H Q + tH ). Hence f = - 2 t - 3 ( H 0 + t H l ) + t~\ or t H. = t ^ + 2H. 1 dt From this together with (66), we get " I I < ^ ) e n £ , n £ = 3E + t(Y | |f | ¥). (67) n£ Using the Hellman-Feynman theorem, namely, we get, from (67), - 4 7 q(n£)e = 3E + t ^ 2 L n n£,n£ ot n£ C O = I ( j + l ) t j " 2 E j-0 2 Expanding the left hand side in powers of t, we complete the proof of the theorem. Our next objective is to calculate the effect of the constraint 00 on the various coefficients e . „, k = 0, 1 and 2. The zero-order n£,n£ parameters e^^ n£, being hydrogenic, do not depend on y. In Sec. 2.2 we found that the effect of the constraint on e^ P^ „ vanishes identically. n£,n£ A similar approach can be made to calculate the effect of the constraint (2) on £ This w i l l require the use of the equation for R(n£, r ) . As m we are interested in the contribution of the constraint towards e „' „, n£, n£ (2) which we w i l l denote by <5e „ we can calculate i t from only those n£,n£' J (2) terms of the equation for R(n£, r) that depend on £ . . and u. T1X* y IiJC - 33 -From equation (33), with the help of (36), we collect a l l 1 j j * (2) (1) j (2) terms of order depending on € n £ n£» V and u • There are four such terms and their sum is ?H(n£'p)+Dr(p)Q(n*£*'p). + ^ W D o ( p ) p ( n * £ * ' p ) + V b y A E i p P <n*£*'p>- <68> H Multiply by P (n£,p) and integrate to get, with the aid of (45), 6enJl!n£ = " "q^niy £ < P H ( N £ ' P > I ^ { 9 ) | Q(n*£*,p)) + AE X (PH(n£, p) | p | PH(n*£*, p) ) ] * (Q(n£,p) I (p) I PM(n*£*,p)). (69) (2) Similarly, we obtain the effect of the constraint on e n*x,* ,n*Jo* to be 6 en*£*,n*£* = ' q^n^T (Q<n***>p) I . ^ ( P ) I PH(n£,p)). (70) It is seen that the constraint affects only the second and higher order terms of the diagonal energy parameters. The effects of the constraint on the second order terms in the expansion of e „ „ and .. n£,n£ £n*£* n*£* a r e S l v e n ^^) and (70) respectively. In some cases the contributions to the total energy E from terms of third and higher order can be quite appreciable. For this reason, we shall calculate the effect of the constraint on the total energy E directly as the difference of constrained total energy and the S.H.F. total energy. In view of the analysis of the present chapter, we do not expect this change to be - 34 -appreciable in general. From the expressions (69) and (70) for 6 e i l ] n i a n d 6 enn*,nn* r e s P e c t i v e l y . ^ i s e a s i l y s e e n t h ^ t | f i € ^ n £ | (2) and 16 € n*£* n*£* I a r e both independent of Z, i.e. the magnitude of the effect of the constraint is the same for a l l members of a particular isoelectronic sequence. However, as a percentage of the total energy, the effect of the constraint on the total energy E tends to zero as Z ->- ». From the nature of the H.F. method, the constraint i s allowed only to raise the total energy E of the ground state, for the S.H.F. wave functions for the ground state minimize the total energy E. Since the energy of a non-ground state is obtained only as a stationary value of the energy functional, the constraint may either lower or raise the total energy in this case. CHAPTER III Sec,. 3.1. Computational Procedure. The f i r s t section of the present chapter w i l l describe the computational methods used to solve the system of equations obtained in Sec. 1.5 and to obtain the value of y, the parameter of the constraint. In the later sections, we w i l l state the results of applying these methods to calculate the total energy E, y and the pf-values for a few transitions of L i - and Be-like isoelectronic sequences. Three types of transitions w i l l be considered for various values of the nuclear charge, Z, mentioned along with the transitions in the following: 2 2 2 2 Is 2s Sj - Is np Pj ; n = 2, 3, and 4. ^ "* (Z = 3,4,5,6,8,10,12,15,47 and 50) 2 2 2 2 Is 3s Sj - Is np Pj ; n = 2 and 3. ^ "* (Z = 3,4,5,6,7 and 8) 2 2 1 2 1 Is 2s S - Is 2s np P ; n = 2, 3, and 4. (Z = 4,5,6,7,8,10,12,16,44 and 50) The computational procedure that has been developed to solve our problem consists of the following steps. Starting with the input data regarding the two configurations, the number and type of electrons in both the upper and lower states and the i n i t i a l estimates for various energy parameters and y, our system of equations is solved. These solutions are employed to calculate the transition energy AE and the value of the error function e(A) (to be defined later) at X = y. A test, whether the constraint i s satisfied or not, is performed. If the constraint is satisfied, the solutions are used to calculate the oscillator strengths. If the constraint is not satisfied, another value of y is predicted by either of the two methods to be described later i n this section. The system of equations is then solved for this new value of y. This process is repeated un t i l the y-iterations converge to yield Hartree-Fock wave functions which also satisfy the constraint. For solving the system of equations, the Hartree-Fock program of C. F. Fischer (Fischer 1968)) has been modified according to our needs. Two methods have been used for the calculation of y. The f i r s t one, referred to as the method of direct estimation, consists of the estimation of y from the differential equations for P(n£, r) and P(n*£*, r ) . The second one, referred to as the method of parametrization by Benston and Chong (1967, 1968), consists of solving the system of equations for a set of values of y and selecting that value for which the wave functions satisfy the constraint. The calculation of the length- and the velocity-form of the oscillator strength reduces to the calculation of the I(n£, n*£*) and J(n£, n*£*) integrals (defined in Sec. 1.4.) respectively. A description of the two methods used for the calculation of y w i l l now be given. "METHOD OF DIRECT ESTIMATION" Let L ^ be the operator defined by n£ " . 2 r " €n£,n£ 2 dr ' r From the equation for P(n£, r ) , namely (25), we have - 37 -rL £ P(n£, r) = 2[Y(n£, r)P(n£, r) - X(n£, r)] (71) + ^nJl.n'A r P ( n ' £ ' r ) + IfiZ) ^ ( r ) P ( n * £ * , r ) , n Similarly, from the equation for P(n*£*, r ) , namely (27), we have rL n A^P(n*£*, r) = 2[Y(n*£*, r)P(n*£*, r) - X(n*£*, r)] (72) + * n*£*, n"£* r P ( n " £ * ' r ) - ^feo rD £* £(r)P(n£, r) Multiply (71) by P(n*£*, r ) , (72) by P(n£, r ) , subtract and integrate to get 2 2 [P(n*£*,r)r P(n£,r) - P(n£,r)r P(n*£*,r)]dr dr dr = ( e n £ , n £ " e n * £ * , n * £ * ) ( P ( n £ ' r ) l r l P ( n * £ * ' r ) ) + 2 U ' £ , , : } ( P ( n £ ' r ) l r 1 'P (n*£* ' r ) > + 2(P(n*£*,r)!Y(n£,r)P(n£,r)-X(n£,r))-2(P(n£,r)|Y(n*£*,r)P(n*£*,r)-X(n*£*,r)) " 2' £n*£* ^ ( H n ' ^ . ^ l r l P ^ . r ) ) + n,£(P<n'£,r)|r|P(n*£*,r)) (73) + q^£j (P(^*,r)|rD £ £*(r)|P(n*£*,r)) + (P(n£,r)|rD£*£(r)|P(n£,r)). Also, from the boundary conditions for P(n£, r) and P(n*£*,r) we have L.H.S. for (73) = 2 ,00 P(n£, r) P(n*£*, r)dr. (74) dr go* From (73), (74) and the definitions of D (r) and {£,£*}, with the aid of the constraint, we get - 38 -(n£, r) |p.* (r)Jp(n*£*,r)) = ( 2 A E + W , n * £ * ' e n l M H H ^ ' r ) I r I ? , r ) ) - 2(P(n*£*,r) |Y(n£,r)P(n£,r)-X(n£,r))+2(P(n£,r) |Y(n*£*,r)P(n*£*,r)-X(n*£*,r)) + l' £n*£* n,tJl*(P(nM^,r)|riP(n£,r)) - j ' . e (P (n ' £,r) | r |P (n*£* ,r) ) n " ' n 1 ' - (P(n*£*,r)|rD££*(r)|P(n*£*,r)) - ( n « & j f c ) (P(n£,r)|rD£*£(r)|P(n£,r)) • - 0. (75) Equation (75) may be used to predict a new value of the parameter u with the aid of the solutions of the system of equations corresponding to the current value of y. The i n i t i a l value of y may be taken to be zero i f no better estimate is available. The solutions of the system of equations w i l l satisfy the constraint when the predicted value of y is the same as the current value. The y-iteration procedure requires a sequence of estimates, t y i * ^ ), from which ic produces a sequence of solutions of (75), ( y ^ i ) • Normally we take y^**^ = uf^ i > i.e. the i t h iterative solution of Lst. Sol. (75) is taken as the (i+l)th estimate. This process may be slow in convergence or may not converge at a l l . In such cases introduction of the accelerating parameters can be of great help (Hartree (1957)). This can be done by taking u^ 1* 1^ = ^ i ^ l + ^ ~ 0> vi^] (Fischer(1968)) for some 9 satisfying 0 < 6 < 1. This technique does accelerate the convergence of y-iterations in some cases and changes the otherwise divergent iterations into convergent ones in some other cases. - 39 -We now describe the method of parametrization for the determination of y. "METHOD OF PARAMETRIZATION" Let P^(n&, r) and P^(n*£,*, r) be the solutions of equations (25) and (27) for u = X. We define an error function e(X) by e(X) = (Px(nA, r) |D U*(r) |Px(n*A*. r)) = -(P x(n*Jl* > r)|D* X ( r ) |Px(n£, r ) ) . In terms of the function e(X), our constraint, namely (20), becomes e(X) =0. The method of parametrization attempts to find the zero of e(X) directly by treating e(X) as a function of X alone. The system of equations is solved and e(X) evaluated for a set of values of X. The set of values of e(X) so obtained is used to determine the zero of e(X) by the variable secant method or an equivalent method. At this stage we must say a few words about the energy difference AE between the two states that so expl i c i t l y appears in the constraint and hence also in our calculation. There are several values of AE (for example, experimental value, S.H.F. value, etc.) that might be used. We have, however, thought i t more consistent with our theory to calculate AE as the difference of H.F. energies at each stage of y-iterations. In the rest of this chapter we state the results of our calculations for L i - and Be-like iso-electronic sequences. - 40 -Sec. 3.2. Results for Three-electron Systems. In this section we state the results of our calculations for the following types of transitions for L i - l i k e iso-electronic sequences. 2 2 2 2 Is 2s Sj - Is np Pi ; n = 2, 3 and 4 * * (Z - 3,4,5,6,8,10,12,15,47 and 50) I s 2 3s 2S1 - I s 2 np 2P,, ; n = 2 and 3 * * (Z = 3,4,5,6,7 and 8 ) . The results of our calculations are presented in tabular form (Tables I to V). One table corresponds to each one of the five transitions given above. A l l members of an iso-electronic sequence are placed in one table. The explanation of various columns is as follows. Column one shows the atom or ion considered. In the second column we have quantities proportional to p f r a ( j a n d p f v e i (defined in Sec. 2.3.), calculated using S.H.F. functions. The pf T - values are placed just below the pf , -values. In column r vel. ^ rad. three we give the common value of pf , and pf .. obtained when C.H.F. r rad. r vel. functions are used. Column four contains the parameter of the constraint. Columns five and six contain the standard and constrained H.F. energies for the two states. The energy of the f i n a l state is placed just below that of the i n i t i a l state. In column seven we give 6E = E(C.H.F.) - E(S.H.F.), for both the i n i t i a l and f i n a l states, which represents the effect of the constraint on the total energy E of the state. DISCUSSION OF RESULTS Having stated the results of our calculations we are now in a TABLE I Results for the transition Is 2s Sj - Is 2p P, for Li-isoelectronic sequence. -2 Ion. •'•Pf (S.H.F.) pf (C.H.F.) y E(S.H.F) E(C.H.F.) <5E L i 0.51041 ./£ 0.52901 v 0.51745 0.00569 -7.432728 -7.365070 -7.432716 -7.365069 0.000012 0.000001 Be + 1 0.34297 'A 0.36668 v 0.34440 0.01600 -14.27740 -14.13086 -14.27734) -14.13083 0.00006 0.00004 B + 2 0.24955 I 0.27173 v 0.24900 0.02061 -23.37599 -23.15372 -23.37592 -23.15364 0.00008 0.00008 0.19458 £ ' 0.21438 v 0.19368 0.02250 -34.72607 -34.42989 -34.72598 -34.42977 0.00009 0.00011 0.13429 & 0.15004 v 0.13354 0.02397 -64.17806 -63.73670 -64.17796 -63.73654 0.00010 . 0.00015 Ne 0.10227 1 0.11520 v 0.10173 0.02450 -102.6311 -102.0462 -102.6310 -102.0461 0.0001 0.0002 Mg + 9 0.08251 I 0.09344 v 0.08213 0.02474 -150.0847 -149.3570 -150.0846 -149.3568 0.0001 0.0002 p+12 0.06394 I 0.07278 v 0.06368 0.02489 -238.1405 -237.1993 -238.1404 -237.1991 0.0001 0.0002 Ag+*4 0.01875 I 0.02165 v 0.01873 0.02520 -2437.409 -2434.202 -2437.409 -2434.201 0 • 0 0.0 S n + 4 7 0.01758 I 0.02031 v 0.01757 0.02521 -2761.716 -2758.296 -2761.716 -2758.296 0.0 0.0 TABLE II Results for the transition Is 2s Sj - Is 3p Py 'for L i isoelectronic sequence. -2 -3 Ion. Pf (S.H.F.) Pf (C.H.F.) u E(S.H.F.) E(C.H.F.) 6E L i 0.00225 I 0.00179 v 0.00181 0.00026 -7.432728 -7.293189 -7.432728 -7.293189 0.0 0.0 Be + 1 0.05198 I 0.04901 v 0.04918 0.00076 -14.27740 -13.83973 -14.27739 -13.83973 0.00001 0.0 B + 2 0.09915 I 0.09532 v 0.09560 0.00115 -23.37599 -22.49936 -23.37599 -22.49936 0.0 0.0 c + 3 0.13279 SL 1 0.12880 v 0.12911 0.00143 -34.72607 -33.27098 -34.72606 -33.27098 0.00001 0.0 0 + 5 0.17483 I 0.17115 v 0.17147 0.00179 -64.17806 -61.14872 -64.17805 -61.14872 0.00001 0.0 Ne + ? 0.19950 I 0.19619 v 0.19650 0.00202 -102.6311.. -97.47161 -102.6311 -97.47161 0.0 0.0 Mg 0.21548 I 0.21260 v 0.21288 0.00217 -150.0847 -142.2393 -150.0847 -142.2393 0.0 0.0 P + 1 2 0.23120 I 0.22870 v 0.22900 0.00231 -238.1405 -225.2244 -238.1405 -225.2244 0.0 0.0 S n + 4 7 0.27186 I 0.27099 v 0.27296 I 0.27215 v 0.27108 0.27224 0.00278 0.00279 -2437.409 -2292.626 -2761.716 -2597.278 -2437.409 -2292.626 -2761.716 -2597.278 0.0 0.0 0.0 6.0 TABLE III Results for the transition Is 2s St - Is 4p ?1 for Li-isoelectronic sequence. -3 -5 Ion. Pf (S.H.F.) Pf (C.H.F.) y E(S.H.F.) E(C.H.F.) 6E L i 0.0023 l 0.0020 v 0.0020 0.00004 -7.432728 -7.268197 -7.432728 -7.268197 0.0 0.0 Be + 1 0.0197 l 0.0187' v 0.0187 0.00011 -14.27740 -13.73896 -14.27740 -13.73896 0.0 0.0 B + 2 0.03231 -I 0.0310 v 0.0310 0.00016 -23.37599 -22.27308 -23.37599 -22.27308 0.0 0.0 c + 3 0.0401 Jl 0.0388 y 0.0388 0.0.0020 -34.72607 -32.87006 -34.72607 -32.87006 0.0 0.0 0.0478 I 0.0472 v 0.0472 0.00025 -64.17806 -60.25202 -64.17806 -60.25202 0.0 0.0 Ne + ? 0.0536s Jl 0.0531 v 0.0531 0.00027 -102.6311 -95.88428 -102.6311 -95.88428 0.0 0.0 Mg + 9 0.0566 Jl 0.0562 v 0.0562 0.00029 -150.0847 -139.7667 -150.0847 -139.7667 0.0 0.0 p+12 0.0593 £ 0.0590 v 0.0591 0.00031 -238.1405 -221.0591 -238.1405 -221.0591 0.0 0.0 A g + 4 4 0.0658 Jl 0.0657 v 0.0657 0.00036 -2437.409 -2243.181 -2437.409 -2243.181 0.0 0.0 S n + 4 7 0.0660 Jl 0.0659 v 0.0659 0.00036 -2761.716 -2541.036 -2761.716 -2541.036 0.0 0.0 TABLE IV 2 2 2 2 Results for the transition Is 3s S! - Is 2p for Li-isoelectronic sequence -3 -2 Ion (S.H.F.) Pf (C.H.F.) V E(S.H.F.) E(C.H.F.) 6E L i 0.2259 £ 0.2329 v 0.2316 0.00084 -7.310214 -7.365071 -7.310214 -7.365069 0.0 0.000002 Be + 1 0.130.6 I 0.1368 y 0.1358 0.00141 -13.87779 -14.13086 -13.87779 -14.13086 0.0 0.0 B + 2 0.0931 £ 0.0978 v 0.0971 0.00171 -22.55805 -23.15372 -22.55805 -23.15371 0.0 0.00001 c + 3 0.0747 £ 0.0784 v 0.0778 0.00186 -33.35004 -34.42989 -33.35004 -34.42988 0.0 0.00001 N + 4 0.0640 £ 0.0671 v 0.0666 0.00196 -46.25344 -47.95779 -46.25344 -47.95777 0.0 0.00002 o + 5 0.0572 £ 0.0598 v 0.0593 0.00202 -61.26813 -63.73670 -61.26813 -63.73668 0.0 0.00002 TABLE. V Results for the transition Is 3s St - Is" 3p P i for Li-isoelectronic s -j " 2 Ion. , P f (S.H.F.) Pf (C.H.F.) y E(S.H.F.) E(C.H.F) <5E L i 0.8206 0.8339 £ V 0.8266 0.00148 -7.310213 -7.293191 -7.310212 -7.293190 0.000001 0.000001 Be + 1 0.5658 0.5809 £ V 0.5666 0.00510 -13.87779 -13.83973 -13.87779 -13.83973 0.0 0.0 B + 2 0.4131 £ 0.4139 0.00772 -22.55805 -22.55804 0.00001 0.4294 V -22.49936 -22.49935 0.00001 c + 3 0.3232 0.3380 £ V 0.3235 0.00932 -33.35004 -33.27098 -33.35003 -33.27097 0.00001 0.00001 N + 4 0.2649 0.2780 £ V 0.2647 0.01036 -46.25344 -46.15417 -46.25343 -46.15415 0.00001 0.00002 0 + 5 0.2241 0.2359 £ V 0.2240 0.01098 -61.26813 -61.14872 -61.26812 -61.14870 0.00001 0.00002 - 46 -position to discuss our results and make certain observations. First of a l l we observe that the asymptotic behaviour of pf-values for a l l the transitions of three-electron systems agrees well with that predicted in Sec. 2.3, namely, that the pf-values tend to a constant (as Z -»• °°) i f the principal quantum number of the active electron changes and tend to zero as otherwise. We next observe that y, the parameter of the constraint, approaches a constant y ^ as Z increases. This is as expected on the grounds that our radial functions tend to hydrogenic functions as Z increases. It has further been observed that for a l l the transitions 2 of the type Is - Ip, where I is the core (Is) , the variation of y with Z is more or less similar. This fact can be used in finding the i n i t i a l estimates, for the y-iterations for an atom or ion, from the values of V for the neighbouring ions. 2 2 Comparison of the total energy of the ground state Is 2s S calculated by the S.H.F. method and by our method of constrained variation shows that the change, due to the constraint, in this energy consists of at most 2 in the fourth decimal place. From the calculation of correlation energy for this ground state by Cooper and Martin (1963), we find that the change in energy, due to constraint, is of the order of one half percent of the correlation energy. From the calculations of Linderberg and Shull (1960), we can estimate the correlation energy for the excited states 2 2 Is np P, n = 2, 3, or 4 to be larger than 0.044, which is the correlation 2 2 2 of the (Is) core. The change in the total energy of the state Is 2p P is found to be at most 3 in the fourth decimal place. The corresponding 2 2 2 2 changes for the states Is 3p P and Is 4p P are even smaller. This - 47 -shows that the effect of the constraint on the total energy of the excited states i s less than one percent of the correlation energy of that state. 2 2 2 2 We next consider transitions of the type Is 3s S - Is np P, n = 2 and 3. A glance at columns seven of Tables IV and V indicates that the effects of the constraint on the total energy E of any state are always smaller than 3 in the f i f t h decimal place. The correlation energies for various states involved are larger than 0.044, which i s 2 the correlation energy of the (Is) core. A simple comparison shows that the change in the total energy of the system due to the constraint is less than 0.1 percent of the correlation energy of the corresponding state. It i s thus concluded that for a l l of the five transitions of three-electron systems the change, due to the constraint, i n the total energy of any state i s less than one percent of the correlation energy of that state. An improvement of worsening of the total energy of a system by one percent of the corresponding correlation energy can easily be neglected, hence i t can be said that in the case of three electron systems the constraint has been imposed at practically no cost of the total energy. - 48 -Sec. 3.3. Results for Four-electron Systems. The results of our calculation for Be-like iso-electronic sequences are presented i n this section. We have considered the following types of transitions: 2 2 1 2 1 Is 2s Sfi - Is 2s np P ; n = 2, 3 and 4. (Z = 4,5,6,7,8,10,12,16,44 and 50) The results are arranged in the form of tables (VI, VII and VIII), which have the same arrangement as given in Sec. 3.2. DISCUSSION OF RESULTS For a l l the transitions mentioned above, the pf-values are found to behave as predicted in Sec. 2.3. The parameter of constraint, y, approaches a constant y ^ ^ as Z-increases. The variation of y with Z is different for a l l the three transitions corresponding to n = 2, 3 and 4. The effect of the constraint on the total energy of various states also 2 2 1 differs from case to case. We consider the transition Is 2s S -I s 2 2s 2p "h? (Table VI) f i r s t . Column seven of Table VI shows that the effect of the constraint 2 2 1 on the total energy of the ground state Is 2s S is smaller than 0.01 for a l l members of the sequence. The constraint raises the energy of the ground state for Be and moderately ionized ions up to Ne+^. For a l l the members of the isoelectronic sequence following Ne+^, the constraint appears to result in a lowering of the energy of the ground state. This lowering of the energy, of magnitude less than 0.001% of the total energy, can be attributed to the non-orthogonality of the calculated constrained TABLE VI Results for the transition Is 2s S - Is 2s 2p P. for Be-isoelectronic s o r 1 Ion. Pf (S.H.F.) Pf (C.H.F.) y E(S.H.F.) E(C.H.F) * <5E * Be 1.71460 1.06388 £ V 1.17214 -0.06156 -14.57303 -14.39474 -14.57083 -14.39423 0.00220 0.00051 B + 1 1.45174 0.76360 £ V 1.33495 -0.25000 -24.23758 -23.91288 -24.22791 -23.90001 0.00967 0.01287 c + 2 1.06834 0.56017 £ V 1.12329 -0.33054 -36.40850 -35.96119 -36.39973 -35.93659 0.00877 0.02460 N + 3 0.84473 0.44857 £ V 0.92499 -0.34785 -51.08233 -50.51708 -51.07687 -50.49796 0.00546 0.01912 o + 4 0.69855 0.37548 £ V 0.77112 -0.34459 -68.25772 -67.57660 -68.25479 -67.54635 0.00293 0.03025 Ne + 6 0.51888 0.28410 £ V 0.56730 -0.32426 -110.1110 -109.2012 -110.1109 -109.1721 0.0001 0.0291 Mg + 8 0.41267 0.22882 £ V 0.44501 -0.30560 -161.9661 -160.8296 -161.9673 -160.8024 -0.0008 +0.0272 s + 1 2 0.29273 0.16490 £ V 0.30940 -0.28080 -295.6787 -294.0916 -295.6809 -294.0670 -0.0022 +0.0246 R u + 4 ° 0.09640 0.05594 £ V 0.09817 -0.23704 -2351.685 -2346.963 -2351.688 -2346.943 -0.003 +0.020 S n + 4 6 0.08428 0.04901 £ V 0.08563 -0.23432 -3047.259 -3041.866 -3047.262 -3041.846 -0.003 +0.020 The ground state values may not be very accurate. See the discussion. TABLE VII Results 2 2 1 for the transition Is 2s S o - I s 2 2s 3p 1 P 1 for Be--isoelectronic sequence. Ion Pf (S.H.F) Pf (C.H.F.) y E(S.H.F.) E(C.H.F.) 6E Be 0.18616 0.07356 £ V 0.08173 -0.00736 -14.57302 -14.33334 -14.57277 -14.33335 +0.00025 -0.00001 B + 1 0.02422 0.05446 £ V 0.04851 -0.01132 -24.23758 -23.62016 -24.23724 -23.62020 +0.00034 -0.00004 c + 2 0.16092 0.19078 £ V 0.18542 -0.00866 -36.40850 -35.27301 -36.40834' -35.27304 +0.00016 -0.00003 N + 3 0.27724 0.29581 £ V 0.29252 -0.00590 -51.08233 -49.28841 -51.08226 -49.28842 +0.00007 -0.00001 o + 4 0.36396 0.37385 £ V 0.37210 -0.00356 -68.25772 -65.66576 -68.25770 -65.66577 +0.00002 -0.00001 Ne + 6 0.48038 0.48042 £ V 0.48041 -0.000025 -110.1110 -105.5052 -110.1110 -105.5052 0.0 0.0 Mg + 8 0.55386 0.54943 £ V 0.55024 0.00246 -161.9661 -154.7902 -161.9661 -154.7902 0.0 0.0 s + 1 2 0.64076 0.63333 £ V 0.63471 0.00569 -295.6787 -281.6951 -295.6786 -281.6951 0.0.001 0.0 R u + 4 ° 0.79173 0.78633 £ V.' 0.78739 0.01212 -2351.685 -2227.819 -2351.685 -2227.819 0.0 0.0 S n + 4 6 0.80139 0.79649 £ V 0.79745 0.01257 -3047.259 -2885.680 -3047.259 -2885.680 0.0 0.0 TABLE VIII 2 2 1 2 1 Results for the transition Is 2s S - Is 2s 4p P1 for Be-isoelectronic sequence. =•—' Ion. pf (S.H.F) Pf (C.H.F.) y E(S.H.F.) E(C.H.F.) 5E Be 0.04902 0.01489 V 0.01553 -0.00125 B + 1 0.01781 0.02932 £ V 0.02890 -0.00127 -24.23758 -23.51060 -24.23756 -23.51060 0.00002 0.0 c + 2 0.06362 0.07064 £ V 0.07040 -0.00073 -36.40850 -35.02703 -36.40850 -35.02703 0.0 0.0 N + 3 0.09466 0.09719 £ V 0.09710 -0.00031 -51.08233 -48.85701 -51.08233 -48.85701 0.0 0.0 0.11536 0.11519 £ V 0.11519 0.00002 -68.25772 -65.00012 -68.25772 -65.00012 0.0 0.0 Ne + 6 0.14055 0.13783 £ V 0.13797 0.00049 -110.1110 -104.2249 -110.1110 -104.2249 0.0 0.0 Mg + 8 0.15504 0.15153 £ V 0.15167 0.00080 -161.9661 -152.7006 -161.9661 -152.7006 0.0 0.0 s + 1 2 0.17084 0.16712 £ V 0.16727 0.00118 -295.6787 -277.4030 -295.6787 -277.4030 0.0 0.0 Ru + A° 0.19485 0.19287 £ V 0.19295 0.00189 -2351.685 -2186.329 -2351.685 -2186.329 0.0 0.0 S n + 4 6 0.19623 0.19446 £ V 0.19454 0.00194 -3047.259 -2831.261 -3047.259 -2831.261 0.0 0.0 - 52 -P(ls, r) and P(2s, r) functions for the ground state. In the S.H.F. scheme i t i s customary to set the off-diagonal energy parameters between two f i l l e d subshells equal to zero, for, in this case these parameters are arbitrary. For a pair of orthogonal solutions, the functions obtained by a unitary transformation from these solutions are also orthogonal solutions of the equations, but with different energy parameters. The total wave function remains unchanged. But the imposition of the constraint with only one function removes the possibility of a unitary transformation since the transformed functions would not satisfy the constraint. This is equivalent to saying that the off-diagonal energy parameter e, „ is unique and in fact, an analysis -LS y like that for 6^£^n»£> e n * £ * n"&* a n d shows this to be the case. In the constrained calculation, the off-diagonal parameter £, _ was r Is,2s l e f t equal to zero and so the computed P(ls,r) and P(2s,r) were not orthogonal as required. As a result, the numerical value obtained for the energy of the constrained ground state using the non-orthogonal functions P(ls,r) and P(2s,r) is in error by a term of order one in the orthogonality integral (P(ls,r) | P(2s,r)). For the sake of comparison, the orthogonality integrals of P(ls,r) and P(2s,r) functions for the constrained ground states are arranged in Table IX. - 53 -TABLE IX Orthogonality integrals for the constrained ground state functions 2 Transition Is 2 1 2 2s S - Is 2snp 1P Ion n = 2 n = 3 n = 4 o + 4 0.009840 0.000051 -0.000004 Mg + 8 0.006849 -0.000025 -0.000005 s + 1 2 0.005012 -0.000044 -0.000006 R u + 4 ° 0.001676 -0.000067 -0.000009 S n + 4 6 0.001465 -0.000073 -0.000009 Watson (1960) and H. P. Kelly (1964) have calculated the 2 2 1 correlation energy for the ground state Is 2s S of Be to be 0.0944. A glance at the Column seven of Table VI shows that the effect of the constraint on the total energy of the ground state i s about ten percent of the corresponding correlation energy for the most affected case of B+^. 2 1 A similar study of the total energy of the state Is 2s 2p P shows a different picture. It i s observed that the change in the energy due to the constraint is about 0.02 for most of the members of the sequence +4 with a maximum of 0.0302 for 0 . The correlation energy for the state 2 L 2 Is 2s 2p TP of Be l i e s between the correlation energies of (Is) and 2 2 (Is) (2s) which are 0.044 and 0.094 respectively. Thus, we estimate 2 1 the correlation energy for Is 2s 2p P of Be to be 0.05. This shows 2 1 that the effect of the constraint on the energy of the state Is 2s 2p P is about forty percent of the corresponding correlation energy for most - 54 -of the moderately ionized members of the sequence except Be and B + \ The behaviour of neutral Be is quite different. The constraint effects on the total energy of the ground state are about three percent of the 2 1 corresponding correlation energy. For the state Is 2s 2p P, the constraint results in a much smaller change i n energy of Be. We next study the changes, due to constraint, i n the total 2 2 1 2 1 energies for the transition Is 2s S - Is 2s 3p P. Column seven of Table VII shows that the changes in the energies of the ground state consist of a maximum of 0.00034 for B + 1. We also find that the effect 2 1 of the constraint on the total energy of the excited state Is 2s 3p P is always smaller than 6 in the f i f t h decimal place. Table VIII shows that the effects of the constraint on the energies of the two states, 2 2 1 2 1 involved i n the transition Is 2s S - Is 2s 4p P, are even smaller. Thus we see that the constraint effects, corresponding to transitions to excited states, on the total energies of the two states are negligible (Being less than one half percent of the corresponding correlation energy) CHAPTER IV 4.1. Comparison of Oscillator Strengths. The present chapter contains the comparison and the accompanying discussion of the oscillator strengths obtained by our method of constrained variation with the values reported by several other workers. Our calculations have been restricted to two-, three- and four-electron systems. Throughout our calculations we have considered oscillator strengths associated with individual lines of the spectrum, i.e. we have considered transitions between two levels determined by the J-values in L-S coupling scheme. Following Shore and Menzel (1965), we define the weighted oscillator strength gf of a line aJ -*• a'J', for absorption, by (using atomic units) gf = oj(J)f ( a J , a ' J ' ) = -| x AE x S(a'J', aJ) (77) where (1) Primes refer to the upper level; ( i i ) w(J) = 2J + 1 is the s t a t i s t i c a l weight of level J ; ( i i i ) AE = E(a'J') - E(aJ), is the absorbed energy; (iv) f(aJ,a'J') i s the oscillator strength; (1) 2 (v) S ( a ' J ' , a J ) is the line strength given by | ( a J | | Q ||a'J')| ; and (vi) is the dipole-length operator (Shore and Menzel (1968)», Chapter 10). Negative absorption or induced emmission, a'J' •*• aJ is described by a negative oscillator strength given by - 56 -u>(J')f (a'J'.aJ) = -u)(J)f (aJ.a'J'). Shore and Menzel, through extensive use of Racah's algebra, have factored the reduced matrix element (aJ | |Q^^| |a'J'), f o r most of the common t r a n s i t i o n s , into three parts, R, . , a l i n e f a c t o r ; r ' l i n e R^ u^ t , a mu l t i p l e t factor; and a r a d i a l i n t e g r a l I(n£, n*£*) (or J(n£, n*£*)) depending only on the quantum numbers of the act i v e electron. Tables of values of l i n e and m u l t i p l e t factors are also worked out. With t h i s f a c t o r i z a t i o n o f ( a j | | Q ( 1 ) I | a ' J ' ) , the gf values defined by (77) can be rewritten as: gf = (-l)V* x / A > x 2|E x R 2 ^ x ^ x l 2 ( n £ j n ^ , ) ( o r J±_^ ( ? 8 ) Here £ denotes the larger of the two integers £ and £* and the two i n t e g r a l s I(n£, n*£*) and J(n£, ri*£*) are the same as defined i n Sec. 1.4. R e c a l l the d e f i n i t i o n s of pf , - and pf n - values defined i n rad. v e l . Sec. 2.3 and observe that these values are proportional to the weighted o s c i l l a t o r strengths gf calculated by using I(n£, n*£*) and J(n£, n*£*) r e s p e c t i v e l y . The tables of Chapter I I I have these gf-values i n the second and t h i r d columns as quantities proportional to pf-values. The c a l c u l a t i o n of the gf-values requires a knowledge of the t r a n s i t i o n energies AE. There are various values f o r AE that can be used, f o r example, the experimental value, or the t h e o r e t i c a l value calculated by our C.H.F. method, or the t h e o r e t i c a l values obtained by various other methods. For three-electron and l a r g e r atoms, the experimental t r a n s i t i o n energies are the most accurately known values - 57 -of AE. For this reason, i t has become a standard practice to use observed energy differences in the theoretical determination of the transition probabilities. The C.H.F. values of AE may differ substantially from their experimental values for two- and four-electron systems, for, in these cases a f i l l e d subshell in the ground state, 2 namely, (ns) , n = 1 and 2,respectively, is s p l i t into two subshells with one electron each in the upper state,and further that the correlation between an s- and a p-electron i s very small compared to that of the 2 (s) subshell. Our calculations indicate that the disagreement between the two values of AE can be as large as twelve percent for neutral Be and six percent for neutral He. This agreement improves as we go towards higher members of the isoelectronic sequence. For the transitions of 2 three-electron systems that we have considered, the core (Is) remains unchanged while the outer electron undergoes a transition. As a result, the correlation effects in both the ground and the upper level are almost equal and the C.H.F. values of AE are expected to be quite close to their experimental values. Our calculations show that the C.H.F. values of AE l i e within one percent of the experimental AE. In our calculations of gf-values for two- and four-electron systems, we have always used both the experimental (Moore (1949)) and the C.H.F. values of AE. The gf-values obtained by the use of experimental AE are referred to as our values. For three-electron systems, because of the good agreement between the calculated and the experimental AE, the gf-values are calculated by the use of only C.H.F. transition energies AE. - 58 -Comparisons of gf-values calculated by our method and a number of other methods w i l l now be given. The gf-values are arranged in tabular forms with one table for each transition. The arrangement of various columns differs from table to table but is clearly indicated in each table. For each transition gf-values obtained by different methods are placed in various columns of the corresponding table for comparison with ours. The heading of each such column gives the reference of the source and the method used for calculation. Among the values reported by several other workers, we shall identify the value considered to be the most accurate by Wiese et. al.(1966). For values appearing after the work of Wiese et. a l . , we shall follow the accuracy classification scheme of Wiese et. a l . whose notation for various accuracy classes i s . AA for uncertainties within 1 percent; A do 3 percent; B . . . . do . . . 10 percent; C . . . . do . . . 25 percent; D do 50 percent; E for uncertainties larger than 50 percent; 2 1 1 For He atom we consider the two transitions Is S - Is np P, n = 2 and 3. For He atom the calculations of Pekeris (1962) and of Schiff et. a l . (1965) are the most accurate determinations of the wave functions' and energies cf txvo electron systems, that have been made so far. Using their highly accurate energies and wave functions for two electron systems, Schiff et. a l . (1964) have calculated the oscillator strengths for various S - P transitions of He atom. We shall refer to these gf-- 59 -values as the exact values. The gf-values obtained by our method of constrained variation are presented in Table X for comparison with other values. Discussion of the Helium transitions (Table X); Comparison of various values of the oscillator strength for 2 1 1 the transition (Is) S - Is 2p P. shows that the S.H.F. radial and velocity forms disagree with each other by eight percent. The constrained value l i e s closer to the radial form than the velocity form but does not l i e between them. The C.H.F. value agrees with the exact value to about 0.5 percent. The comparison of our value with those obtained by Chong et. a l . (1968) and Yue et. a l . (1969) indicates that our value is at least as accurate as the best among their values. It appears that our imposition of the constraint symmetrically on both the states and the unrestricted variational determination of the radial function more than make up for their use of several configurations. Our constrained value agrees to better than 0.5 percent with that obtained by Weiss (1967). 2 1 1 A similar comparison for the transition (Is) S - 12 3p P shows that the S.H.F. radial and velocity values disagree with each other to about nine percent. The constrained value li e s within one percent of the radial value but does not l i e between the radial and velocity values. The C.H.F. value agrees to better than 0.5 percent with the exact value. Our constrained value also agrees with that of Weiss to about 0.8 percent. The constrained gf-values obtained' by using calculated AE, for both the above mentioned transitions f a l l short of the exact value by about six TABLE X Comparison of oscillator strengths for He transitions r 1 (a) (b) (c) (d) (e) Transition AE c a l . AE exp. AA, AA (S.H.F.) (C.H.F.) (S.H.F.) (C.H.F.) i s 2 Vls2 P h> 0.25322 £ 0.26124 v 0.26045 0.26711 £ 0.24766 v 0.27478 0.27616 ±0.00001 0.27953 0.28004 0.2760 2 1 1 Is S-ls3p P t-0.06940 £ 0.07018 v 0.07016 0.07297 £ 0.06675 v 0.07367 0.0734 ±0.0001 0.0732 (a) - Present calculations (d) - Yue and Chong (1969) (b) - Schiff and Pekeris (1964) (e) - Weiss, A. W. (1967) (c) - Chong and Benston (1968) A E c a l ~ C a l c u l a t e d transition energy AE - Experimental transition energy, exp • - 61 -percent. As we have seen above, large parts of these discrepancies l i e i n the use of calculated AE. For both the t r a n s i t i o n s discussed above, our gf-values show an excellent agreement with the values obtained by the use of highly accurate solutions of the SchrBdinger equation ( S c h i f f et. a l . (1964)) and also with the values obtained by the use of Hylleraas type (with 52 terms) wave function by Weiss (1967). Such an agreement i s highly encouraging, f o r , i t demonstrates that the discrepancy between the length and v e l o c i t y forms of S.H.F. functions has been removed and the common value agrees c l o s e l y with highly accurate values provided accurate values of AE can be obtained by other methods, po s s i b l y from experimental data. 2 2 2 2 Discussion of Is 2s S Is 2p P (Table XI) Comparison of various values of gf given i n the table shows a good o v e r a l l agreement between values obtained by d i f f e r e n t methods. Our S.H.F. r a d i a l forms agree with the r a d i a l forms of Cohen and K e l l y and also with those of Weiss to w i t h i n two percent. Our v e l o c i t y form agrees with the v e l o c i t y forms of Weiss and those of Cohen and K e l l y to within four percent. Furthermore, there i s e x c e l l e n t agreement between our constrained values and those computed by Weiss using 45-configuration wave functions. Weiss's r a d i a l and v e l o c i t y form agree with each other to three percent i n 45-configuration scheme. Our constrained values l i e between these values i n a l l but one case. Our constrained values are TABLE XI Comparison of Oscillator Strengths for Is 2s Sj - Is 2p Px -2 ^ Ion gf (S.H.F.) gf (C.H.F.) (a) (b) (c) (d) (e) (f) L i 0.51041 £ 0.52901 v. 0.51745 0.5113 0.5270 0.5021 0.5150 A 0.50925 0.52728 0.5408 0.53 Be* 1 0.34297 £ 0.36668 v 0.34440 0.3408 ) 0.3661 0.3367 0.3470 A 0.34164 0.36987 0.3607 0.35 B + 2 0.24955 I 0.27173 v 0.24900 0.2474 0.2700'. 0.2443 0.2507 A :•• 0.24953 0.27647 0.2611 c + 3 0.19458 I 0.21438 v 0.19368 0.1930 0.2118 0.1905 0.1943 A 0.19543 0.21943 0.2025 0.1903 0.20 0 + 5 0.13429 £ 0.15004 v 0.13354 0.1335 0.1468 0.1307 0.1340 A 0.1322+ 0.1382 0.1326 0.14 Ne + ? 0.10227 £ 0.11520 v 0.10173 0.1020 0.1119 0.1014 B+ 0.10* 0.1045 0.11 (a) Weiss, A., (1963) - Anal. Self. Cons. (e) Warner (1968) - Scaled Thomas-Fermi (b) Weiss, A., (1963) - Configuration interaction (f) Allen (1963) - Misc. (c) Cohen & Kelly (1967) - Frozen Core t Kelly, P. S. (1964) (d) Varsausky, (1961) - Var. Screening Constants * Hinnov.e, E. (1966) - 63 -closer to our S.H.F. radial values but do not necessarily l i e between the radial and velocity form. Values obtained by Warner using scaled Thomas-Fermi functions l i e very close to our radial form. 2 2 2 2 Discussion of Is 2s S + Is 3p P (Table XII): Our S.H.F. radial forms agree to within five percent with the radial forms of Weiss and those of Cohen and Kelly (in a l l cases except L i ) . Our S.H.F. velocity values agree with the velocity values of Weiss and of Cohen and Kelly to within four percent. Our radial- and velocity-values l i e within four percent of each other except for L i . Our constrained gf-values are very close to our S.H.F. velocity value and always l i e between the radial and velocity forms. Comparison of our values with those of Kelly and those of Allen shows much larger discrepancies. This can be due to the Coulomb approximation and H.F.-Slater methods used by them. Values obtained by Warner, using scaled Thomas-Fermi functions, l i e within one percent of our radial form and three percent of our constrained values. Discussion of I s 2 3s 2S ->- I s 2 3p 2P (Table XIII) : Comparison of our values with those given by Weiss shows that both our S.H.F. radial and velocity forms agree with the corresponding forms of Weiss to within two percent. Values obtained by A l i and Crossley, using the length form, agree with our length form values to within five percent. Our S.H.F. radial and velocity forms agree with each other to about six percent. Our constrained values l i e very close to our S.H.F. radial values but do not always l i e between the length and the velocity TABLE XII Comparison of Oscillator Strengths for I s 2 2s 2S^ - I s 2 3p 2P X Ion gf (S.H.F.) gf (C.H.F.) (a) (b) (c) (d) L i 0.00225 0.00179 i V 0.00181 0.0018 0.0017 0.0024 0.0017. Be + 1 0.05198 0.04901 I V 0.04918 0.0536 0.0487 B 0.0519 0.0483. B + 2 0.09915 0.09532 I V 0.09560 0.1006 0.0949 B 0.0987 0.0946 c + 3 0.13279 0.12880 I V 0.12911 0.1336 0.1282 B+ 0.1329 0.1281 0.15 0.1337 0 + 5 0.17483 0.17115 I V 0.17147 0.1745 0.1705 B + 0.206+ 0.25 0.1746 Ne + 7 0.19950 0.19619 V 0.19650 0.1986 0.1956 B (a) Weiss, A. (1963) - Anal. Self. Cons. (d) Warner (1968) - Thomas-Fermi (b) Cohen & Kelly .(1967) - Frozen Core. t Kelly, P. S. (1964) (c) Allen (1963) - Misc. TABLE XIII Comparison of Oscillator 2 2 strengths for Is 3s St - Is 2 3 p \ Ion gf (S.H.F.) gf (C.H.F.) (a) (b) (c) (d) L i 0.8206 I 0.8339 v 0.8266 0.8566 0.8172 0.8374 B+ 0.8314 0.8457 Be + 1 0.5658 I 0.5809 v 0.5666 0.5628 0.5590 0.5865 B 0.5666 0.5845 B + 2 0.4131 I 0.4294 v 0.4139 0.4100 0.4094 0.4331 B 0.4170 0.4345 C + 3 0.3232 I 0.3380 v 0.3235 0.3208 0.3208 0.3404 B 0.3223 0.3370 0.3193 0.2649 I 0.2780 v 0.2647 0.2633 0.2633 0.2797 B 0.2627 o + 5 0.2241 I 0.2359 v 0.2240 0.2230 0.2230 0.2372 B 0.2230 (a) A l i & Crossley (1968) (b) Weiss,, A., (1963) (c) Cohen & Kelly (1967) (d) Warner (1968) - 66 -form. Both our S.H.F. radial- and velocity-values agree with the corresponding forms of Cohen and Kelly to within two percent. 2 2 2 7 Discussion of Is 3s S ->• Is 2p P (Table XIV): Our S.H.F. radial form values agree to within three percent with the radial values computed by Weiss and those by Cohen and Kelly. Our velocity form values compared with the velocity form values of Weiss and those of Cohen and Kelly show an agreement to within three percent. Our S.H.F. radial and velocity values agree with each other to within five percent. The constrained values are found to l i e very close to our S.H.F. velocity values and always l i e between the radial and velocity values. Discussion of I s 2 2s 2 1S ->- I s 2 2s 2p (Table XV): For this case we have a large number of values for comparison. As indicated by the accuracy rating (C +, D and E) of Wiese et. a l . , not many of these values can be expected to be very reliable. A glance at column four of the table shows that our S.H.F:' radial- and velocity-values differ with each other very widely, to a factor of 2. Our radial values are in agreement upto six percent with the radial values of Pfennig et. a l . (except for Be). Our S.H.F. velocity values also agree with the velocity form pf Pfennig et. a l . up to seven percent (except for Be.) Our constrained values l i e close to S.H.F. radial values (except for Be) but do not always l i e between the length and velocity forms. Wiese et. a l . (1966) have suggested gf-values with accuracy rating of C + for Be and D for the rest of the members. Comparison of our C.H.F. gf-values with these values shows an agreement to better than six percent for Be TABLE XIV 2 2 2 2 Comparison of O s c i l l a t o r Strengths f o r Is 3s S, - Is 2p F1 2 3_ Ion gf gf (a) Cb) Cc) (S.H.F.) (C.H.F) L i 0.2259 £ 0.2316 0.2305 B+ 0.2269 0.2329 v 0.2278 0.2342 B e + 1 0.1306 £ 0.1350 0.1330 B+ 0.1311 0.1368 v 0.1342 0.1371 B + 2 0.0931 £ 0.0971 0.0939 B+ 0.0932 0.0978 v 0.0962 0.0980 c + 3 0.0747 £ 0.0784 v 0.0778 0.0752 0.0772 B+ 0.0748 0.0786 0.0737 N + 4 0.0640 £ 0.0671 v 0.0666 0.0644 0.0625 B + 0.0633 o + 5 0.0572 £ 0.0598 v 0.0593 0.0574 0.0590 B + 0.0570 (a) Weiss, A. (1963) (c) Warner (1968) (b) Cohen & Kelly (1967) TABLE XV Comparison of Oscillator strengths for the transition lo 2s S - Is 2s 2p P Calculated AE Experimental AE Ion gf (S.H.F.) gf (C.H.F.) gf (S.H.F.) gf (C.H.F.) (a) (b) (d) Be 1.71460 1.06388 I V 1.17214 1.86521 0.97797 I V 1.28729 1.6420 1.0410 1.7 1.36 C+ 1.254+ 1.311 1.066 1.19 B+ B + 1 1.45174 0.76360 I V 1.33495 1.49526 0.74137 % V 1.36155 1.1 D c + 2 1.06834 0.56017 I V 1.12329 1.11385 0.53728 I V 1.13113 1.0711 0.5634 0.6 0.81 D 0.793 0.605 0.69 C+ N + 3 0.84473 0.44857 I V 0.92499 0.88997 0.42577 Jl V 0.93537 0.8466 0.4510 0.5 0.64 D 0.628 0.480 0.54 C+ o 0.69855 0.37548 Jl V 0.77112 0.74208 0.35345 Jl V 0.78759 0.7002 0.3771 0.5 0.53 D 0.524 0.403 0.45 C+ Ne 0.51888 0.28410 I V 0.56730 0.55857 0.26391 Jl V 0.59185 0.60* 0.57 E (a) Pfennig et. a l . , (1965) (d) Steele et. a l . , (1966) (b) Wiese, Smith and Glennon, (1966) * Hinnowe, E., (1966) t Kelly, H. P., (1964) 69 -and Ne+^. For the rest of the members the agreement i s very poor. Steele et. a l . have calculated the oscillator strengths for this sequence by including the correlation effects of the ground state 2 2 1 Is 2s S. We tend to regard these values as quite reliable, for, we 2 2 know that the ground state Is 2s has a very strong orbital degeneracy 2 2 and must be treated by mixing Is 2p with i t . Comparison of our constrained values with those suggested by Steele et. a l . shows that our constrained values are about seventy percent higher than those of Steele et. a l . in a l l cases except Be. For the Be atom we also have H. P. Kelly's value obtained by including various correlation effects. Our constrained value for Be lie s within eight percent of the value of Steele et. a l . and about three percent of H. P. Kelly's value. This shows that for a l l members (except Be and B +) of the iso-electronic sequence the constrained values are a poorer estimate of oscillator strengths than either the velocity- or radial values. 2 2 1 2 1 Discussion of Is 2s S -»- Is 2s 3p P (Table XVI): Comparison of our S.H.F. radial values with those of Pfennig et. a l . shows the two sets of values agree with each other to within ten percent. A similar comparison of velocity-values shows an agreement within five percent of each other except for Be. Comparison of our S.H.F. radial values with our velocity values shows a very poor agreement for +1 +2 Be and B , but this improves very rapidly for C and rest of the sequence. For this transition we do not have any values obtained by including the correlation effects 5so we can not judge the r e l i a b i l i t y of our constrained values. The values given by Wiese et. a l . have an accuracy classification TABLE XVI Comparison of oscillator strengths for the 2 2 1 transition Is 2s S o Is 2s 3p 1 P 1 Calculated AE Experimental AE Ion gf (S.H.F.) gf (C.H.F.) gf (S.H.F.) gf (C.H.F.) (a) (b) (c) •(d) Be 0.18616 £ 0.07356 -v 0.08173 0.21301 £ 0.06429 v 0.09362 0.1961 . 0.0735 0.299 B + 1 0.02422 £ 0.05446 v 0.04851 0.25757 £ 0.05121 v 0.05162 0.045 c + 2 0.16092 £ 0.19078 v 0.18542 0.16720 £ 0.18361 v 0.19269 0.1573 „ 0.1897 , 1 0.26 D 0.180 0.2 N + 3 0.27724 £ 0.29581 v 0.29252 0.28486 £ 0.28790 v 0.30057 0.2734 0.2947 U , / y 0.55 D 0.279 0.3 o + 4 0.36396 i 0.37385 v 0.37210 0.37163 £ 0.36614 v 0.37994 0.3621 0.3729 0.59 D 0.350 0.3 (a) Pfenning et. a l . , (1965) (b) Wiese, Smith and Glennon (1966): (c) Odabasi (1969) (d) Allen (1963) - 71 -+2 +3 +4 (D) for C , N and 0 . Pfennig et. a l . , however, have recommended some values lying between their radial and velocity values as the gf-values for this transition. Our constrained values are found to f a l l within seven percent of these recommended values. The discrepancies between the same form of gf-values obtained by the use of the experimental and the calculated transition energies are largest, as expected, for the Be case where the values differ by about twelve percent. These discrepanci decrease quite rapidly as we go along the isoelectronic sequence to higher members. The comparisons of the oscillator strengths of various transitions of three-electron systems show excellent agreement of our constrained values with those obtained by several other authors by a large variety of methods, especially that of configuration interaction by Weiss. The success of our method can be attributed to the fact that the three-electron systems are essentially of one-electron type. The constraint that we impose on the system acts mainly on the valence electron and leaves the 2 f i l l e d core (Is) unchanged. This also suggests that we can expect to obtain reliable values of oscillator strengths, by our method combined with the frozen core method of Cohen and Kelly (1967), even for larger atoms with one electron outside the f i l l e d subshells. An immediate 2 2 application can be the Na atom with (Is) (2s) treated as the frozen core in both the upper and the lower state and the constrained variation applied to the 2p and 3s radial functions of the ground state and 2p and np radial function of the higher state. - 72 -Calculations for four-electrons systems show that, for 2 2 1 2 1 the transition Is 2s S Is 2s 3p P, our constrained gf-values are an improvement over the S.H.F. values mainly because they remove the large discrepancies between the length and velocity forms. However, 2 2 1 2 1 for the transition Is 2s S Is 2s 2p P, the gf-values obtained by our method of constrained variation do not show good agreement with quite reliable values obtained by Steele et. a l . by the use of configuration interaction. This disagreement may be the result of strong correlation effects. In both transitions, the ground state is a mixture of the 2 2 2 2 2 2 2 configurations Is 2s and Is 2p . For the Is 2s Is 2s 2p transition, both the configurations of the ground state contribute appreciably to the transition probabilities, whereas, for the 2 2 2 2 2 Is 2s ->• Is 2s 3p transition, the contribution of the Is 2p configuration of the ground state towards the transition probability is very small. - 73 -CONCLUSIONS A method of constrained variation is developed to improve the quality, in connection with the calculation of the transition probabilities, of the Hartree-Fock wave functions for atomic systems. The improvement of wave functions results in the removal of the discrepancy between the dipole-length and the dipole-velocity forms of the oscillator strengths calculated from S.H.F. functions. The results of our calculation show that our method is capable of giving us values of the oscillator strengths which are more accurate than those given by the S.H.F. method. For the simple systems like He and L i , which have no degeneracy correlation effects, the imposition of our constraint may prove almost as useful as the configuration mixing. It is very encouraging to find that a d i f f e r e n t i a l constraint, involving wave functions belonging to different configurations,can be incorporated in the variation scheme of Hartree-Fock without any significant cost of the total energy. Furthermore, this constraint f i t s quite nicely into the Z-expansion scheme and gives a reasonably meaningful expansion of the parameter y associated with i t . The comparison of the relative magnitudes of the length-, velocity- and the constrained form of the oscillator strengths obtained by using calculated transition energies, for a l l the transitions we have considered, indicates that the length form agrees more closely with the constrained value than the velocity form for a l l the transitions for which the principal quantum number of the active electron does not change. - 74 -We thus suggest the use of dipole length form for such transitions. The same conclusion has been arrived at by Layzer (1968) and Crossley (1969). For the transitions for which the principal quantum number of the active electron changes, we find that the velocity form of the gf-values agrees better with the constrained values than the length form. The use of the dipole velocity form i s suggested for such transitions. The same suggestion has been made by Layzer (1968) and Crossley (1969). - 75 -BIBLIOGRAPHY A l i , M.A. and Crossley, R.; "Oscillator strengths by perturbation theory I. Transitions between l s ^ 3L ^-L states of the lithium isoelectronic sequence", Mol. Phys. 15 ~4 (1968), 397-404. Allen, C , "Astrophysical Quantities", Athlone Press (1963). Bates, D. R. and Damgaard, A., "The calculation of the absolute strengths of spectral lines", Phil. Tran. Roy. Soc. 242 (1949). 101-122. Benston, M. L. and Chong, D. P., "Off-diagonal constrained variation in open shell SCF theory", Mol. Phys. 12 (1967), 487-492 and 13 (1967), 199. Bethe, H. A. and Salpeter, E., "Quantum Mechanics of one and two Electron Atoms", Springer Verlag, Berlin, (1957). Chen, J. C. Y., "Off-diagonal hypervirial theorem and i t s applications", J. Chem. Phys. 40 (1964), 615-621. Chong, D. P. and Benston, M. L., "Off-diagonal hypervirial theorems as constraints", Chem. Phys. 49 (1968), 1302-1306. Cohen, M. and Dalgarno, "An expansion method for calculating atomic properties IV. Transition probabilities", Proc. Roy. Soc. (London) A-280 (1964), 258-270. Cohen, M. and Kelly, P. S., "Hartree-Fock wave functionsfor excited state III. Dipole transitions in three electron systems", Canad. J. Phys. 45 (1967),1661-1673. Cooper, J. W. and Martin, J. B., "Correlation effects in two- and three-electron systems", Phys. Rev. 131 (1963), 1183-1186. Coulson, C. A., "The use of commutator relationships in determining Schrtfdinger wave functions", Quart. J l . Math. Oxford 1_6 (1965), 279-287. Crossley, R. , "The calculation of atomic transition probabilities", Adv. At. and Mol. Phys. 5_ (1969), 237-296. Edlen, B., "Atomic Spectra" in Handbuch d. Physik XXVII, Springer Verlag, Berlin,(1964). Edmonds, "Angular Momentum in Quantum Mechanics", Princeton University Press (1957). Epstein, S. T. and Hirschfelder J. 0., "Hypervirial theorems for variational wave functions", Phys. Rev. 123 (1961), 1495-1502. - 76 -Fischer, C. F., "Description of a H-F Program with Configuration Mixing", Unpublished (1968). Froese, C , "The limiting behaviour of atomic wave functions for large atomic number ", Proc. Roy. Soc. A-239 (1957), 311-319. Froese, C , "The limiting behaviour of atomic wave functions for large atomic number II", Proc. Roy. Soc. A-244 (1958), 390-397. Froese, C , "The effect of configuration interaction on the transition integral", Ap. J. 141 (1965), 1206-1221. Godfredsen, E., "Atomic term energies for atoms and ions with 11 to 28 electrons", Ap. J. 145_ (1966), 308-332. Hartree, D., "The Calculation of Atomic Structures", Wiley (1957). Hartree, D., "Variation of atomic wave functions with atomic number", Rev. Mod. Phys. 30 (1958), 63-68. Hinnove, E., "Excitation and ionization rates of Neon ions in a stellarator discharge", J. Op. Soc. Am. 56 (1966), 1179-1188. Hirschfelder, J., "Classical and quantum mechanical hypervirial theorems", J. Chem. Phys. 33 (1960), 1462-1466. Hirschfelder, J. and Coulson, C. A., "Hypervirial theorems applied to molecular quantum mechanics", J. Chem. Phys. 3>6_ (1962), 941-946. Kelly, P. S., "Transition probabilities in Nitrogen and oxygen from Hartree-Fock-Slater wave functions", J. Quan. Spec. Rad. Trans. 4_ (1964), 117-148. Kelly, H. P., "Many body perturbation theory applied to atoms", Phys. Rev. 136 (1964), B 896-B 912. Layzer, D., "On a screening theory of atomic spectra", Ann. Phys. 8_ (1959), 271-296. Layzer, D., "Z-expansion of Hartree-Fock wave functions", Phys. Rev. 132 (1963), 735-737. Layzer, D., "Theoretical Atomic Transition Probabilities", Annual review of Astronomy and Astrophysic ,6_, 449, (1968). Linderberg, J. and Shull, H., "Electronic correlation energy in 3- and 4-electron atoms", J. Mol. Spec. 5_ (1960), 1 - 16. - 77 -Linderberg, J., "Perturbation treatment of Hartree-Fock equations", Phys. Rev. 121 (1961), 816-819. Lowden, P. 0., "Studies of atomic self-consistent fields II. Interpolation problems", Phys. Rev. 94 (1954), 1600-1609. Messiah, A., "Quantum Mechanics", Vol. I & II, Wiley (1962). Moore, C., "Atomic Energy levels", Nat. Bur. Stds. Circular 467, (1949). Odabasi, H., "Configuration mixing in N = 4 electron isoelectronic sequence", J l . Opt. Soc. America 59-5 (1969), 583-588. Pekeris, C. L., "Ground states of two-electron atoms", Phys. Rev. 112 (1958), 1649-1658. Pekeris, C. L., "1 1 s and 2 3S states of Helium", Phys. Rev. 115 (1959), 1216-1221. 1 1 3 -Pekeris, C. L., "1 s, 2 s and 2 S states of H and of He", Phys. Rev. 126 (1962),1470-1476. Pfennig, H., Steele, R., and Trefftz, E., "Wave functions and oscillator strengths of Be-like ions (Be I, CIII, N IV, 0 V)", J. Quan. Spec. Rad. Trans. 5_ (1965) , 335-357. Racah, G., "Theory of Complex spectra I", Phys. Rev. 61 (1942), 186-197. Schiff, B., and Pekeris, C. L., "f values for transitions between the 1 1S, 2 1S and 2 3S and the 21?, 2 3P, 3 LP and 3 3P states in Helium", Phys. Rev. 134 (1964), A638-A640. Schiff, B., Lifson, Pekeris, and Rabinowitz," 2 ' P, 3 ' P and 4 ' P states of He and the 2 4 state of L i + " , Phys. Rev. 140 (1965), A1104-A1121. Schiff, L., "Quantum Mechanics", McGraw H i l l (1955). Sharma, C. S. and Coulson, C. A., "Hartree-Fock and correlation energies for Is 2s ^S and S states of Helium-like ions", Proc. Phys. Soc. (London) 80 (1962), 81-96. Shore, B. and Menzel, D., "Generalized tables for the calculation of dipole transition probabilities", Ap. J. (Supp) 1^2 (1965), 187-214. Shore, B. and Menzel, D., "Principles of Atomic Spectra", Wiley (1968). Slater, J. C. , "Quantum Theory of Atomic Structure", Vol. I & II, McGraw H i l l (1960). - 78 -Steele, R. and Trefftz, E., "Wave functions and oscillator strengths of some Iron and Be-like ions with configuration mixing", J. Quan. Spec. Rad. Trans. 6 (1966), 833-846. 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Constrained Hartree-Fock wave functions for atoms Qureshi, Hilal Ahmed 1970
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Title | Constrained Hartree-Fock wave functions for atoms |
Creator |
Qureshi, Hilal Ahmed |
Publisher | University of British Columbia |
Date Issued | 1970 |
Description | This thesis deals with the calculation of Hartree-Fock wave functions satisfying an off-diagonal hypervirial relation as a constraint. The constraint in this case implies that the dipole length form and the dipole velocity form of the transition probability give identical values. Mathematically, this is equivalent to forcing the approximate eigenfunctions of the Hamiltonian of the system to satisfy a relation which is true for exact eigenfunctions. The method of constrained variation is used to solve this problem. The constrained Hartree-Fock system of equations is solved numerically. The Z-expansions of radial wave functions, the diagonal and the off-diagonal energy parameters and the parameter of constraint are carried out. The effect of the constraint on the total energy E of the system, defined as the change in the Hartree-Fock total energy due to the constraint, is estimated. The method of constrained variation is then applied to a few two, three and four electron systems to calculate the constrained total energy E of the system and also the oscillator strengths of a few of the transitions of the system. The results indicate that the oscillator strengths can be calculated more accurately, at practically no cost of the total energy E, with the aid of the constrained Hartree-Fock functions than with the standard Hartree-Fock functions in all those cases where the correlation effects are not too strong to invalidate the single configuration approximation. |
Subject |
Wave functions |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080506 |
URI | http://hdl.handle.net/2429/35394 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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