Wave functions and matrix elements

It has already been said at the beginning of this chapter that the wave functions of complex configurations are built by vectorial coupling of orbital and spin momenta of individual shells. Then, 

Bearing in mind that second-quantization operators from different shells anticommute, we can represent the conjugate wave function as follows  

The submatrix elements of creation operators will thus be 

The submatrix elements that enter into the left sides of these equations can be expressed in terms of complex CFP [112]. Using these coefficients the many-shell wave function for N electrons (N = Ni + N2 +...+ Nu) is composed of the antisymmetric wave functions of (N — 1) electrons 

In Chapter 15 the one-shell CFP was related to the pertinent submatrix element of the creation operator (formula (15.21)) using an expansion of antisymmetric wave functions in terms of one-determinant functions. This 

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Elsum, I. R., and Gordon, R. G. (1982), Accurate Analytic Approximations for the Rotating Morse Oscillator Energies, Wave Functions, and Matrix Elements, J. Chem. Phys. 76, 5452. 

Formula (3) of the Introduction was the first example of how to use transformation matrices. The simplest case of transformation matrices is presented by formulas (10.4) and (10.5), while considering the relationships between the wave functions of coupled and uncoupled momenta. The Clebsch-Gordan coefficients served there as the corresponding transformation matrices. However, the theory of transformation matrices is most widely utilized for transformations of the wave functions and matrix elements from one coupling scheme to another (Chapters 11,12) as well as for their calculations. 

In previous chapters we considered the wave functions and matrix elements of some operators without specifying their explicit expressions. Now it is time to discuss this question in more detail. Having in mind that our goal is to consider as generally as possible the methods of theoretical studies of many-electron systems, covering, at least in principle, any atom or ion of the Periodical Table, we have to be able to describe the main features of the structure of electronic shells of atoms. In this chapter we restrict ourselves to a shell of equivalent electrons in non-relativistic and relativistic cases. 

In many physical problems we come across excited configurations consisting of several open shells or at least one electron above the open shell. Therefore, we have to be able to transform wave functions and matrix elements from one coupling scheme to the other for such complex configurations. If K denotes the configuration, and m, fifi stand for the quantum numbers of two different coupling schemes, then for the corresponding wave functions formulas of the kind (12.1), (12.2) hold, whereas the matrix element of some scalar operator D transforms as ... 

The latter statement means that the matrix elements of Eqs. (3) and (A.8) coincide. However, Eq. (A.8) has a more symmetrical form on the right and on the left there is a single molecule and a single lepton. This is convenient for writing down the wave functions and matrix elements of decay. 

Here (P)0 7 ) is the second-quantized operator producing the relevant normalized wave function out of vacuum, i.e. it simply generalizes the relationship (15.4) to the case of tensors with an additional (isospin) rank. Since the vacuum state is a scalar in isospin space (unlike quasispin space), the expressions for wave functions and matrix elements of standard quantities in the spaces of orbital and spin momenta can readily be generalized by the addition of a third (isospin) rank to two ranks in appropriate formulas of Chapter 15. 

Using the same semiempirical VB scheme as in (I), but with inclusion of overlap in all wave functions and matrix elements. 

The eigenvalues of this Hamitonian can calculated by numerical diagonalization of the truncated matrix of the quantum system in the basis of the harmonic oscillator wave functions. The matrix elements of Hq and V are... 

With such an elaborate form for the helium wave function, the matrix elements of the operator L between any two short-range terms in the trial function of the product form (I>,u.(/)t and j may be more conveniently expressed, after integrating by parts, as... 

The tensorial structure of the spin-orbit operators can be exploited to reduce the number of matrix elements that have to be evaluated explicitly. According to the Wigner-Eckart theorem, it is sufficient to determine a single (nonzero) matrix element for each pair of multiplet wave functions the matrix element for any other pair of multiplet components can then be obtained by multiplying the reduced matrix element with a constant. These vector coupling coefficients, products of 3j symbols and a phase factor, depend solely on the symmetry of the problem, not on the particular molecule. Furthermore, selection rules can be derived from the tensorial structure for example, within an LS coupling scheme, electronic states may interact via spin-orbit coupling only if their spin quantum numbers S and S are equal or differ by 1, i.e., S = S or S = S 1. 

To calculate expectation values, we have to get rid of the determinantal wave function in matrix elements over a many-electron operator O such as the many-electron Hamiltonian and reduce them to computable one-electron and two-electron matrix elements. This can be done easily using creators and annihilators since we can expand a Slater determinant according to Eq. (8.127)... 

With the approximation of Abarenkov and Heine, the orthogonalisation hole enters in the expression for in a fairly simple way. A detailed calculation of the orthogonalisation hole from the core wave functions involves matrix elements < k + q P k > [Ref. 1]. But as mentioned before, the difference between this full treatment and the one proposed by Abarenkov and Heine is quite small in practice. And even in principle, the difference... 

Provided we can find some way of evaluating the effect of the operators A and Q on the wave function, the matrix elements of this Hamiltonian would not be too difficult to calculate. This question is addressed later on in this chapter. But is this a useful place to stop the transformation process, and how accurate are the results we get from this Hamiltonian ... 

The last equation shows that the second-order energy correction may be written in terms of the first order wave function (c,) and matrix elements over unperturbed states. The second-order wave function correction is... 

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. 

Without loss of generality y = y can be assumed. If the dipole moment can be assumed to be a linear function of coordinate within the spread of the frozen Gaussian wave packet, the matrix element (gy,q,p, Pjt(r) Y,q, p ) can be evaluated analytically. Since the integrand in Eq. (201) has distinct maxima usually, we can introduce the linearization approximation around these maxima. Namely, the Taylor expansion with respect to bqp = Qq — Qo and 8po = Po — Po is made, where qj, and pj, represent the maximum positions. The classical action >5qj, p , ( is expanded up to the second order, the final phase-space point (q, p,) to the first order, and the Herman-Kluk preexponential factor Cy pj to the zeroth order. This approximation is the same as the ceUularization procedure used in Ref. [18]. Under the above assumptions, various integrations in U/i(y, q, p ) can be carried out analytically and we have... 

A set of nonlinear parameters Aj, in general case, is unique for each function To satisfy the requirement of square integrability of the wave function, each matrix must be positively defined. It imposes certain restrictions on the values that the elements of matrix A may take. To ensure the positive definiteness and to simplify some calclations, it is very convenient to represent matrix A in a Cholesky factored form. 

Abstract. The elements of the second-order reduced density matrix are pointed out to be written exactly as scalar products of specially defined vectors. Our considerations work in an arbitrarily large, but finite orthonormal basis, and the underlying wave function is a full-CI type wave function. Using basic rules of vector operations, inequalities are formulated without the use of wave function, including only elements of density matrix. 

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