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Studies in constrained variation Yue, Tony Chee Ping

Abstract

Three different constraints are considered in this thesis, namely: the integral electron cusp condition as a constraint; the off-diagonal hyper-virial theorems as constraints; and perturbation-induced constraints. Eleven approximate configuration-interaction wavefunctions for the ground state of helium are used to test the application of the integral electron cusp condition as a constraint. The results indicate that, if the approximate wavefunction is flexible enough, the calculated electron density at the nucleus is improved when the cusp constraint is imposed. However, the expectation values of rˉ ¹and r ˉ² do not change significantly. Further investigation is made on the use of off-diagonal hypervirial theorems as constraints. The transition from the 1¹ S state to the 2¹ P state of helium is chosen as an example. Firstly, it is found that an energy-independent off-diagonal constraint is not useful in improving calculation of transition probabilities. Secondly, when the approximate wavefunction is flexible enough, iteration on the transition energy converges very rapidly. Finally, the study is extended to the isoelectronic species Li⁺ and Be⁺⁺. A five-term approximate configuration-interaction wavefunction for the ground state of helium is used to test the validity of the perturbation-induced constraints scheme. Different constraint operators are constructed for different properties. The properties studied are expectation values of rˉ² , rˉ¹ , r and r⁺² . Two methods of the perturbation-induced constraints are tested: in one, the first-order wavefunction is fixed and constant for all constrained properties; in the other, the first-order wavefunction varies with the constrained properties. It is disappointing to find that for this particular wavefunction chosen, both methods fail to improve the properties studied when one imposes the constraints.

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