Many-body perturbation theory coupled cluster methods

It is quite common in correlated methods (including many-body perturbation theory, coupled-cluster, etc., as well as configuration interaction) to invoke the frozen core approximation, whereby the lowest-lying molecular orbitals, occupied by the inner-shell electrons, are constrained to remain doubly-occupied in all configurations. The frozen core for atoms lithium to neon typically consists of the Is atomic orbital, while that for atoms sodium to argon consists of the atomic orbitals Is, 2s, 2px, 2py and 2pz. The frozen molecular orbitals are those made primarily from these inner-shell atomic orbitals. 

Costa et al. present results of many-body perturbation theory, coupled cluster and quadratic Cl methods apphed to the calculation of the polarizability and first hyperpolarizability of NaH. It is shown that the nuclear relaxation contribution is substantial for this molecule and that it is appreciably affected by electron correlation effects. 

There are three main methods for calculating electron correlation Configuration Interaction (Cl), Many Body Perturbation Theory (MBPT) and Coupled Cluster (CC). A word of caution before we describe these methods in more details. The Slater determinants are composed of spin-MOs, but since the Hamilton operator is independent of spin, the spin dependence can be factored out. Furthermore, to facilitate notation, it is often assumed that the HF determinant is of the RHF type. Finally, many of the expressions below involve double summations over identical sets of functions. To ensure only the unique terms are included, one of the summation indices must be restricted. Alternatively, both indices can be allowed to run over all values, and the overcounting corrected by a factor of 1/2. Various combinations of these assumptions result in final expressions which differ by factors of 1 /2, 1/4 etc. from those given here. In the present book the MOs are always spin-MOs, and conversion of a restricted summation to an unrestricted is always noted explicitly. 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

For Three Molecules in Valence Double-Zeta Basis Sets, a Comparison of Energies in Hartrees (H) from the 2-RDM Method with the T2 Condition (DQGT2) with the Energies from Second-Order Many-Body Perturbation Theory (MP2), Coupled-Cluster Method with Single-Double Excitations and a Perturbative Triples Correction (CCSD(T)), and Full Configuration Interaction (FCI)... 

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

Since the Dirac equation is written for one electron, the real problem of ah initio methods for a many-electron system is an accurate treatment of the instantaneous electron-electron interaction, called electron correlation. The latter is of the order of magnitude of relativistic effects and may contribute to a very large extent to the binding energy and other properties. The DCB Hamiltonian (Equation 3) accounts for the correlation effects in the first order via the Vy term. Some higher order of magnitude correlation effects are taken into account by the configuration interaction (Cl), the many-body perturbation theory (MBPT) and by the presently most accurate coupled cluster (CC) technique. 

The partitioned equation-of-motion second-order many-body perturbation theory [P-EOM-MBPT(2)] [67] is an approximation to equation-of-motion coupled-cluster singles and doubles (EOM-CCSD) [17], which will be fully described in Section 2.4. The EOM-CCSD method diagonalizes the coupled-cluster effective Hamiltonian H = [HeTl+T2) in the singles and doubles space, i.e.,... 

Tel. 904-392-1597, fax 904-392-8722, e-mail bartlett qtp.ufl.edu Ab initio molecular orbital calculations using coupled-cluster and many-body perturbation theory methods. 

In volume 1 of this series, I compared the use of second-order many-body perturbation theory in its MP2 form with that of density functional theory and coupled cluster theory. I recorded how the number of hits in a literature search on the string MP2 rises from 3 in 1989 to 854 in 1998. The corresponding results for DFT, the most widely used semi-empirical method, are 7 in 1989 growing to 733 by 1998. By 1998, the number of hits recorded for CCSD stood as 244. 

G. Fitzgerald, R. J. Harrison, and R. J. Bartlett,/. Chem. Phys., 85, 5143 (1986). Analytic Energy Gradients for General Coupled-Cluster Methods and Fourth-Order Many-Body Perturbation Theory. 

T. D. Crawford, Ph.D. Thesis, University of Georgia, 1996. Many-Body Perturbation Theory and Perturbational Triple Excitation Corrections to the Coupled-Cluster Singles and Doubles Method for High-Spin Open-Shell Systems. 

In Volume 5 of this series, R. J. Bartlett and J. E Stanton authored a popular tutorial on applications of post-Hartree-Fock methods. Here in Chapter 2, Dr. T. Daniel Crawford and Professor Henry F. Schaefer III explore coupled cluster theory in great depth. Despite the depth, the treatment is brilliantly clear. Beginning with fundamental concepts of cluster expansion of the wavefunction, the authors provide the formal theory and the derivation of the coupled cluster equations. This is followed by thorough explanations of diagrammatic representations, the connection to many-bodied perturbation theory, and computer implementation of the method. Directions for future developments are laid out. 

It is well known that electron correlation plays a key role in understanding the most interesting phenomena in molecules. It has been the focal point of atomic and molecular theory for many years [1] and various correlated methods have been developed [2]. Among them are many-body perturbation theory [3] (MBPT) and its infinite-order generalization, coupled cluster (CC) theory [4,5], which provides a systematic way to obtain the essential effects of correlation. Propagator [6-9] or Green s function methods (GFM) [10-14] provide another correlated tool to calculate the electron correlation corrections to ionization potentials (IPs), electron affinites (EAs), and electronic excitations. 

Foundations to the CC methods were laid by Coester and Kuemmel,1 Cizek,2 Hubbard,3 Sinanoglu,4 and Primas,5 while Cizek2 first presented the CC equations in explicit form. Also Hubbard3 called attention to the equivalence of CC methods and infinite-order many-body perturbation theory (MBPT) methods. From this latter viewpoint, the CC method is a device to sum to infinity certain classes of MBPT diagrams or all possible MBPT diagrams when the full set of coupled-cluster equations is solved. The latter possibility would require solving a series of coupled equations involving up to IV-fold excitations for N electrons. Practical applications require the truncation of the cluster operators to low N values. 

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