Energy perturbed

in the HF ground-state F0, the energy expectation value of the perturbed system is 

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This suggests that only in cases where the perturbation energy Vba is very small on average (on the order oikT 1 kcal/mol) will Eq. (8) be directly applicable. More often, the transformation from A to B must be divided into several discrete steps, say n, to each of which corresponds a perturbation energy on the order of which can be made... 

For variationally optimized wave functions (HF or MCSCF) there is a 2n -I- 1 rule, analogous to the perturbational energy expression in Section 4.8 (eq. (4.34)) knowledge of the Hth derivative (also called the response) of the wave function is sufficient for... 

If the unperturbed system is degenerate, so that several linearly independent eigenfunctions correspond to the same energy value, then a more complicated procedure must be followed. There can always be found a set of eigenfunctions (the zeroth order eigenfunctions) such that for each the perturbation energy is given by equation 9 and the perturbation theory provides the... 

The perturbation energy is here just the electrostatic energy calculated for the electron distribution given by perturbation theory which led Schrodinger to... 

This resonance energy leads to molecule formation only if the eigenfunction is symmetric in the two nuclei. The perturbation energy for the antisymmetric eigenfunction is... 

An attempt was made by Unsold (33) to evaluate to the second-order the interaction of a proton and a hydrogen atom. He found, neglecting the resonance phenomenon, that the second-order perturbation energy is given approximately by the expression... 

In this case, too, molecule formation results from the symmetric eigenfunction. The corresponding perturbation energy W1 is obtained from an equation of the type of Equation 20 involving I hj and the wave equation 28. It is... 

We shall next consider whether or not the antisymmetric eigenfunction Hl for two hydrogen atoms (Equation 29b) would lead to an excited state of the hydrogen molecule. The perturbation energy is found to be... 

Substitution of this eigenfunction in an expression of the type of Equation 21 permits the evaluation of the perturbation energy W1, in the course of which use is made of the properties of orthogonality arid normalization of the spin eigenfunctions namely,... 

If the perturbation function shows cubic symmetry, and in certain other special cases, the first-order perturbation energy is not effective in destroying the orbital magnetic moment, for the eigenfunction px = = i py leads to the same first-order perturbation terms as pi or pv or any other combinations of them. In such cases the higher order perturbation energies are to be compared with the multiplet separation in the above criterion. 

Perturbed Energy Equation for Small Peclet Number... 

The perturbed energy equation for moderate Peclet number has (at = 0) the following form ... 

The non-bonded interaction is now introduced as a perturbation V(r) on this double oscillator, and the perturbation energy becomes... 

The eonclusion to be drawn from equation (6) is that the perturbation energy is equal to the value of the perturbing potential at the equilibrium separation plus terms which are proportional to the even derivatives of V(r) at the equilibrium separation, and also proportional to increasing powers of the mean square of the total deviation from this separation. It is via this mean square that the isotopic mass will affect the perturbation energy. 

When two similar systems, one containing protium and the other deuterium attached to a carbon atom, are subject to the same perturbation V r) by the approach of a non-bonded atom, we obtain by means of expression (6) the following difference in perturbation energy ... 

The analysis of the regioselective reactivity of olefins in identical topochemical environments by three computational methods concludes that both steric factors (cavity and potential energy) and electronic factors (perturbation energy from orbital interactions) play important cooperative roles in determining which C—C double bond in a molecule reacts first in [2-1-2] photodimerization. The steric factor is considered to be effective in the movement of olefins at an early stage of the reaction, whereas the electronic factors are effective in the adduction of olefins at a later stage of the reaction. 

The perturbed energies and wavefunctions for the i-th system state can be expressed in a similar way as in scalar perturbation theory ... 

Hydrogen atom, in its ground state, can be treated in an entirely analytic approach. The ealeulation of the second-order perturbed energy gives the well known values ... 

The present paper is aimed at developing an efficient CHF procedure [6-11] for the entire set of electric polarizabilities and hyperpolarizabilities defined in eqs. (l)-(6) up to the 5-th rank. Owing to the 2n+ theorem of perturbation theoiy [36], only 2-nd order perturbed wavefunctions and density matrices need to be calculated. Explicit expressions for the perturbed energy up to the 4-th order are given in Sec. IV. 

It should be noted that, in this notation, the order of the superscripts is irrelevant ab... is the entire perturbed energy term linear in ab. .. z and there is no... 

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