Jerry Goodisman

Calculation of Diamagnetic Shielding in Molecules

The average diamagnetic shielding at a nucleus can be computed from the free‐atom shieldings plus a knowledge of the molecular structure. The validity of computing this molecular quantity from free‐atom values is related to the isoelectronic principle or the negligible change in dissociation energy with change in nuclear charge (total numbers of electrons and internuclear distances remain constant). The theory is also applied to the determination of the signs of nuclear spin–rotation constants. The signs of the proton spin–rotation constants in methane and acetylene are shown to be positive.

Numerical Calculation of H2 and HeH+ Wavefunctions

Use of purely numerical techniques to evaluate integrals allows for greater flexibility in the trial functions used for variational calculations. For two‐electron problems, we have used crossed Gauss quadratures in four dimensions. Accuracy can be assessed in terms of the effect of increasing the number of integration points. Previously given formulas are modified, for singular integrands, so that results are monotonic with the number of points in any dimension, and the new formulas are tested on H2 and on the HeH+ ground state. The meaning of linear dependence of basis functions in the framework of Gauss integration is discussed. The first excited 1Σg+ state of HeH+ is investigated, and found to be nonbonding. The basis functions used here were James—Coolidge functions modified to include polar character. Another example of the flexibility of the numerical method is the use of correlation functions of simple analytical form. This is illustrated with the Gaussian exp(γr122), while formulas are given for c...

Perturbation Treatment of the Ground State of the Hydrogen Molecule

Perturbation theory is used to obtain approximate energies and wavefunctions for the hydrogen molecule ground state. A function of simple form is chosen as zero‐order function, the zero‐order Hamiltonian is constructed, and the variational equation for the first‐order wavefunction is solved approximately by expansion in James—Coolidge basis functions, yielding the second‐ and third‐order energies. All integrals are calculated numerically. The binding energy and equilibrium internuclear distance are fair, but not as good as what one obtains from a comparable variational calculation.

Minimization of the Width as an Alternative to the Conventional Variation Method

The choice of param eters in a trial function to minimize the width Q2 (Q2 = 〈H2〉 − 〈H〉2 for a normalized wavefunction) is an alternative procedure to the usual minimization of 〈H〉 for generation of approximations to the true wavefunction. The minimization of the width is related to solving the eigenvalue equation by making the local energy HΦ/Φ as constant as possible, and other arguments can be put forth which suggest that this procedure should yield a wavefunction closer to the exact than that given by the usual variational method. For the ground state of H2 near its equilibrium internuclear distance, we choose the parameters ci in Φ = ΣiciΦi to (a) minimize the width and (b) minimize the energy. The expectation values from the function obtained by (a) are almost always worse than those from (b). Modifications of procedure (a) designed to improve the wavefunction in a specific region of configuration space were also attempted, without encouraging results.

Calculation of the Barrier to Internal Rotation of Ethane

It is proposed to calculate the barrier to internal rotation from the torque on a CH3 top when the molecule is in an intermediate rotational configuration, rather than from the energies of the staggered and eclipsed forms. Advantages of this approach, which requires only an approximation to the electronic distribution for this one configuration, are discussed. Three easily derived approximate molecular orbital functions for ethane yield barriers between 0.7 and 2.1 kcal/mole.

Use of Numerical Integration in the Computation of the Expectation Value of H2 with Applications to H2

The use of numerical integration for evaluating the expectation values of the Hamiltonian and the square of the Hamiltonian is difficult due to the presence of singularities in the integrand. A basis set is discussed which removes one of the principal singularities from Hψ in the case of the hydrogen molecule. This basis set is used and all integrals are evaluated numerically to obtain variational upper bounds and Weinstein and Temple lower bounds for the ground state of H2. First‐excited‐state upper bounds are obtained by a method involving the expectation value of the square of the Hamiltonian and the possibility of computing excited‐state wavefunctions by minimization of the width, W=〈H2〉—〈H〉2, is discussed.

Weinstein Calculation on Hydrogen Molecular Ion

A Weinstein calculation for the two‐parameter James function for H2+ has been performed. The lower bound to the energy was maximized and a value of —1.14724 a.u. was obtained. The improvement with additional variation parameters is greater than for the upper bound.

Bare‐Nucleus Perturbation Theory: Excited States of Hydrogen Molecule

Ground and excited Σ+ states of R = 1.4a0 are treated by perturbation theory. The full interelectronic repulsion is taken as the perturbation, and the energy of each of nine states is calculated through third order. The equation for the first‐order wavefunction is treated by the Hylleraas variational principle, using a linear variational function including up to 30 James–Coolidge basis functions. Comparisons with conventional linear variational calculations using the same basis functions, and with related calculations of other workers, are given. The perturbation results are in most cases superior to variation, especially when small basis sets are used, but it is noted that the inadequacy of the present basis set for excited states leads to slow convergence of the energies of both methods with addition of basis functions. The bare‐nucleus Hamiltonian also becomes a worse starting point for higher states because of electronic shielding, as evidenced by slower convergence of the perturbation energy series. ...

Bare‐Nucleus and Screened‐Nucleus Perturbation Theory for He2

In the bare‐nucleus perturbation theory, where the full interelectronic repulsion is taken as the perturbing part of the Hamiltonian, determination of the first‐order wavefunction for a closed‐shell system reduces to a set of two‐electron problems. Only in the evaluation of the energy do three‐electron integrals appear. One can thus produce a correlated wavefunction (containing interelectronic distances explicitly) without evaluation of three‐electron integrals as arise in the variational method. After a review of the necessary formulas, we present calculations for He2. The convergence of the energy series is disappointing. However, the same formalism can be used when any one‐electron local operator is taken as zero‐order Hamiltonian. In particular we present screened‐nucleus calculations, the zero‐order Hamiltonian corresponding to noninteracting electrons moving in the field of nuclei of charge 1.5. Here, our energy through second order is as good as the best variational calculations.

Explicitly Correlated Wavefunction for LiH by Perturbation Theory

Start with a local one‐electron Hamiltonian and construct a determinantal wavefunction from its eigenfunctions. Taking this as zero‐order function, calculate the first‐order correction by perturbation theory. This reduces to several two‐electron problems, approximate solutions to which may be found variationally, using trial functions involving interelectronic coordinates explicitly with no necessity for evaluation of multielectron integrals. An exception to this is discussed and dealt with. The energy through second order may be calculated as a sum of pair contributions. For LiH at 3.02a0, calculations with screened nucleus and another zero‐order potential are performed. It is shown that this energy can be as good as that from the best variational calculations. Problems of convergence are discussed.