Dilemma of the “Best Wavefunction”: Comparing Results of the STO-NG Procedure versus the Linear Variational Method

Abstract

The aim of this article is to present a set of simple exercises that help the students of a second course in quantum chemistry to understand that some molecular properties do not improve even when the molecular energy improves with an increasing number of basis functions in the expansion of the wavefunction. The essential idea is to create a dilemma for the students by showing that a wavefunction that gives a better energy (i.e., lower, according to the variational principle) may not yield a better value for another molecular property. The first exercise is to calculate the ground-state energy of the hydrogen atom for the contracted 1s STO-2G basis function; the second is to compute the ground-state energy for the STO-2G basis function reformulated as a linear variational function; the third is to calculate the expectation value of the electron–nuclear distance for the two approximate wavefunctions. Comparison of the approximate results with the exact one reveals starkly the dilemma of the “best wavefunction”. The set of exercises presented here are interesting to any quantum mechanics course.

A simple approximate solution for the H3+ ion

“…The fundamental breathing frequency ν (in cm −1 ) was computed from the curvature of the PES (black curve in Figure 6) at the equilibrium position using the equation [22] , where c is the speed of light in vacuum and m = 1837.152 a.u. is the effective mass associated to this normal mode.…”

“…According to the variational principle [20,22,23], for each internuclear distance R we must minimize Equation ( 9) with respect to the Gaussian exponent α. The task now is a non-linear optimization problem which must be solved numerically as Equation ( 9) is too complicated to take derivatives analytically.…”

“…The fundamental breathing frequency ν (in cm À1 ) was computed from the curvature of the PES (black curve in Figure 6) at the equilibrium position using the equation [22]…”

Integrating Computational Chemistry into Secondary School Lessons

“…The areas of chemistry, where the CC approach can be utilized, are diverse, ranging from general and inorganic to physical and organic chemistry. 24 Looking at the utilization of quantum mechanical calculations and MO theory, Peŕez-Torres 25 and Cruzeiro et al 26 explore the STO-NG procedure and the linear variational method, respectively, to compare wave functions, aiming to clarify the concept of the "best wave function" for students. Murphy et al 27 provide a student-friendly guide to molecular integrals, making complex quantum mechanical concepts more accessible.…”

Estimating the Expected Time to Enter and Leave a Common Target Area in Robotic Swarms

“…There are interesting applications of this method in multiagent research such as Chen et al [52], Lasry and Lions [53]. The variational method [54,55] is a solution similar to that used in quantum mechanics for approximating the desired energy state by selecting wave functions whose parameters minimise the expected value of the energy. The neural network quantum states [56] approximate the solution using machine learning to minimise the variational energy.…”