Single determinant reference

If we consider all possible excited configurations that can be generated from the HF determinant, we have a full CI, but such a calculation is typically too demanding to accomplish. However, just as we reduced the scope of CAS calculations by using RAS spaces, what if we were to reduce the CI problem by allowing only a limited number of excitations How many should we include To proceed in evaluating this question, it is helpful to rewrite Eq. (7.1) using a more descriptive notation, i.e.. 

If we assume that we do not have any problem with non-dynamical correlation, we may assume that there is little need to reoptimize the MOs even if we do not plan to carry out the expansion in Eq. (7.10) to its full CI limit. In that case, the problem is reduced to determining the expansion coefficients for each excited CSF that is included. The energies E of N different CI wave functions (i.e., corresponding to different variationally determined sets of coefficients) can be determined from the N roots of the CI secular equation 

H is the Hamiltonian operator and the numbering of the CSFs is arbitrary, but for convenience we will take I l = I hf and then all singly excited determinants, all doubly excited, etc. Solving the secular equation is equivalent to diagonalizing H, and permits determination of the CI coefficients associated with each energy. While this is presented without derivation, the formalism is entirely analogous to that used to develop Eq. (4.21). 

To solve Eq. (7.11), we need to know how to evaluate matrix elements of the type defined by Eq. (7.12). To simplify matters, we may note that the Hamiltonian operator is composed only of one- and two-electron operators. Thus, if two CSFs differ in their occupied orbitals by 3 or more orbitals, every possible integral over electronic coordinates hiding in the r.h.s. of Eq. (7.12) will include a simple overlap between at least one pair of different, and hence orthogonal, HF orbitals, and the matrix element will necessarily be zero. For the remaining cases of CSFs differing by two, one, and zero orbitals, the so-called Condon-Slater rules, which can be found in most quantum chemistry textbooks, detail how to evaluate Eq. (7.12) in terms of integrals over the one- and two-electron operators in the Hamiltonian and the HF MOs. 

A somewhat special case is the matrix element between the HF determinant and a singly excited CSF. The Condon-Slater rules applied to this situation dictate that 

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Applications of electron propagator methods with a single-determinant reference state seldom have been attempted for biradicals such as ozone, for operator space partitionings and perturbative corrections therein assume the dominance of a lone configuration in the reference state. Assignments of the three lowest cationic states were inferred from asymmetry parameters measured with Ne I, He I and He II radiation sources [43]. 

Further pragmatic moves are described in details in numerous books and reviews of which we cite the most concise and recent Ref. [82], Two further hypotheses are an important complement to the above cited theorems. One is the locality hypothesis, another is the Kohn-Sham representation of the single determinant reference state in terms of orbitals. The locality has been seriously questioned by Nesbet in recent papers [83,84], however, it remains the only practically implemented solution for the DFT. The single determinant form of the reference state in its turn guarantees that all the averages of the electron-electron interaction appearing in this context are in fact calculated with the two-electron density given by the determinant term in Eq. (5) with no cumulant. 

We emphasize that the present discussion focuses only on high-spin open-shell systems to which a single-determinant reference wavefunction is applicable. Coupled cluster techniques for low-spin cases, such as open-shell singlets, have been pursued in the literature for many years, however, and provide a fertile area of research (Refs. 158, 167-170). 

D. Watts,. Gauss, and R.. Bartlett,/. Chem. Phys., 98, 8718 (1993). Coupled-Cluster Methods with Noniterative Triple Excitations for Restricted Open-Shell Hartree-Fock and Other General Single Determinant Reference Functions. Energies and Analytical Gradients. 

J. F. Stanton, R. J. Bartlett, and C. M. L. Rittby,/. Chem. Phys., 97,5560 (1992). Fock Space Multireference Coupled-Cluster Theory for General Single Determinant Reference Functions. 

Watts, J.D., Gauss, J., Bartlett, R.J. Coupled-cluster methods with noniterative triple excitations for restricted open-shell Hartree-Fock and other general single determinant reference functions—energies and analytical gradients. J. Chem. Phys. 1993, 98(11), 8718-33. 

Cluster expansion representation of a wave-function built from a single determinant reference function [1] has been eminently successful in treating electron correlation effects with high accuracy for closed shell atoms and molecules. The cluster expansion approach provides size-extensive energies and is thus the method of choice for large systems. The two principal modes of cluster expansion developments in Quantum Chemistry have been the use of single reference many-body perturbation theory (SR-MBPT) [2] and the non-perturbative single reference Coupled Cluster (SRCC) theory [3,4]. While the former is computationally economical for the first few orders of the perturbation expansion... 

To improve on a single determinant reference we must develop a superior treatment of the two-electron interaaion. Logically, this would involve explicit use of the two-particle operator in a trial wavefunaion, and such Hylleraas-type trial wavefunctions have been used to obtain the best results (for, e.g.. He, Li, Be, H2, and LiH). Analysis of certain other analytic properties of the correlation cusp l/( r,- — r, ) have also been exploited to develop better descriptions without having to use wavefunctions that are explicitly dependent on r j, but such methods also have many computational restriaions. 

We can take the dynamic correlation route and always start with a single determinant reference, even when it is a poor approximation, and introduce higher categories of excitations until we believe our wavefunaion is flexible enough to obtain satisfactory results. 

The failing of MBPT is that it is basically an order-by-order perturbation approach. For difficult correlation problems it is frequently necessary to go to high orders. This will be the case particularly when the single determinant reference function offers a poor approximation for the state of interest, as illustrated by the foregoing examples at 2.0 R. A practical solution to this problem is coupled-cluster (CC) theory. In fact, CC theory simplifies the whole concept of extensive methods and the linked-diagram theorem into one very simple statement the exponential wavefunction ansatz. 

The Brueckner orbital variant of CC should also be mentioned. CCSD puts in all single excitation effects via the wavefunction exp(Ti + T2) o- We can instead change the orbitals ip, in Oo in this wavefunction until Tj = 0. These orbitals are called Brueckner orbitals and define a single determinant reference B instead of o that has maximum overlap with the correlated wave-function. Since B-CCD " (or BD) effectively puts in Tj, it will give results similar but not identical to those from CCSD (they differ in fifth order). For BH, the corresponding B-CCD errors are 1.81, 2.88, and 5.55, compared to 1.79, 2.64 and 5.05, for CCSD as a function of R. . See also B-CCD for symmetry breaking problems. ... 

The considerations discussed above clearly show that multireference Cl approaches to the study of excited states require great care and are not well suited to black-box applications. Ideally, one would like a method that retains the conceptual simplicity of the ACI approach while providing a better description of excited states. Since the problems associated with the ACI method are due to the use of a single determinant reference state for all electronic states, one might naturally wonder whether a better (more flexible) reference state can be used. In other words, we assume the same form of the excited state wavefunction... 

Multireference Coupled-Cluster Theory for C)eneral Single Determinant Reference Functions. 

This completes our discussion of the third order diagrams and associated algebraic expressions for a single determinant reference function. 

If the renormalization of the wave function is also taken into account, the (1 - al) quantity is divided by ai, and the corresponding correction is called the renormalized Davidson correction. The effect of higher order excitations is thus estimated from the correlation energy obtained at the CISD level times a factor that measures how important the single-determinant reference is at the CISD level. The Davidson correction does not yield zero for two-electron systems, where CISD is equivalent to full Cl, and it is likely that it overestimates the higher order corrections for systems with few electrons. More complicated correction schemes have also been proposed, but are rarely used. 

It is well-known, for example, that in a perturbation theory analysis of the method of configuration interaction when restricted to single- and double-excitations with respect to a single determinant reference function includes many terms, which correspond to unlinked diagrams, which are exactly canceled by terms involving higher order excitations. 

The treatment of some classes of multi-reference problems can be found by exploiting single-determinant reference states composed of a different number of particles from that of the MR target state, with subsequent addition or removal... 

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