Ab initio quantum chemistry methods

Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry.^[1] The term ab initio was first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.^[2][3] The background is described by Parr.^[4] In its modern meaning ('from first principles of quantum mechanics') the term was used by Chen^[5] (when quoting an unpublished 1955 MIT report by Allen and Nesbet), by Roothaan^[6] and, in the title of an article, by Allen and Karo,^[7] who also clearly define it.

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.

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 and when the finite set of basis functions tends toward the limit of a complete set. In this case, configuration interaction, where all possible configurations are included (called "Full CI"), tends to the exact non-relativistic solution of the electronic Schrödinger equation (in the Born-Oppenheimer approximation). 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 identify zero and extremely small integrals and neglect them. Correlated calculations scale even less favorably: MP2 as N⁵, MP4 as N⁶ and coupled cluster as N⁷. DFT methods using functionals which include Hartree–Fock exchange scale in a similar manner to Hartree–Fock but with a larger proportionality term and are thus more expensive than an equivalent Hartree–Fock calculation. DFT methods that do not include Hartree–Fock exchange can scale better than Hartree–Fock.

Linear scaling approaches

The problem of computational expense can be alleviated through simplification schemes.^[8] 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^[9] (many authors use lower-case to prevent confusion with DFT). In the local approximation,^[10][11] 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 orbitals are neglected in the correlation calculation. This 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.^[9][11] Both schemes can be employed together, as in the df-LMP2^[9] 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.

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 (MPn)
• Configuration interaction (CI)
• Coupled cluster (CC)
• Quadratic configuration interaction (QCI)
• Quantum chemistry composite methods

Multi-reference methods

• Multi-configurational self-consistent field (MCSCF)
• Multi-reference configuration interaction (MRCI)
• n-electron valence state perturbation theory (NEVPT)
• Complete active space perturbation theory (CASPTn)

Methods in detail

Hartree–Fock and Post-Hartree–Fock methods

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 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.^[12] 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.^[13][14] 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 (MCSCF) and methods have been developed that use these multi-determinant references for improvements.^[13]

Example

Is Si₂H₂ like acetylene (C₂H₂)?

A series of ab initio studies of Si₂H₂ 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, Si₂H₂ had the same structure as ethyne (acetylene), C₂H₂. In early studies, by Binkley and Lischka and Kohler, it became clear that linear Si₂H₂ 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.^[15][16] Interest then moved to look at whether structures equivalent to vinylidene (Si=SiH₂) existed. This structure is predicted to be a local minimum, i. e. an isomer of Si₂H₂, 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.^[17] 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.^[18] Similar results were later obtained for Ge₂H₂.^[19] Al₂H₂ and Ga₂H₂ have exactly the same isomers, in spite of having two electrons less than the Group 14 molecules.^[20][21] 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 Si₂H₂ and Al₂H₂. 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.^[22]

Valence bond methods

Valence bond (VB) methods are generally ab initio although some semi-empirical versions have been proposed. Current VB approaches are^[1]:-

• 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.

See also

• Density functional theory
• Quantum chemistry computer programs - see columns for Hartree–Fock and post-Hartree–Fock methods

References

1. ^ ^{a b} Levine, Ira N. (1991). Quantum Chemistry. Englewood Cliffs, New jersey: Prentice Hall. pp. 455–544. ISBN 0-205-12770-3. 
2. ^ Parr, Robert G.. "History of Quantum Chemistry". http://www.quantum-chemistry-history.com/Parr1.htm. 
3. ^ Parr, Robert G.; Craig D. P,. and Ross, I. G (1950). "Molecular Orbital Calculations of the Lower Excited Electronic Levels of Benzene, Configuration Interaction included". Journal of Chemical Physics 18 (12): 1561–1563. doi:10.1063/1.1747540. 
4. ^ Parr, R. G. (1990). "On the genesis of a theory". Int. J. Quantum Chem. 37 (4): 327–347. doi:10.1002/qua.560370407. 
5. ^ Chen, T. C. (1955). "Expansion of Electronic Wave Functions of Molecules in Terms of 'United‐Atom' Wave Functions". J. Chem. Phys. 23 (11): 2200–2201. doi:10.1063/1.1740713. 
6. ^ Roothaan, C. C. J. (1958). "Evaluation of Molecular Integrals by Digital Computer". J. Chem. Phys. 28 (5): 982–983. doi:10.1063/1.1744313. 
7. ^ Allen, L. C.; Karo, A. M. (1960). "Basis Functions for Ab Initio Calculations". Revs. Mod. Phys. 32 (2): 275-. doi:10.1103/RevModPhys.32.275. 
8. ^ Jensen, Frank (2007). Introduction to Computational Chemistry. Chichester, England: John Wiley and Sons. pp. 80–81. ISBN 0-470-01187-4. 
9. ^ ^{a b c} Werner, H-J; Manby, F. R.; Knowles, P. J. (2003). "Fast linear scaling second-order Møller-Plesset perturbation theory (MP2) using local and density fitting approximations". Journal of Chemical Physics 118 (18): 8149–8161. doi:10.1063/1.1564816. 
10. ^ Saebø, S.; Pulay, P. (1987). "Fourth-order Møller–Plessett perturbation theory in the local correlation treatment. I. Method". Journal of Chemical Physics 86 (2): 914–922. doi:10.1063/1.452293. 
11. ^ ^{a b} Schütz, M.; Hetzer, G.; Werner, H-J (1999). "Low-order scaling local electron correlation methods. I. Linear scaling local MP2". Journal of Chemical Physics 111 (13): 5691–5705. doi:10.1063/1.479957. 
12. ^ Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd.. pp. 153–189. ISBN 0-471-48552-7. 
13. ^ ^{a b} Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd.. pp. 191–232. ISBN 0-471-48552-7. 
14. ^ Jensen, Frank (2007). Introduction to Computational Chemistry. Chichester, England: John Wiley and Sons. pp. 98–149. ISBN 0-470-01187-4. 
15. ^ Binkley, J. S. (1983). "Theoretical studies of the relative stability of C₂H₂ of Si₂H₂". Journal of the American Chemical Society 106 (3): 603. doi:10.1021/ja00315a024. 
16. ^ Lischka, H.; H-J Kohler (1983). "Ab initio ivestigation on the lowest singlet and triplet state of Si₂H₂". Journal of the American Chemical Society 105 (22): 6646. doi:10.1021/ja00360a016. 
17. ^ Colegrove, B. T.; Schaefer, Henry F. III (1990). "Disilyne (Si₂H₂) revisited". Journal of Physical Chemistry 94 (14): 5593. doi:10.1021/j100377a036. 
18. ^ Grev, R. S.; Schaefer, Henry F. III (1992). "The remarkable monobridged structure of Si₂H₂". Journal of Chemical Physics 97 (11): 7990. doi:10.1063/1.463422. 
19. ^ Palágyi, Zoltán; Schaefer, Henry F. III, Kapuy, Ede (1993). "Ge₂H₂: A Molecule with a low-lying monobridged equilibrium geometry". Journal of the American Chemical Society 115 (15): 6901–6903. doi:10.1021/ja00068a056. 
20. ^ Stephens, J. C.; Bolton, E. E.,Schaefer, H. F. III, and Andrews, L. (1997). "Quantum mechanical frequencies and matrix assignments to Al₂H₂". Journal of Chemical Physics 107 (1): 119–223. doi:10.1063/1.474608. 
21. ^ Palágyi, Zoltán; Schaefer, Henry F. III, Kapuy, Ede (1993). "Ga₂H₂: planar dibridged, vinylidene-like, monobridged and trans equilibrium geometries". Chemical Physics Letters 203 (2-3): 195–200. doi:10.1016/0009-2614(93)85386-3. 
22. ^ DeLeeuw, B. J.; Grev, R. S. and Schaefer, Henry F. III (1992). "A comparison and contrast of selected saturated and unsaturated hydrides of group 14 elements". Journal of Chemical Education 69 (6): 441. doi:10.1021/ed069p441.