Operator one-particle

The problem of finding the best approximation of this type and the best one-electron set y2t. . ., y>N is handled in the Hartree-Fock scheme. Of course, a total wave function of the same type as Eq. 11.38 can never be an exact solution to the Schrodinger equation, and the error depends on the fact that the two-electron operator (Eq. 11.39) cannot be exactly replaced by a sum of one-particle operators. Physically we have neglected the effect of the "Coulomb hole" around each electron, but the results in Section II.C(2) show that the main error is connected with the neglect of the Coulomb correlation between electrons with opposite spins. 

Here we have deduced the equivalence of the one-particle operator forms (8-147) and (8-149) with the iV-particle forms (8-143) and (8-148). It is worthwhile to invert the argument as follows ... 

The set of A -particle Trace Class operators that contract to the one-particle operator X. 

The operator Hd) may be written as the sum of one-particle operators that act on and are symmetric with respect to all particles, —... 

Cumulants of any order can be defined via a generating function [17, 18]. Consider the expectation value of the exponential of an arbitrary one-particle operator k ... 

Note that a dot ( ) always means a matrix element of the antisymmetrized electron interaction g, a cross (x) a matrix element of the one-particle operator /, while an open square ( ) collects the free labels in any of these contractions. If the reference function is a single Slater determinant, all cumulants X vanish one is then left with particle and hole contractions, like in traditional MBPT in the particle-hole picture. 

Thus we see that Hartree-Fock theory is identical to a canonical transformation theory retaining only one-particle operators with an optimized reference, and the canonical transformation model retaining one- and two-particle operators employed in the current work, if employed with an optimized reference, is a natural extension of Hartree-Fock theory to a two-particle theory of correlation. 

One-particle operators Hi and H3 cause relativistic corrections to the total energy. Two-particle operators H2, H3 and H s define more precisely the energy of each term, whereas H4 and H" describe their splitting (fine structure), i.e. they cause a qualitatively new effect. These operators are also often called describing magnetic interactions. 

Operators corresponding to physical quantities, in second-quantization representation, are written in a very simple form. In the quantum mechanics of identical particles we normally have to deal with two types of operators symmetric in the coordinates of all particles. The first type includes N-particle operators that are the sum of one-particle operators. An example of such an operator is the Hamiltonian of a system of noninteracting electrons (e.g. the first two terms in (1.15)). The second type are iV-particle operators that are the sum of two-particle operators (e.g. the energy operator for the electrostatic interaction of electrons - the last term in (1.15)). In conventional representations these operators are... 

Let us now look at one-particle operators in the second-quantization representation, defined by (13.22). Substituting into (13.22) the one-electron matrix element and applying the Wigner-Eckart theorem (5.15) in orbital and spin spaces, we obtain by summation over the projections... 

Similar data for the dN configuration may be found in [91]. Applying such tables we can directly find the matrix elements of one-particle operators corresponding to physical quantities. The matrix elements of two-particle... 

In Chapter 14 we have shown how an expansion in terms of irreducible tensors in the spaces of orbital and spin angular momenta for one shell can be obtained for the operators corresponding to physical quantities. The tensors introduced above enable the terms of a similar expansion to be also defined in the space of a two-shell configuration. So, for the one-particle operator of the most general tensorial structure (14.51) we find, instead of (14.52),... 

Just as in the case of one-particle operator (17.14), expressions (17.21) and (17.22) embrace already in operator form, the interaction terms, both the diagonal ones relative to the configurations, and the non-diagonal ones. Coming under the heading of diagonal terms are, first, the one-shell operators of electrostatic interaction of electrons, discussed in detail in Chapter 14. Second, the operators of direct... 

In the previous section we discussed the Hermitean operator -a (t) It is just a one particle operator with a complete, discrete spectrum, and the following relations hold, for <.0 i... 

The accurate parameterization of the effective core potential has shown that the reduction of the pseudopotential to the form of a one-particle operator is adequate. The scaling of the two-body potentials by the use of an operator65... 

Finally we may consider a one-particle operator F = Yj.fi- ts interaction with the shell is represented by a matrix of one-electron elements The... 

This result reflects the fact that the photon operator, as a one-particle operator, interacts with the active ls-electron only, ejecting it into a wave characterized asymptotically by Ka or, alternatively, Kb, while the passive ls-electron leads to... 

The Is and 2s orbitals which are affected by neither the photoionization nor the Auger process are omitted for simplicity.) If these wavefunctions are constructed from single-electron orbitals of a common basis set (the frozen atomic structure approximation), the photon operator as a one-particle operator allows a change of only one orbital. Hence, the photon operator induces the change 2p to r in these matrix elements ... 

To understand these seemingly opposite facts, we have to leave the global S expression and rather write the spin-orbit Hamiltonian as a sum of one-particle operators... 

We note that z is here a complex composite variable corresponding to the real composite variable x = (r, ). It is evident that the complex scaling operator U defined by Eq. (2.20) is of the product form Eq. (2.23), provided that the one-particle operator u is defined through the relation... 

For many-electron states (energy states), the spin-orbit operator Hso is given as a sum of one-particle operators, i.e., the sum of hSOi operators for the single electron i ... 

These rules are a consequence of the fact that the spin-orbit operator for the many-electron states is a sum of one-particle operators according to (5) and the Slater-Condon rules for matrix elements between states of such operators [121]. 

As in the conventional Hartree-Fock method, the approximate eigenvalue I is hence essentially different from the sum of the N eigenvalues of the one-particle operator. 

The question is now what happens to the effective one-particle operator Tueff associated with the transformed many-particle operator Tu = UTU 1. Using (2.40) and (3.4), one obtains... 

It is then easily shown that also the effective one-particle operator Tefj is self-adjoint ... 

The many-particle operator U defined by the product (3.2) defines a restricted similarity transformation provided that the one-particle operators u satisfy the condition ... 

---