Ab initio quantum chemistry methods

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Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry.

The simplest type of ab initio electronic structure calculation is the Hartree-Fock (HF) scheme, in which the Coulombic electron-electron repulsion is not specifically taken into account. Only its average effect is included in the calculation. This is a variational procedure, therefore the obtained approximate energies, expressed in terms of the system's wave function, are always equal to or greater than the exact energy, and tend to a limiting value called the Hartree-Fock limit as the size of the basis is increased. Many types of calculations begin with a Hartree-Fock calculation and subsequently correct for electron-electron repulsion, referred to also as electronic correlation. Møller-Plesset perturbation theory (MPn) and coupled cluster theory (CC) are examples of these post-Hartree-Fock methods. In some cases, particularly for bond breaking processes, the Hartree-Fock mehtod is inadequate and this single-determinant reference function is not a good basis for post-Hartree-Fock methods. It is then necessary to start with a wave function that includes more than one determinant such as Multi-configurational self-consistent field and methods have been developed that use these multi-determinant references for improvements.

Almost always the basis set (which is usually built from the LCAO ansatz) used to solve the Schrödinger equation is not complete, and does not span the Hilbert space associated with ionization and scattering processes (see continuous spectrum for more details). In the Hartree-Fock method and the Configuration interaction method, this approximation allows one to treat the Schrödinger equation as a "simple" eigenvalue equation of the electronic molecular Hamiltonian, with a discrete set of solutions.

Contents 1 Classes of methods 1.1 Hartree-Fock methods 1.2 Post-Hartree-Fock methods 1.3 Multi-reference methods 2 Accuracy and scaling 2.1 Linear scaling approaches 3 Valence bond methods 4 Quantum Monte Carlo methods

[edit] Classes of methods

The most popular classes of ab initio electronic structure methods:

[edit] Hartree-Fock methods

• Hartree-Fock (HF)
• Restricted Open-shell Hartree-Fock (ROHF)
• Unrestricted Hartree-Fock (UHF)

[edit] Post-Hartree-Fock methods

• Møller-Plesset perturbation theory (MPn)
• Configuration interaction (CI)
• Coupled cluster (CC)
• Quadratic configuration interaction (QCI)

[edit] Multi-reference methods

• Multi-configurational self-consistent field (MCSCF)
• Multi-reference configuration interaction (MRCI)
• N-Electron Valence state Perturbation Theory (NEVPT)
• CASPT2

[edit] Accuracy and scaling

Ab initio electronic structure methods have the advantage that they can be made to converge to the exact solution, when all approximations are sufficiently small in magnitude. In particular configuration interaction where all possible configurations are included (called "Full CI") tends to the exact non-relativistic solution of the Schrödinger equation. The convergence, however, is usually not monotonic, and sometimes the smallest calculation gives the best result for some properties. The downside of ab initio methods is their computational cost. They often take enormous amounts of computer time, memory, and disk space. The HF method scales nominally as N⁴ (N being the number of basis functions) – i.e. a calculation twice as big takes 16 times as long to complete. However in practice it can scale closer to N³ as the program can indentify zero and extremely small integrals and neglect them. Correlated calculations scale even less favorably - MP2 as N⁵; MP4 as N⁶ and Couple cluster as N⁷. DFT methods scale in a similar manner to Hartree-Fock but with a larger proportionality term. Thus DFT calculations are always more expensive than an equivalent Hartree-Fock calculation.

[edit] Linear scaling approaches

The problem of computational expense can be alleviated through simplification schemes. In the density fitting scheme, the four-index integrals used to describe the interaction between electron pairs are reduced to simpler two- or three-index integrals, by treating the charge densities they contain in a simplified way. This reduces the scaling with respect to basis set size. Methods employing this scheme are denoted by the prefix "df-", for example the density fitting MP2 is df-MP2 (lower-case is advisable to prevent confusion with DFT). In the local orbital approximation, the molecular orbitals, which are formally spread across the entire molecule, are restricted to localised domains. This eliminates the interactions between distant electron pairs and hence sharply reduces the scaling with molecular size, a major problem in the treatment of biologically-sized molecules. Methods employing this scheme are denoted by the prefix "L", e.g. LMP2. Both schemes can be employed together, as in the recently developed df-LMP2 method.

[edit] Valence bond methods

Valence bond (VB) methods are generally ab initio although some semi-empirical versions have been proposed. Current VB approaches are:-

• Generalized valence bond (GVB)
• Modern valence bond theory (MVBT)

[edit] Quantum Monte Carlo methods

A method that avoids making the variational overestimation of HF in the first place is Quantum Monte Carlo (QMC), in its variational, diffusion, and Green's function forms. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo integration. Such calculations can be very time-consuming, but they are probably the most accurate methods known today.