Fock operator eigenstates

As mentioned previously, the specifics of the central atoms in CCs are determined by the structure of the supermatrix II, which is in its turn predefined by the structure of the carrier space of the CLS group and by the number of electrons in it. Indeed, the supermatrix II of the polarization propagator is particularly simple in the basis of the eigenstates of the Fock operator Fq. Its matrix elements then are ... 

The nontrivial one-electron eigenstates of the effective Fock operator for this CLS have the form ... 

We concentrate on the expression for the covalent part of the crystal field Wffff and transform it to the form coinciding with the AOM and relate the parameters of the latter with the electronic structure of the ligands. To do so, we perform a unitary transformation of the canonical MOs of the /-system Z) - the eigenstates of the Fock operator - to the localized MO L) separately for the occupied and vacant canonical MO ... 

The Fock operator f and the one-particle density matrix 7 commute, i.e. have common eigenfunctions. This allows an iterative construction of 7 from the eigenstates of f. The leading relativistic corrections for the Dirac-Coulomb operator are ... 

As for the Fock operator, the eigenstates of the zero-order Epstein-Nesbet Hamiltonian are determinants. Although occasionally used, the Epstein-Nesbet approach has been considerably less successful than that of MPPT. 

In M0ller-Plesset theory, / can be brought to a diagonal form with the single-reference zero-order state as an eigenstate. We shall now examine the CASSCF Fock operator and see how it may be modified to constitute a zero-order Hamiltonian with the CASSCF state as an eig state. 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

The definition of the pair part or, equivalently the no-pair part of Hmat is not unique. The precise meaning of no-pair implicitly depends on the choice of external potential, so that the operator Hm t depends implicitly on the external potential, whereas the sum Hmat Hm t + Hm t is independent of the choice of external potential. Since the no-pair part conserves the number of particles (electrons, positrons and photons) we can look for eigenstates of Hm j in the sector of Fock space with N fermions and no photons or positrons. Following Sucher [18,26,28], the resulting no-pair Hamiltonian in configuration space can be written as... 

The Fock-space Hamiltonian H is equivalent to the configuration-space Hamiltonian H insofar as both have the same matrix elements between n-electron Slater determinants. The main difference is that H has eigenstates of arbitrary particle number n it is, in a way, the direct sum of aU // . Another difference, of course, is that H is defined independently of a basis and hence does not depend on the dimension of the latter. One can also define a basis-independent Fock-space Hamiltonian H, in terms of field operators [11], but this is not very convenient for our purposes. 

Using the block-diagonal nature of the density matrix, besides reducing the requirement in CPU time, also allows one to label the DMEV by the appropriate particle number(ne) and the z— component of the total spin of the block [Mg,a)- The Fock space of the individual sites that are added at each iteration are eigenstates of the site spin and number operators. This allows targeting a definite particle number Nf.) and a definite projected spin (M ) state of the total system. 

Typical structures are specified in Table 1 which uses the labelling of carbon atoms in Cjo defined in Fig. 1. The restricted open-shell Hartree-Fock (ROHF) method was used in all geometry optimizations using a minimal basis set of orbitals (STO-3G) [13]. These calculations are therefore exploratory in nature. Here we have chosen to use the standard ab initio ROHF method since it is well-known that the UHF method (as used in the PRDDO approximation [9]) does not give wave functions which are eigenstates of the total spin operator S. The effect of spin contamination on molecular properties is uncertain, particularly if the contamination is high (the... 

The operator sets q = (qi qj 3 ) used in the last section contain products of 1,2,3,... creation/annihilation operators, which act on an N-electron reference state 0), say, to produce other elements of Fock space, which describe the system (or its ions) with N, N 1,... electrons. These elements are not eigenstates of the Hamiltonian, but of course may be combined to give approximations to actual state vectors, for both the neutral system and its ions. In the EOM approach generalized operators are introduced, which work on the ground state 0) to produce any desired excited state n), neutral or ionic, and an attempt is made to determine these operators directly. 

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