Ab initio quantum chemistry methods: Difference between revisions
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'''Ab initio quantum chemistry methods''' are [[computational chemistry]] methods based on [[quantum chemistry]]. |
'''Ab initio quantum chemistry methods''' are [[computational chemistry]] methods based on [[quantum chemistry]]. |
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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 method|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. [[post Hartree-Fock|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]] (MP''n'') and [[coupled cluster]] theory (CC) are examples of these [[post-Hartree-Fock]] methods. |
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 method|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. [[post Hartree-Fock|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]] (MP''n'') and [[coupled cluster]] theory (CC) are examples of these [[post-Hartree-Fock]] methods. |
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Almost always the [[basis set (chemistry)|basis set]] (which is usually built from the [[linear combination of atomic orbitals molecular orbital method|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 spectrum|discrete]] set of solutions. |
Almost always the [[basis set (chemistry)|basis set]] (which is usually built from the [[linear combination of atomic orbitals molecular orbital method|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 spectrum|discrete]] set of solutions. |
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The most popular classes of ab initio electronic structure methods: |
The most popular classes of ab initio electronic structure methods: |
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* [[Quadratic configuration interaction]] (QCI) |
* [[Quadratic configuration interaction]] (QCI) |
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''[[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. The convergence, however, is usually not [[monotonic function|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 as ''N<sup>4</sup>'' (''N'' being the number of basis functions) – i.e. a calculation twice as big takes 16 times as long to complete &ndash |
''[[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 function|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 ''N<sup>4</sup>'' (''N'' being the number of basis functions) – i.e. a calculation twice as big takes 16 times as long to complete &ndash. However in practice it can scale closer to ''N<sup>3</sup>'' as the program can indentify zero and extremely small integrals and neglect them. Correlated calculations scale even less favorably - MP2 as ''N<sup>5</sup>''; MP4 as ''N<sup>6</sup>'' and Couple cluster as ''N<sup>7</sup>''. DFT methods scale similar to Hartree-Fock although they always take longer than an equivalent Hartree-Fock calculation. |
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==Linear scaling approaches== |
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The problem of computational expense can be alleviated through simplification schemes. In the ''density fitting'' scheme, the four-index [[integral]]s 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 [[Møller-Plesset perturbation theory|MP2]] is df-MP2 (lower-case is advisable to prevent confusion with [[density functional theory|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 [[biomolecule|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. |
The problem of computational expense can be alleviated through simplification schemes. In the ''density fitting'' scheme, the four-index [[integral]]s 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 [[Møller-Plesset perturbation theory|MP2]] is df-MP2 (lower-case is advisable to prevent confusion with [[density functional theory|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 [[biomolecule|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. |
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Revision as of 12:18, 9 April 2006
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.
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.
The most popular classes of ab initio electronic structure methods:
Hartree-Fock methods
- Hartree-Fock (HF)
- Restricted Open-shell Hartree-Fock (ROHF)
- Unrestricted Hartree-Fock (UHF)
Post-Hartree-Fock methods
- Møller-Plesset perturbation theory (MPn)
- Multi-configurational self-consistent field (MCSCF)
- Configuration interaction (CI)
- Multi-reference configuration interaction (MRCI)
- Coupled cluster (CC)
- Quadratic configuration interaction (QCI)
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 &ndash. 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 similar to Hartree-Fock although they always take longer than an equivalent Hartree-Fock calculation.
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.
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)
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.