Unperturbed wave function

The expansion coefficients determine the first-order correction to the perturbed wave function (eq. (4.35)), and they can be calculated for the known unperturbed wave functions and energies. The coefficient in front of 4>o for 4 i cannot be determined from the above formula, but the assumption of intermediate normalization (eq. (4.30)) makes Co = 0. 

The formulas for higher-order conections become increasingly complex. The main point, however, is that all corrections can be expressed in terms of matrix elements of the perturbation operator over the unperturbed wave functions, and the unperturbed energies. 

The van der Waals interaction energy of two hydrogen atoms at large intemuclear distances is discussed by the use of a linear variation function. By including in the variation function, in addition to the unperturbed wave function, 26 terms for the dipole-dipole interaction, 17 for the dipole-quadrupole interaction, and 26 for the quadrupole-quadrupole interaction, the... 

In the integral of Eq. (21), one can substitute the unperturbed wave function V 1/2 exp(ik r ) for if k(r ) since 4> is a small perturbation. With the resulting expression for (/, one calculates the electron density as a sum of densities for the occupied orbitals (with k < kF). Replacing the sum by an integral and subtracting the unperturbed electron density, one finds50 for the change of electron density the Lindhard result... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

It is evidently additive the additive operator V is calculated over unperturbed wave functions, which means that at this approximation the interacting charges are rigid. The exchange energy, 1 nonadditive for all exchanges mixing electrons of three or more atoms. 

EDE in the external Coulomb field in Fig. 1.6. The eigenfunctions of this equation may be found exactly in the form of the Dirac-Coulomb wave functions (see, e.g, [10]). For practical purposes it is often sufficient to approximate these exact wave functions by the product of the Schrodinger-Coulomb wave functions with the reduced mass and the free electron spinors which depend on the electron mass and not on the reduced mass. These functions are very convenient for calculation of the high order corrections, and while below we will often skip some steps in the derivation of one or another high order contribution from the EDE, we advise the reader to keep in mind that almost all calculations below are done with these unperturbed wave functions. 

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation... 

Notice that with the case 1 assumptions the energies after the interaction are raised and lowered by equal amounts, iAB, relative to the unperturbed energy, sA(=eB), and the resulting wave functions are an equal admixture of the unperturbed wave functions. 

The last integral is zero because of the orthogonality of the unperturbed wave functions. Equation (A.101) simplifies to... 

Let the first-order correction to the perturbed wave function be expanded as a linear combination of unperturbed wave functions, that is,... 

Thus, the first-order correction to the zero-order (unperturbed) wave function is obtained by substituting equation (A.104) into equation (A.103) and changing the summation index ... 

Using orthonormality of the unperturbed wave functions, and then multiplying by exp(/ 0V//j), we simplify (3.11) to... 

When the secular determinant in (1.207) is in diagonal form (all off-diagonal elements equal to zero), it follows that the initially chosen unperturbed wave functions ipj0) of the degenerate level are the correct... 

Consider now a high-Z two-electron atom. For simplicity, we will assume that the unperturbed wave function is a one-determinant function... 

Generalizations.—The expansion of Brueckner10 and Goldstone11 can only be applied when the unperturbed wave function can be described by a single Slater... 

We have established that, for a fully variational wave function, we may calculate the first-order properties from the zero-order response of the wave function (i.e., from the unperturbed wave function) and the second-order properties from the first-order response of the wave function. In general, the 2n -f 1 rule is obeyed For fully variational wave functions, the derivatives (i.e., responses) of the wave function to order n determine the derivatives of the energy to order 2n+ 1. This means, for instance, that we may calculate the energy to third order with a knowledge of the wave function to first order, but that the calculation of the energy to fourth order requires a knowledge of the wave-function response to second order. 

Constructing the perturbed Hamiltonian in the usual manner and carrying out some simple algebra, we find that the unperturbed wave function xim is correct to first order in the magnetic field B ... 

Consider now the interaction energies at medium and short distances. Here the application of a standard perturbation theory is not free of complications. The main reason for this is the fact that the unperturbed wave function, is antisymmetric only with respect... 

Segmented contraction of basis sets. 157 Stochastical dynamics, 389 Unperturbed wave function, 123 model, 209 ... 

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