Zeroth-order wave functions

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... 

Since H° is the sum of hydrogenlike Hamiltonians, the zeroth-order wave function is the product of hydrogenlike functions, one for each electron. We call any one-electron spatial wave function an orbital. To allow for electron spin, each spatial orbital is multiplied by a spin function (either a or 0) to give a spin-orbital. To introduce the required antisymmetry into the wave function, we take the zeroth-order wave function as a Slater determinant of spin-orbitals. For example, for the Li ground state, the normalized zeroth-order wave function is... 

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... 

The term symbols 1S and 3S are defined in the next section. For the states of the s2p configuration, we get zeroth-order wave functions similar to those of (1.256), except that there are 12 (=4x3) zeroth-order functions instead of 4 the factor 3 arises from the spatially degenerate 2px, 2py> and 2pz orbitals. 

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). 

In a first step, all the fundamental VB structures are involved, and a VBSCF wave function is calculated and taken as a zeroth-order wave function. 

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 ... 

The point is that the vectors k 4 satisfying the unperturbed Schrodinger equation, if used to expand 44 make the right hand side disappear and the equation becomes a uniform one. The only thing we can do is to use it to determine the proper expansion coefficients of the zeroth order wave function b p in terms of the degenerate subspace as well as the first order energy. (The first order wave function is usually not calculated/considered in the degenerate case.)... 

For convenience, we use the Dirac notation > to represent the zeroth-order wave functions. For simplicity, the electronic hamiltonian Jtf0 will be referred to later as H. 

Neglecting v in (10.4.15), the mean field equation is solved by the zeroth order wave functions... 

The energy shift AEmn in first order perturbation theory is given by the expectation value of v with the zeroth order wave functions (10.4.16). An exact calculation of the expectation value is diflBcult. But since we are only interested in the scaling of AEmn in fn and n for large m, n, we evaluate it semiclassically. Semiclassically, iprn is given by ... 

Next are methods based on zeroth-order wave functions that are linear combinations of Slater determinants that again may or may not include a treatment of residual electron correlation effects. In these multiconfigurational (MC) approaches, the starting wave function is presupposed to be given by a determinantal expansion... 

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