Wavefunction first-order correction

It is often of interest to calculate the corresponding first-order correction to the wavefunctions. The necessary expression can be obtained by returning to... 

In other words, the diagonal elements of the perturbing Hamiltonian provide the first-order correction to the energies of the spin manifold, and the nondiagonal elements give the second-order corrections. Perturbation theory also provides expressions for the calculation of the coefficients of the second-order corrected wavefunctions l / in terms of the original wavefunctions (p)... 

The first-order correction can be thought of as arising from the response of the wavefunction (as contained in its LCAO-MO and Cl amplitudes and basis functions %v) plus the response of the Hamiltonian to the external field. Because the MCSCF energy functional has been made stationary with respect to variations in the Cj and Cj a amplitudes, the second and third terms above vanish ... 

The first of these is already solved, by assumption. If the second can be solved, we can find the first-order corrections to the wavefunction and the energy Solution of the third equation gives the second-order corrections, and so on. It is shown in the standard textbooks (e.g. Eyring, Walter and Kimball, 1944) that the solutions are... 

An important implication of Brillouin s theorem is the guarantee of no first-order errors for all one-electron operators provided the exact Hartree-Fock wavefunctions are used.426 This follows because the first-order correction to the correlation effect is derived from two-electron excitations. 

In Eq. (34) first-order corrections to the vibrational wavefunctions and energies are determined by the terms in Vs and V" that arc cubic in the normal coordinates. This equation is solved imposing the first-order normalization condition (i.e. = 0). 

The zeroth-order wavefunction yields the first-order perturbation to the energy when combined with the operator V, which describes the interactions between electrons and nuclei on the two different molecules. This first-order correction, known as the Heitler-London interaction energy,may be thought of as consisting of two terms. The first is the classical Coulombic interaction between the charge clouds of the (undistorted) subunits, commonly known as the electrostatic energy, and computed as... 

Fig. 5.22. First-order corrections to the wavefunction (a) two-body correlation p and q are excited to k and fc (b) p is excited to k by interacting with unexcited states n (c) is the exchange interaction from p to k via n (d) is the excitation from ptok through the perturbation. The dashed line is the Coulomb interaction while the cross indicates an interaction via the perturbation. For a Hartree-Fock potential, (b), (c) and (d) sum to zero (after H.P. Kelly [241]).

The first-order correction to the energy is determined completely by the unperturbed wavefunction and the perturbing Hamiltonian. 

In order to calculate the first-order correction to the wavefunction recall the first-order equation eqn ( 10.A.9) ... 

How does the first-order correction alterthe wavefunction Recall that the perturbation raises the potential energy near the top of the box (near L) much more than near the bottom (near x = 0) therefore, we expect the probability of finding the particle near the bottom to be enhanced compared with that of finding it near the top. Because the zero-order ground-state wavefunction is positive throughout the interior of the box, we thus expect the wavefunction itself to be raised near the bottom of the box and lowered near the top. In fact, the correcticHi terms do just this. First, note that the basis wavefunctions with odd n are symmetric with respect to the center of the box therefore, they would have the same effect near the top of the box as near the bottom. The coefficients of these terms are zero they do not contribute to the correction. The even- basis functions all start positive near x = 0 and end negative near x = L therefore, such terms must be multiplied by positive coefficients (as the result provides) to enhance the wavefunction near the bottom and diminish it near the top. 

To calculate the effects of an electric field it is necessary to add the term pot( )V n( ) to the equation that describes the exciton wavefunction, ipnif) (namely, eqn (D.17)). For sufficiently small fields, the effect of V ot( ) on the exciton wavefunctions and energies can be calculated by perturbation theory. Now, since V pot(r) is an odd function of r and ipnir) are either even or odd functions of r it immediately follows that the first order corrections to the energy are zero. Thus, the change in energy to n) to second order in perturbation theory... 

The first order correction to the wavefunction is also identical to the first order Moller-Plesset (MPl) perturbation... 

In a recent work [36], Bak et al presented a new method to obtain first-order corrected radial conplings from MCSCF wavefunctions and applied it to three E states of... 

Our final value for the first-order correction is the average value theorem (Eq. 2.10) applied to the perturbation Hamiltonian H and integrated over the zero-order wavefunction if/Q. 

In the case at hand—the many-electron atom—we can justify the qualitative behavior of the first- and second-order energy corrections as follows. The first-order correction must be positive, because it is the average of the electron-electron repulsion. However, because that repulsion is averaged over the densities of non-interacting electrons (using the zero-order wavefunctions), it overestimates... 

As you can see, the computations necessary for these corrections become increasingly complicated as we go to higher-ordei When perturbation theory works well, the relatively little effort required for the first- and second-order corrections accounts for most of the perturbation. We can see that perturbation theory converges fairly quickly for the two electrons in ground state helium because the second-order correction to the energy is much smaller than the first-order correction. Similarly, Fig. 4.10 shows that the first-order, two-electron wavefunction is about 67% Is and about 30% ls 2s with much smaller contributions from other states. 

A FIGURE 4.8 Radial distribution functions for the perturbation theory solution to helium. The average radial distribution functions for each of the two electrons in the zero-order and first-order wavefunctions are shown. The zero-order case is the same distribution as occupied by the single electron in Is He, because we have turned off the electron-electron repulsion. Once we turn on the repulsion (by adding the first-order correction to the energy), each electron pushes the other away. As a result, the electron density expands to a larger average distance from the nucleus. 

Write, but do not evaluate, the integral in terms of all the relevant coordinates for first-order correction to the energy of the ground state lithium atom, f tf/lH iffodr. Do not normalize or symmetrize the wavefunction, do not include the spin terms, but do include the limits of integration. 

To get the first-order correction to this energy, we apply the average value theorem to get the average perturbation energy using the zero-order wavefunction ... 

Thus we have arrived at an expression for the first-order correction to the energy in terms of known quantities. It is the expectation value for the perturbation operator calculated using the wavefunction of the unperturbed system. The analogy between this formula and the use of the population of the unperturbed city to calculate additional revenues from a new tax should be apparent. H corresponds to the tax per wage earner, and /irffi dz corresponds to the sum of wage earners in the city before the tax was imposed... 

To find an expression for P, the first-order correction to the wavefunction for the ith state, we first recognize that we can expand in terms of the complete set of eigenfunctions... 

This formula prescribes the way the first-order correction to the wavefunction is to be built up from eigenfunctions of the unperturbed system. We discuss this formula in detail later when considering an example. 

Since the constant first-order contribution to the energies of all the states is not physically interesting, let us examine the second-order contribution to the ground-state energy. Equation (12-23) shows that this is related to the first-order correction to the wavefunction,, which, as we have already seen, causes the wavefunction to become skewed toward the low-potential end of the box. It is clear from Eq. (12-22) that, in calculating the coefficients for we have already done most of the work needed to... 

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