Fock space theory expansion theories

FULL CLUSTER EXPANSION THEORIES IN FOCK SPACE 7.1 Preliminaries for a Fock Space Approach... 

FULL CLUSTER EXPANSION THEORIES IN FOCK SPACE... 

In this section, we shall motivate towards the need for a Fock-space approach to generate core-valence extensive cluster expansion theories (i.e., of type (a3)), introduced in Sec.2. Since we have to maintain size-extensivity of the energy... 

As in the ordinary EOMCC theory, in the EOMXCC method we solve the electronic Schrodinger equation (1) assuming that the excited states are represented by Eq. (7). We use the exponential representation of the ground-state wave function I S o), Eq. (8), but no longer assume that the cluster components Tn result from standard SRCC calculations (see below). The many-body expansions of the excitation operator Rk have the same form as in the ordinary EOMCC formalism. In particular, the three different forms of Rk discussed in the previous section [fi -E, R A, and REqs. (28), (30), and (26), respectively] are used to define the EE-EOMXCC, EA-EOMXCC, and IP-EOMXCC methods. As in the standard EOMCC method, by making suitable choices for the operators Qa, which define Rk, we can always extend the EOMXCC theory to other sectors of the Fock space. 

We can also formulate this in a different manner and say that the self-consistent field procedure plays a crucial role in 4-component theory because it serves to define the spinors that isolate the n-electron subspaces from the rest of the Fock space. In this manner it determines in effect the precise form of the electron-electron interaction used in the calculations. Both aspects are a consequence of the renormalization procedure that was followed when fixing the energy scale and interpretation of the vacuum. The experience with different realizations of the no-pair procedure has learned that the differences in calculated chemical properties (that depend on energy differences and not on absolute energies) are usually small and that other sources of errors (truncation errors in the basis set expansion, approximations in the evaluation of the integrals) prevail in actual calculations. 

The basic expression for the quantization of the electromagnetic field is the expansion Eq(54). In the quantized theory the numbers Ck,, C x become operators of the creation C x and the annihilation Ck,x of photons. These operators are acting on the state vector < ) that is defined in the Fock space (occupation number space). The C xt Ck, operators satisfy the commutation relations ... 

Of the many quantum chemical approaches available, density-functional theory (DFT) has over the past decade become a key method, with applications ranging from interstellar space, to the atmosphere, the biosphere and the solid state. The strength of the method is that whereas conventional ah initio theory includes electron correlation by use of a perturbation series expansion, or increasing orders of excited state configurations added to zero-order Hartree-Fock solutions, DFT methods inherently contain a large fraction of the electron correlation already from the start, via the so-called exchange-correlation junctional. 

In this figure we have used both a harmonic-oscillator (HO) and a Brueckner-Hartree-Fock (BHF) basis for the single-particle wave functions in order to study the behavior of the RS perturbation theory at low orders. What can be seen from this figure is that the BHF basis yields a smaller overlap between states in the excluded space and the model space, reflected in the small change when going from second order to third order in the perturbation expansion. However, the BHF spectra are too compressed and in poor agreement with experiment. This is probably related to the fact that the radii obtained for the self-consistent single-particle wave functions are much smaller than the empirical ones [53]. 

MOs and the configuration expansion. To be successful, we must choose the parametrization of the MCSCF wave function with care and apply an algorithm for the optimization that is robust as well as efficient. The first attempts at developing MCSCF optimization schemes, which borrowed heavily from the standard first-order methods of single-configuration Hartree-Fock theory, were not successful. With the introduction of second-order methods and the exponential parametrization of the orbital space, the calculation of MCSCF wave functions became routine. Still, even with the application of second-order methods, the optimization of MCSCF wave functions can be difficult - more difficult than for the other wave functions treated in this book. A large part of the present chapter is therefore devoted to the discussion of MCSCF optimization techniques. 

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