Ab initio quantum chemistry methods 4683300 215564615 2008-05-28T19:32:34Z DOI bot 6652755 Citation maintenance. You can [[WP:DOI|use this bot]] yourself! Please [[User:DOI_bot/bugs|report any bugs]]. '''''Ab initio'' quantum chemistry methods''' are [[computational chemistry]] methods based on [[quantum chemistry]].<ref name=levine> {{cite book | last = Levine | first = Ira N. | title = Quantum Chemistry | publisher = Prentice Hall | date = 1991 | location = Englewood Cliffs, New jersey | pages = 455 - 544 | isbn = 0-205-12770-3}} </ref> The term ''ab initio'' indicates that the calculation is from first principles and that no empirical data is used. [[Robert Parr]] claims in an interview that the term was first used in a letter to him by [[David P. Craig|David Craig]] and was put into the manuscript of their paper on the excited states of benzene published in 1950.<ref> [http://www.quantum-chemistry-history.com/Parr1.htm History of Quantum Chemistry: Robert G. Parr] </ref> <ref> {{cite journal | last = Parr | first = Robert G. | author-link = Robert Parr | coauthors = Craig D. P,. and Ross, I. G | title = Molecular Orbital Calculations of the Lower Excited Electronic Levels of Benzene, Configuration Interaction included | journal = Journal of Chemical Physics | volume = 18 | pages = 1561–1563 | date = 1950 | doi = 10.1063/1.1747540}} </ref> The simplest type of ''[[ab initio]]'' electronic structure calculation is the [[Hartree-Fock]] (HF) scheme, in which the instantaneous Coulombic electron-electron repulsion is not specifically taken into account. Only its average effect (mean field) 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.<ref> {{cite book | last = Cramer | first = Christopher J. | title = Essentials of Computational Chemistry | publisher = John Wiley & Sons, Ltd. | date = 2002 | location = Chichester | pages = 153 - 189 | isbn = 0-471-48552-7}} </ref> [[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.<ref name=cramer2> {{cite book | last = Cramer | first = Christopher J. | title = Essentials of Computational Chemistry | publisher = John Wiley & Sons, Ltd. | date = 2002 | location = Chichester | pages = 191 - 232 | isbn = 0-471-48552-7}} </ref> <ref> {{cite book | last = Jensen | first = Frank | title = Introduction to Computational Chemistry | publisher = John Wiley and Sons | date = 2007 | pages = 98 - 149 | location = Chichester, England | isbn = 0470011874}} </ref> In some cases, particularly for bond breaking processes, the Hartree-Fock method 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.<ref name=cramer2/> 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. ==Classes of methods== 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]] (MP''n'') * [[Configuration interaction]] (CI) * [[Coupled cluster]] (CC) * [[Quadratic configuration interaction]] (QCI) * [[Quantum chemistry composite methods]] ===Multi-reference methods=== * [[Multi-configurational self-consistent field]] (MCSCF) * [[multireference configuration interaction|Multi-reference configuration interaction]] (MRCI) * [[N-Electron Valence state Perturbation Theory]] (NEVPT) * [[Complete Active Space Perturbation Theory]] (CASPT''n'') ==Example: Is Si<sub>2</sub>H<sub>2</sub> like acetylene (C<sub>2</sub>H<sub>2</sub>)?== {{Inappropriate tone|date=February 2008}} A series of ''ab initio'' studies of Si<sub>2</sub>H<sub>2</sub> is an example of how ''ab initio'' computational chemistry can predict new structures that are subsequently confirmed by experiment. They go back over 20 years, and most of the main conclusions were reached by 1995. The methods used were mostly [[post-Hartree-Fock]], particularly [[configuration interaction]] (CI) and [[coupled cluster]] (CC). Initially the question was whether [[disilyne]], Si<sub>2</sub>H<sub>2</sub> had the same structure as [[ethyne]] (acetylene), C<sub>2</sub>H<sub>2</sub>. In early studies, by Binkley and Lischka and Kohler, it became clear that linear Si<sub>2</sub>H<sub>2</sub> was a transition structure between two equivalent trans-bent structures and that the ground state was predicted to be a four-membered ring bent in a 'butterfly' structure with hydrogen atoms bridged between the two silicon atoms.<ref> {{cite journal | first = J. S. | last = Binkley | coauthors = | year = 1983 | title = Theoretical studies of the relative stabiity of C<sub>2</sub>H<sub>2</sub> of Si<sub>2</sub>H<sub>2</sub> | journal = Journal of the American Chemical Society | volume = 106 | pages = 603 }}</ref> <ref> {{cite journal | first = H. | last = Lischka | coauthors = H-J Kohler | year = 1983 | title = ''Ab initio'' ivestigation on the lowest singlet and triplet state of Si<sub>2</sub>H<sub>2</sub> | journal = Journal of the American Chemical Society | volume = 105 | pages = 6646 | doi = 10.1021/ja00360a016 }}</ref> Interest then moved to look at whether structures equivalent to vinylidene - Si=SiH<sub>2</sub> - existed. This structure is predicted to be a local minimum, i. e. an isomer of Si<sub>2</sub>H<sub>2</sub>, lying higher in energy than the ground state but below the energy of the trans-bent isomer. Then a new isomer with an unusual structure was predicted by Brenda Colegrove in [[Henry F. Schaefer, III]]'s group.<ref> {{cite journal | first = B. T. | last = Colegrove | coauthors = [[Henry F. Schaefer, III|Schaefer, Henry F. III]] | year = 1990 | title = Disilyne (Si<sub>2</sub>H<sub>2</sub>) revisited | journal = Journal of Physical Chemistry | volume = 94 | pages = 5593 | doi = 10.1021/j100377a036 }}</ref> It requires [[post Hartree-Fock]] methods to obtain a local minimum for this structure. It does not exist on the [[Hartree-Fock]] energy hypersurface. The new isomer is a planar structure with one bridging hydrogen atom and one terminal hydrogen atom, cis to the bridging atom. Its energy is above the ground state but below that of the other isomers.<ref> {{cite journal | first = R. S. | last = Grev | coauthors = Schaefer, Henry F. III | year = 1992 | title = The remarkable monobridged structure of Si<sub>2</sub>H<sub>2</sub> | journal = Journal of Chemical Physics | volume = 97 | pages = 7990 | doi = 10.1063/1.463422 }}</ref> Similar results were later obtained for Ge<sub>2</sub>H<sub>2</sub>.<ref> {{cite journal | first = Zoltán | last = Palágyi | coauthors = Schaefer, Henry F. III, Kapuy, Ede | year = 1993 | title = Ge<sub>2</sub>H<sub>2</sub>: A Molecule with a low-lying monobridged equilibrium geometry | journal = Journal of the American Chemical Society | volume = 115 | pages = 6901–6903 | doi = 10.1021/ja00068a056 }}</ref> Al<sub>2</sub>H<sub>2</sub> and Ga<sub>2</sub>H<sub>2</sub> have exactly the same isomers, in spite of having two electrons less than the Group 14 molecules.<ref> {{cite journal | first = J. C. | last = Stephens | coauthors = Bolton, E. E.,Schaefer, H. F. III, and Andrews, L. | year = 1997 | title = Quantum mechanical frequencies and matrix assignments to Al<sub>2</sub>H<sub>2</sub> | journal = Journal of Chemical Physics | volume = 107 | pages = 119–223 | doi = 10.1063/1.474608 }}</ref> <ref> {{cite journal | first = Zoltán | last = Palágyi | coauthors = Schaefer, Henry F. III, Kapuy, Ede | year = 1993 | title = Ga<sub>2</sub>H<sub>2</sub>: planar dibridged, vinylidene-like, monobridged and trans equilibrium geometries | journal = Chemical Physics Letters | volume = 203 | pages = 195–200 | doi = 10.1016/0009-2614(93)85386-3 }}</ref> The only difference is that the four-membered ring ground state is planar and not bent. The cis-mono-bridged and vinylidene-like isomers are present. Experimental work on these molecules is not easy, but matrix isolation spectroscopy of the products of the reaction of hydrogen atoms and silicon and aluminium surfaces has found the ground state ring structures and the cis-mono-bridged structures for Si<sub>2</sub>H<sub>2</sub> and Al<sub>2</sub>H<sub>2</sub>. Theoretical predictions of the vibrational frequencies were crucial in understanding the experimental observations of the spectra of a mixture of compounds. This may appear to be an obscure area of chemistry, but the differences between carbon and silicon chemistry is always a lively question, as are the differences between group 13 and group 14 (mainly the B and C differences). The silicon and germanium compounds were the subject of a Journal of Chemical Education article.<ref> {{cite journal | first = B. J. | last = DeLeeuw | coauthors = Grev, R. S. and Schaefer, Henry F. III | year = 1992 | title = A comparison and contrast of selected saturated and unsaturated hydrides of group 14 elements | journal = Journal of Chemical Education | volume = 69 | pages = 441 }}</ref> ==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 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) &ndash; 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 identify 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 coupled cluster as ''N<sup>7</sup>''. 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. ===Linear scaling approaches=== The problem of computational expense can be alleviated through simplification schemes.<ref> {{cite book | last = Jensen | first = Frank | title = Introduction to Computational Chemistry | publisher = John Wiley and Sons | date = 2007 | pages = 80 - 81 | location = Chichester, England | isbn = 0470011874}} </ref> 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 approximation'', the molecular orbitals are first localized by a unitary rotation in the orbital space (which leaves the reference wave function invariant, i.e., is not an approximation) and subsequently interactions of distant pairs of localized orbtials are neglected in the correlation calculation. This 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 and df-LCCSD(T0) methods. In fact, df-LMP2 calculations are faster than df-Hartree-Fock calculations and thus are feasible in nearly all situations in which also DFT is. ==Valence bond methods== Valence bond (VB) methods are generally ''ab initio'' although some semi-empirical versions have been proposed. Current VB approaches are<ref name=levine/>:- * [[Generalized Valence Bond|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 method|Monte Carlo]] integration. Such calculations can be very time-consuming, but they are probably the most accurate methods known today. == See also == * [[Quantum chemistry computer programs]] - see columns for Hartree-Fock and Post-Hartree-Fock methods == References == {{Reflist}} [[Category:Computational chemistry]] [[fr:Méthode ab initio de chimie quantique]] [[it:Metodo ab initio (chimica)]] [[pl:Metody ab initio]]