Energy first-order correction

The first-order energy correction with respect to the unperturbed problem is then... 

The zeroth-order energy level is twofold degenerate. The corresponding vibronic basis functions are ur K+2 0 0 —) = 11) and luj- A"—2 0 0 +) = 2). The first-order energy correction is... 

The zero-order wave function is the HF determinant, and the zero-order energy is just a sum of MO energies. The first-order energy correction is the average of the perturbation operator over the zero-order wave function (eq. (4.36)). 

For example, the first-order energy correction in Eq. (1.5 a) is given by... 

The first-order energy correction is given in terms of the zeroth-order (i.e., unperturbed) wavefunction as ... 

The perturbation-theory first-order energy correction is (see also Problem 8.35)... 

This secular equation is an algebraic equation of degree n with n roots ,(1),. .., (,) that give the n first-order energy corrections. Substitution of each (,) in turn into (1.205) allows one to solve for the set of coefficients ctj (/=I,..., ) that go with the root, ,(1). Having found the n correct zeroth-order wave functions <)py(0) (j = 1,..., ), we can then proceed to find Ej<2 ipjl and so on the formulas for these corrections turn out to be essentially the same as for the nondegenerate case, provided that (pj0) is used in place of 0). 

Show that the sum of the n first-order energy corrections for an rt-fold degenerate energy level equals the sum of the n matrix elements H u in (1.207). 

The first-order term is identical to eg. (4.36), while the second-order equation corresponds to eq. (4.38) with an additional term involving the expectation value of P2 over the unperturbed wave function. The first-order energy correction is identified with the first-order property, the second-order correction with the second-order property, etc. 

The ground state density will depend on the field strength parameter x, which is denoted by a subscript. The functional has both an explicit and an implicit dependence on x and the first-order energy correction, the total derivative is formed by a partial derivative and a functional derivative... 

Since the correction to the energy is W k, it is convenient to include the parameter X in the symbols for the first-order perturbation and the first-order energy correction, so that to the first order it is usual to write the relations... 

Problem 23-1. Calculate the first-order energy correction for a onedimensional harmonic oscillator upon which the perturbation H (x) acts, where H (x) is zero unless z < t and H (x) = b for x < e, with e a quantity which is allowed to approach zero at the same time that b approaches infinity, in such a way that the product 2e5 = c. Compare the effect on the odd and even levels of the oscillator. What would be the effect of a perturbation which had a very large value at some point outside the classically allowed range of the oscillator and a zero value elsewhere ... 

Using this result we see at once that the first-order energy correction is zero, for... 

A compact formulation of stationary perturbation theory in the non-relativistic theory has been given [72] in a Lie-algebraic language. One of the essential messages of stationary perturbation theory is that under a certain condition the essential theorems of exact perturbation theory, (e.g. that the first order energy correction is equal to the expectation value of the perturbation with the unperturbed wave function) remain valid. The condition is that all perturbation corrections are formulated in terms of a variational group , with respect to which the unperturbed energy expectation value is stationary. 

As seen in equation (39), the block consists of three contributions. Since —f is of the order aP, whereas at the quasi-relativistic level X, B, and A are proportional to an approximate first-order energy correction can be obtained by neglecting the B and X terms in a = c is the fine structure constant. However, this approximation has not been tested numerically. 

This is an important quantity. Our first development of it followed the above, derivative argument [ 103,104,109]. This simply means we use first-order perturbation theory with the CC reference as the unperturbed problem to get the first-order energy correction—the gradient. It was also suggested from a different field as a generalization of CC theory [110, 111], but in chemistry we consider energies and their derivatives (properties) to be synonymous as above [111]. Finally, it can also be deduced as a consequence of a Lagrangian multiplier constraint [112]. 

It is assumed that the state vectors are represented by orthonormal kets of spatial variables, viz. K) = a, L, ML). In terms of perturbation theory the first-order energy correction is... 

The First-Order Energy Correction. To find we multiply (9.18) by and integrate over all space, which gives... 

Having found the d first-order energy corrections, we go back to the set of equations (9.84) to find the unknowns C , which determine the correct zeroth-order wave functions. To find the correct zeroth-order function... 

Solving this equation, we find the first-order energy corrections ... 

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