Wave functions and operators

The main effect of taking into account spin-orbit coupling in excited state calculations will be an admixture of triplet character to the singlet states S and of singlet character to the triplet states T. We will confine the following discussion to this situation and use the configuration interaction (Cl) approach to describe singlet and triplet wave functions 

The spin-adapted configurational state functions (CSF s) I y(l.iV) may in general be expressed in terms of Slater determinants 

Numerical ab initio calculations for selected examples with polarized basis sets and Cl of reasonably size confirmed that the size of the matrix elements within the active space matrix is negligible. In contrast, the elements of that involve both the active and inner shells are large, since is primeuily due to the shielding of nuclei by inner-shell electrons [11]. It is therefore common practice in many semiquantitative applications, to account for the effect of the fixed-core electrons by replacing the factor gPgZ r in by the empirical value of the atomic spin-orbit coupling constant valence p orbitals on 

(11) the operator chain in the last term of the svun corresponds to the spin-density operator [13]. 

There are a number of different methods for the construction of spin eigen- 

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It is very convenient, in quantum mechanics, to consider the wave functions and operators as matrices (matrix form of quantum mechanics). Then the wave function xpj(x) of a set of states of the system under consideration is represented by a one-column matrix... 

For the basis (18.27) to be used effectively in practical computations an adequate mathematical tool is required that would permit full account to be taken of the tensorial properties of wave functions and operators in their spaces. In particular, matrix elements can now be defined using the Wigner-Eckart theorem (5.15) in all three spaces, so that the submatrix element will be given by... 

To sum up the potentialities of the isospin method are not exhausted by the results stated above. There is a deep connection between orthogonal transformations of radial orbitals and rotations in isospin space (see (18.40) and (18.41)). This shows that the tensorial properties of wave functions and operators in isospin space must be dominant in the Hartree-Fock method. This issue is in need of further consideration. 

Eq. (1-3) is the principal assertion of the quantum mechanics needed in this book. Assertions (a) and (b) simply define wave functions and operators, but assertion (c) makes a connection with experiment. It follows from Eq. (1-3), for example, that the probability of finding an electron in a small region of space, (fir, is i// (r)i//(r)d r TItus is the probability density for the electron. 

We are already familiar with the second quantized wave functions and operators. In what follows we shall study how to work with them. First, it appears to be worthwhile to check an important property, the Hermiticity of quantum mechanical operators. 

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