Zeroth-order wave function, correct

After this step one may already identify the paramagnetic and diamagnetic terms as arising from the pp and pn first order orbital corrections to the zeroth order wave function. The contribution is then further approximated by neglecting all off-diagonal terms in the pn-pn part of the Hessian... 

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). 

An example is the perturbation treatment of the He atom in which we neglect the interelectronic repulsion e2/rl2. The correct zeroth-order wave function for the ground state is... 

Since the value of M does not affect E ° the unperturbed levels are (2y+l)-fold degenerate. Hence, before forging ahead, we must be sure that we have the correct zeroth-order wave functions for the perturbation (4.36). (See Section 1.10.) It was noted in Section 1.10 that when the secular determinant is diagonal, we have the right zeroth-order functions. We now show that the functions (4.38) give a diagonal secular determinant. An off-diagonal element has the form... 

The energy correction (4.52) takes account of the effect of the terms (bq2 + dq4) in the perturbation (4.36). However, since the first-order correction due to the terms (aq + cq3) vanished, we must go to the second-order energy correction to take these terms into account. The contribution of (aq + cq1) to (2) will turn out to be of the same order of magnitude as the contribution of (bq2 + dq4) to (,). We have the correct zeroth-order wave functions (4.38), and we can use (1.202) for the second-order energy correction for state n ... 

In the first-order approximation to NMR spectra when H° is taken as the first two terms in (8.78), the nuclear-spin energy levels are, in general, degenerate. Show for the A2X2 case (where A and X are protons) that the functions (8.79) are the correct zeroth-order wave functions for the perturbation H where H is the last term in (8.78). 

This is a system of inhomogeneous linear equations for the functions (vectors) T m ) (the mixed notation for the perturbation corrections to eigenvalues and eigenvectors is used above). The 0-th order in A yields the unperturbed problem and thus is satisfied automatically. The others can be solved one by one. For this end we multiply the equation for the first order function by the zeroth-order wave function and integrate which yields ... 

This equation 24—19 illustrates, in addition, that the integrals H mn depend on the set of zeroth-order functions secular equation. In particular, in case that the perturbation is a function of one variable (x, say) alone, and each function of the initial set of unperturbed wave functions can be expressed as the product of a function of x and a function of the other variables, the individual functions being mutually orthogonal, then these product functions are correct zeroth-order wave functions for this perturbation. This situation arises whenever the unperturbed wave equation can be separated in a set of variables in which x is included. 

The correct zeroth-order wave functions are obtained by multiplying the correct positional wave functions obtained in the preceding section by the four spin functions. For the configuration ls2s alone they are... 

The correct zeroth-order wave functions for the two levels with n = 1 are found to be... 

From the Pauli principle follows that the projected function J4ab o. rather than should be considered as the correct zeroth-order wave function in the perturbation theory of intermolecular interactions. Here J4ab is the usual intermolecular antisymmetrization operator and is (the lowest) eigenfunction of, the sum of... 

An alternative use of perturbation theory together with the idea of electron pairs is the method of Malrieu, Diner and Clavery abbreviated as PCILO ) (perturbative configuration interaction based on focalized orbitals). One uses a zeroth-order wave function, one in which each electron pair is in a fully localized LCAO-MO. All delocalization and correlation corrections are then treated by perturbation theory (not limited to second order). 

Our first task is thus to determine the correct zeroth-order wave functions (9.75) for the perturbation H. Calling these correct zeroth-order functions f we have... 

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... 

For each root E n = 1,2,..., d, we have a different set of coefficients Ci, C2,..., c, giving a different correct zeroth-order wave function. 

Using procedures similar to those for the nondegenerate case, one can now find the first-order corrections to the correct zeroth-order wave functions and the second-order energy corrections. For the results, see Bates, Volume I, pages 197-198 Hameka, pages 230-231. 

Now we want to find the correct zeroth-order wave functions. We shall assume that the roots (9.92) are all different. For the root <9 = H n, the system of equations (9.84) is... 

Since we are assuming unequal roots, the quantities H j2 H[i,..., H 44 - are all nonzero.Therefore,C2 = 0, C3 = 0,..., q = O.ITie normalization condition (9.88) gjves Cl = l.The correct zeroth-order wave function corresponding to the first-order perturbation energy correction /fj, is then [Eq. (9.76)] For the root H 22, the same... 

The secular determinant in (9.94) has the same form as the secular determinant in the linear-variation secular equation (8.66) with Sij = By the same reasoning used to show that two of the variation functions are linear combinations of/i and and two are linear combinations of f-j and f [Eq. (8.70)], it follows that two of the correct zeroth-order wave functions are linear combinations of and and two are linear combinations of and... 

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