Single-particle operators

In his pioneering work Baetzold used the Hartree-Fock (HF) method for quantum mechanical calculations for the cluster structure (the details are summarized in Reference 33). The value of the HF procedure is that it yields the best possible single-determinant wave function, which in turn should give correct values for expectation values of single-particle operators such as electric moments and... 

To understand this more clearly, consider a simpler model where A consists of single excitations, only single-particle operators are retained in the effective Hamiltonian, and we choose the reference function iho to be a single determinant. Then, from a cumulant decomposition of the two-particle terms, the effective Hamiltonian becomes... 

The derivation of the second line of equation (A.81) follows the same reasoning as was used to obtain the one-electron part of the electronic energy [equation (A.21)], since both fi and h are sums of single-particle operators. The dipole moment integrals over basis functions in the last line of equation (A.81) are easily evaluated. Within the HF approximation, dipole moments may be calculated to about 10% accuracy provided a large basis set is used. 

The unit tensor operators are irreducible-tensor operators with reduced matrix elements of unity. They are a valid choice to use as a basis in order to express any arbitrary tensor operator as a linear combination, since they are linearly independent. Attention is restricted to these for the sake of simplicity. Hence the definition of unit tensor single-particle operator U (aK, a L, r) is... 

Ceulemans showed that the time reversal character of a single-particle operator was linked to its quasispin rank. A time-even operator Ti contained a quasispin triplet plus a term of the form 5 which is totally scalar (in... 

However, because of the correlated motion of the electrons, many-electron processes will also occur. (Looking at the many-particle effects in this way, the photon operator is a single-particle operator and electron-electron interactions have to be incorporated explicitly into the wavefunction. It is, however, also possible to describe the combined action of the electrons as an induced field which adds to the external field of the photoprocess, i.e., the transition operator becomes modified. Generally, the influence of the electron-electron interaction can be represented by modifying the wavefunction or the operator or by modifying both the wavefunction and the operator [DLe55, CWe87].) Of all the possible processes, only the important two-electron processes restricted to electron emission will be considered here. In many cases they can be divided into two different classes (see Fig. 1.3) t... 

The many-body perturbation theory is developed in terms of some set of single particle states, Pi which are eigenfunctions of some single-particle operator, /,... 

The final term in eqn (6.79) arises from the projection of the electron-electron repulsive potential energy from F by the single-particle operator... 

Three known facts are essentially important in the development of a divide-and-conquer strategy. First, the KS Hamiltonian is a single particle operator that depends only on the total density, not on individual orbitals. This enables one to project the energy density in real space in the same manner in which one projects the density (see below). Second, any complete basis set can solve the KS equation exactly no matter where the centers of the basis functions are. Thus, one has the freedom to select the centers. It is well known that for a finite basis set the basis functions can be tailored to better represent wavefunctions, and thus the density, of a particular region. The inclusion of basis functions at the midpoint of a chemical bond is the best known example. Finally, the atomic centered basis functions used in almost all quantum chemistry computations decay exponentially. Hence both the density and the energy density contributed by atomic centered basis functions also decrease rapidly. All these... 

The usefulness of the single-particle density matrix becomes apparent when we consider how one would calculate the expectation value of a multiplicative single-particle operator A = Y, a i) (such as the potential V =... 

This equation follows immediately from comparing (8) with (46). We can define an alternative density operator, n, by requiring that the same equation must also be obtained by substituting n(r) into Eq. (51), which holds for any single-particle operator. This requirement implies that n(r) = 4(r — ry).24... 

For nonmultiplicative single-particle operators (such as the kinetic energy, which contains a derivative) one requires the full single-particle matrix 7(2, x ) and not only 7(2,2). 

As was the case when solving the Hartree-Fock equations, also the eigenfunctions to the Kohn-Sham equations are expanded in some basis set as in Eq. (13). And since the single-particle operator heff depends on the density, which in turn depends on the orbitals, in this case also the single-particle equations have to be solved self-consistently. 

Thus, we have, say, an 7V-electron operator in first quantization, which is a sum of single-particle operators... 

The T =0 time-dependent mean-field theory currently provides the best description of nuclear dynamics at low energies [5,6]. We consider two single-particle operators, Q, P interpreted as a collective coordinate and a collective momentum. Their nature depends on the kind of motion that we want to focus upon. We require that Q Q and P-r — P under time reversal and that IP, Q] 0. We then form a constrained Hartree-Fock (CHF) calculation on the many-body Hamiltonian H by minimizing the functional... 

In the context of atomic spectroscopy, with special emphasis on the lanthanide atoms and ions, most configurations will consist of more than one electron and the corresponding operators of interest will contain a sum of single-particle operators, e.g.. 

The multi-dimensional integrations can be circumvented if the Hamiltonian is written as a sum of products of single-particle operators,... 

There are only single-particle operators in (7.11). Therefore, the solution to the Schrodinger equation for a model system of noninteracting electrons can be written exactly as a single Slater determinant S = p, p2,.. , PnV where the single-particle orbitals pi are determined as solutions of the single-particle equation... 

For the calculation of the conductivity we will rely on the general result for the expectation value of a many-body operator 0 ri ), which can be expressed as a sum of single-particle operators o(r ),... 

As far as the relevant single-particle operator o(r) is concerned, it must describe the response of the physical system to the external potential, which is of course the induced current. The single-particle current operator is... 

Using this expression as the single-particle operator o(r), and combining it with the expression for the single-particle density matrix above, we obtain for... 

This expression involves exclusively matrix elements of the single-particle operators o(r) and the single-particle density matrix y(r, rO in the single-particle states (p (r), which is very convenient for actual calculations of physical properties (see, for example, the discussion of the dielectric function in chapter 6). 

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