Hartree Fock coupled-cluster theory

Another category of approaches that avoids the symmetry breaking problem of the orbitals for the target state is based on using a related, well-behaved HF reference instead. Examples of such techniques include quasi-restricted Hartree-Fock coupled-cluster (QRHF CC) (11,19), symmetry adapted cluster configuration interaction (SAC-CI) (22,23), coupled-cluster linear response theory (CCLRT) (24-26) or equivalently equation-of-motion coupled-cluster (EOM-CC) (27-32), Fock space multi-reference coupled-cluster (FSMRCC) (33-37), and similarity transformed equation-of-motion coupled-cluster (STEOM-CC) (38-40). In the case of NO3 and N03, the restricted Hartree-Fock (RHF) orbitals of the closed-shell N03 anion system can be used as the reference. The anion orbitals are stable with respect to symmetry perturbations, and the system does not suffer from the artifactual symmetry breaking of the neutral and cation. 

There is also a hierarchy of electron correlation procedures. The Hartree-Fock (HF) approximation neglects correlation of electrons with antiparallel spins. Increasing levels of accuracy of electron correlation treatment are achieved by Mpller-Plesset perturbation theory truncated at the second (MP2), third (MP3), or fourth (MP4) order. Further inclusion of electron correlation is achieved by methods such as quadratic configuration interaction with single, double, and (perturbatively calculated) triple excitations [QCISD(T)], and by the analogous coupled cluster theory [CCSD(T)] [8],... 

However, until today no systematic comparison of methods based on MpUer-Plesset perturbation (MP) and Coupled Cluster theory, the SOPPA or multiconfigurational linear response theory has been presented. The present study is a first attempt to remedy this situation. Calculations of the rotational g factor of HF, H2O, NH3 and CH4 were carried out at the level of Hartree-Fock (SCF) and multiconfigurational Hartree-Fock (MCSCF) linear response theory, the SOPPA and SOPPA(CCSD) [40], MpUer-Plesset perturbation theory to second (MP2), third (MP3) and fourth order without the triples contributions (MP4SDQ) and finally coupled cluster singles and doubles theory. The same basis sets and geometries were employed in all calculations for a given molecule. The results obtained with the different methods are therefore for the first time direct comparable and consistent conclusions about the performance of the different methods can be made. 

M. Urban, P. Neogrady, and I. Hubac, Spin Adaptation in the Open-Shell Coupled-Cluster Theory with a Single Determinant Restricted Hartree-Fock Reference. In R. J. Bartlett (Ed.) Recent Advances in Coupled-Cluster Methods. Recent Advances in Computational Chemistry, Vol. 3. (World Scientific, Singapore, 1997), pp. 275-306. 

Tel. 904-392-1597, fax. 904-392-8722, e-mail aces2 qtp.ufl.edu Ab initio molecular orbital code specializing in the evaluation of the correlation energy using many-body perturbation theory and coupled-cluster theory. Analytic gradients of the energy available at MBPT(2), MBPT(3), MBPT(4), and CC levels for restricted and unrestricted Hartree-Fock reference functions. MBPT(2) and CC gradients. Also available for ROHE reference functions. UNIX workstations. 

The exponential ansatz given in Eq. [31] is one of the central equations of coupled cluster theory. The exponentiated cluster operator, T, when applied to the reference determinant, produces a new wavefunction containing cluster functions, each of which correlates the motion of electrons within specific orbitals. If T includes contributions from all possible orbital groupings for the N-electron system (that is, T, T2, . , T ), then the exact wavefunction within the given one-electron basis may be obtained from the reference function. The cluster operators, T , are frequently referred to as excitation operators, since the determinants they produce from fl>o resemble excited states in Hartree-Fock theory. Truncation of the cluster operator at specific substi-tution/excitation levels leads to a hierarchy of coupled cluster techniques (e.g., T = Ti + f 2 CCSD T T + T2 + —> CCSDT, etc., where S, D, and... 

P. Neogrady, M. Urban, and I. Hubac, /. Chem. Phys., 100, 3706 (1994). Spin Adapted Restricted Hartree-Fock Reference Coupled-Cluster Theory for Open-Shell Systems. 

The set of atomic orbitals Xk is called a basis set, and the quality of the basis set will usually dictate the accuracy of the calculations. For example, the interaction energy between an active site and an adsorbate molecule might be seriously overestimated because of excessive basis set superposition error (BSSE) if the number of atomic orbitals taken in Eq. [4] is too small. Note that Hartree-Fock theory does not describe correlated electron motion. Models that go beyond the FiF approximation and take electron correlation into account are termed post-Flartree-Fock models. Extensive reviews of post-HF models based on configurational interaction (Cl) theory, Moller-Plesset (MP) perturbation theory, and coupled-cluster theory can be found in other chapters of this series. ... 

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. 

Whenever the Hartree-Fock wave function provides a good zero-order description of the electronic system, it is natural to investigate the possibility of treating dynamical correlation by perturbation theory rather than by coupled-cluster theory. In this manner, we may hope to recover the most important effects of dynamical correlation at a cost lower than that of coupled-cluster theory. 

Along the ordinate, a sequence of correlation-consistent cc-pVXZ basis sets with X > 2 is depicted. Along the abscissa, the FCI limit is approached—beginning with Hartree-Fock theory and followed by the first correlated level, at which the single and double excitations are described by MP2 perturbation theory. The same excitations are subsequently treated by coupled-cluster theory at the CCSD level, which is then further improved upon by a perturbation treatment of the triple excitations at the CCSD(T) level. At the CCSDT level, the triple excitations are fully treated by coupled-cluster theory, and so on. In this manner, the hierarchy Hartree-Fock -> MP2 — CCSD -> CCSD(T) > CCSDT —---------------> FCI is established. 

By ab initio we refer to quantum chemical methods in which all the integrals of the theory, be it variational or perturbative, are exactly evaluated. The level of theory then refers to the type of theory employed. Common levels of theory would include Hartree-Fock, or molecular orbital theory, configuration interaction (Cl) theory, perturbation theory (PT), coupled-cluster theory (CC, or coupled-perturbed many-electron theory, CPMET), etc. - We will use the word model to designate approximations to the Hamiltonian. For example, the zero differential overlap models can be applied at any level of theory. The distinction between semiempirical and ab initio quantum chemistry is often not clean. Basis sets, for example, are empirical in nature, as are effective core potentials. The search for basis set parameters is not usually considered to render a model empirical, whereas the search for parameters in effective core potentials is so considered. 

One of the most dramatic changes in the standard theoretical model used most widely in quantum chemistry occurred in the early 1990s. Until then, ab initio quantum chemical applications [1] typically used a Hartree-Fock (HF) starting point, followed in many cases by second-order Moller-Plesset perturbation theory. For small molecules requiring more accuracy, additional calculations were performed with coupled-cluster theory, quadratic configuration interaction, or related methods. While these techniques are still used widely, a substantial majority of the papers being published today are based on applications of density functional theory (DFT) [2]. Almost universally, the researchers use a functional due to Becke, whose papers in 1992 and 1993 contributed to this remarkable transformation that changed the entire landscape of quantum chemistry. 

Table 16 Results for hyperpolarizabilities (in a.u.) for N2- TDHF and MBPT(2) are results from time-dependent Hartree-Fock and perturbation-theory calculations, respectively, whereas CCSD and CCSD(T) are coupled-cluster results. Exp. denotes experimental results, and LDA, GGA, and LB94 are results from time-dependent density-functional calculations with different density functionals. For a description of the quantities, see the text. The results are from ref. 95...

Ab-initio methods based on solution of the Hartree-Fock equations are well established (Ref 14), and are used in routine molecular and electronic calculations on small organic and inorganic molecules. For such molecules, extremely accurate predictions of many spectroscopic properties can be made using methods such as CAS-CI, MCSCF, coupled-cluster theory, and multireference methods. Recent advances in supercomputer technology coupled with improved algorithms have made it possible to perform full Cl calculations for small systems. Due to their size and complexity, such calculations have been limited mostly to diatomic molecules. However, where cost is not a problem, it is quite feasible to perform full Cl calculations on quite large systems, and such calculations have been carried out on small energetic molecules by Haskins and Cook at RARDE (Ref 16). 

---