Ab initio quantum chemistry methods

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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] Ab initio means "from first principles" or "from the beginning", implying that the only inputs into an ab initio calculation are physical constants. [5] Ab initio quantum chemistry methods attempt to solve the electronic Schrödinger equation given the positions of the nuclei and the number of electrons in order to yield useful information such as electron densities, energies and other properties of the system. The ability to run these calculations has enabled theoretical chemists to solve a range of problems and their importance is highlighted by the awarding of the Nobel prize to John Pople and Walter Kohn. [6]

Contents

Accuracy and scaling

Ab initio electronic structure methods aim to calculate the many electron function which is the solution of the non-relativistic electronic Schrödinger equation (in the Born–Oppenheimer approximation). The many electron function is generally a linear combination of many simpler electron functions with the dominant function being the Hartree-Fock function. Each of these simple functions are then approximated using only one-electron functions. The one-electron functions are then expanded as a linear combination of a finite set of basis functions. This approach has the advantage that it can be made to converge to the exact solution, when the basis set tends toward the limit of a complete set and where all possible configurations are included (called "Full CI"). However this convergence to the limit is computationally very demanding and most calculations are far from the limit. Nevertheless important conclusions have been made from these more limited classifications.

One needs to consider the computational cost of ab initio methods when determining whether they are appropriate for the problem at hand. When compared to much less accurate approaches, such as molecular mechanics, ab initio methods often take larger amounts of computer time, memory, and disk space, though, with modern advances in computer science and technology such considerations are becoming less of an issue. The Hartree-Fock (HF) method scales nominally as N4 (N being a relative measure of the system size, not the number of basis functions) – e.g., if one doubles the number of electrons and the number of basis functions (double the system size), the calculation will take 16 (24) times as long per iteration. 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 less favorably, though their accuracy is usually greater, which is the trade off one needs to consider. One popular method is Møller–Plesset perturbation theory (MP). To second order (MP2), MP scales as N4. To third order (MP3) MP scales as N6. To fourth order (MP4) MP scales as N7. Another method, coupled cluster with singles and doubles (CCSD), scales as N6 and extensions, CCSD(T) and CR-CC(2,3), scale as N6 with one noniterative step which scales as N7. Hybrid Density functional theory (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. Local DFT methods that do not include Hartree–Fock exchange can scale better than Hartree–Fock.[ citation needed ]

Linear scaling approaches

The problem of computational expense can be alleviated through simplification schemes. [7] 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 [8] (many authors use lower-case to prevent confusion with DFT). In the local approximation, [9] [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., 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. [12] [13] Methods employing this scheme are denoted by the prefix "L", e.g. LMP2. [8] [10] Both schemes can be employed together, as in the df-LMP2 [8] 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.[ citation needed ]

Classes of methods

The most popular classes of ab initio electronic structure methods:

Hartree–Fock methods

Post-Hartree–Fock methods

Multi-reference methods

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. [15] 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. [16] [17] 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. [16] However, if one uses coupled cluster methods such as CCSDT, CCSDt, CR-CC(2,3), or CC(t;3) then single-bond breaking using the single determinant HF reference is feasible. For an accurate description of double bond breaking, methods such as CCSDTQ, CCSDTq, CCSDtq, CR-CC(2,4), or CC(tq;3,4) also make use of the single determinant HF reference, and do not require one to use multi-reference methods.

Example
Is the bonding situation in disilyne Si2H2 the same as in 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. [18] [19] 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. [20] 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. [21] Similar results were later obtained for Ge2H2. [22] Al2H2 and Ga2H2 have exactly the same isomers, in spite of having two electrons less than the Group 14 molecules. [23] [24] 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. [25]

Valence bond methods

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

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. The accuracy of QMC depends strongly on the initial guess of many-body wave-functions and the form of the many-body wave-function. One simple choice is Slater-Jastrow wave-function in which the local correlations are treated with the Jastrow factor.

Sign Learning Kink-based (SiLK) Quantum Monte Carlo (website): [14] The Sign Learning Kink (SiLK) based Quantum Monte Carlo (QMC) method is based on Feynman's path integral formulation of quantum mechanics, and can reduce the minus sign problem when calculating energies in atomic and molecular systems.

See also

Related Research Articles

Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of molecules, groups of molecules, and solids. It is essential because, apart from relatively recent results concerning the hydrogen molecular ion, the quantum many-body problem cannot be solved analytically, much less in closed form. While computational results normally complement the information obtained by chemical experiments, it can in some cases predict hitherto unobserved chemical phenomena. It is widely used in the design of new drugs and materials.

Coupled cluster (CC) is a numerical technique used for describing many-body systems. Its most common use is as one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry, but it is also used in nuclear physics. Coupled cluster essentially takes the basic Hartree–Fock molecular orbital method and constructs multi-electron wavefunctions using the exponential cluster operator to account for electron correlation. Some of the most accurate calculations for small to medium-sized molecules use this method.

Q-Chem is a general-purpose electronic structure package featuring a variety of established and new methods implemented using innovative algorithms that enable fast calculations of large systems on various computer architectures, from laptops and regular lab workstations to midsize clusters and HPCC, using density functional and wave-function based approaches. It offers an integrated graphical interface and input generator; a large selection of functionals and correlation methods, including methods for electronically excited states and open-shell systems; solvation models; and wave-function analysis tools. In addition to serving the computational chemistry community, Q-Chem also provides a versatile code development platform.

Psi is an ab initio computational chemistry package originally written by the research group of Henry F. Schaefer, III. Utilizing Psi, one can perform a calculation on a molecular system with various kinds of methods such as Hartree-Fock, Post-Hartree–Fock electron correlation methods, and density functional theory. The program can compute energies, optimize molecular geometries, and compute vibrational frequencies. The major part of the program is written in C++, while Python API is also available, which allows users to perform complex computations or automate tasks easily.

NWChem is an ab initio computational chemistry software package which includes quantum chemical and molecular dynamics functionality. It was designed to run on high-performance parallel supercomputers as well as conventional workstation clusters. It aims to be scalable both in its ability to treat large problems efficiently, and in its usage of available parallel computing resources. NWChem has been developed by the Molecular Sciences Software group of the Theory, Modeling & Simulation program of the Environmental Molecular Sciences Laboratory (EMSL) at the Pacific Northwest National Laboratory (PNNL). The early implementation was funded by the EMSL Construction Project.

The Vienna Ab initio Simulation Package, better known as VASP, is a package written primarily in Fortran for performing ab initio quantum mechanical calculations using either Vanderbilt pseudopotentials, or the projector augmented wave method, and a plane wave basis set. The basic methodology is density functional theory (DFT), but the code also allows use of post-DFT corrections such as hybrid functionals mixing DFT and Hartree–Fock exchange, many-body perturbation theory and dynamical electronic correlations within the random phase approximation (RPA) and MP2.

Møller–Plesset perturbation theory (MP) is one of several quantum chemistry post-Hartree–Fock ab initio methods in the field of computational chemistry. It improves on the Hartree–Fock method by adding electron correlation effects by means of Rayleigh–Schrödinger perturbation theory (RS-PT), usually to second (MP2), third (MP3) or fourth (MP4) order. Its main idea was published as early as 1934 by Christian Møller and Milton S. Plesset.

Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.

In computational chemistry, post–Hartree–Fock (post-HF) methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. They add electron correlation which is a more accurate way of including the repulsions between electrons than in the Hartree–Fock method where repulsions are only averaged.

In theoretical and computational chemistry, a basis set is a set of functions that is used to represent the electronic wave function in the Hartree–Fock method or density-functional theory in order to turn the partial differential equations of the model into algebraic equations suitable for efficient implementation on a computer.

Restricted open-shell Hartree–Fock (ROHF) is a variant of Hartree–Fock method for open shell molecules. It uses doubly occupied molecular orbitals as far as possible and then singly occupied orbitals for the unpaired electrons. This is the simple picture for open shell molecules but it is difficult to implement. The foundations of the ROHF method were first formulated by Clemens C. J. Roothaan in a celebrated paper and then extended by various authors, see e.g. for in-depth discussions.

The Davidson correction is an energy correction often applied in calculations using the method of truncated configuration interaction, which is one of several post-Hartree–Fock ab initio quantum chemistry methods in the field of computational chemistry. It was introduced by Ernest R. Davidson.

Jaguar is a computer software package used for ab initio quantum chemistry calculations for both gas and solution phases. It is commercial software marketed by the company Schrödinger. The program was originated in research groups of Richard Friesner and William Goddard and was initially called PS-GVB.

<span class="mw-page-title-main">Spartan (chemistry software)</span>

Spartan is a molecular modelling and computational chemistry application from Wavefunction. It contains code for molecular mechanics, semi-empirical methods, ab initio models, density functional models, post-Hartree–Fock models, and thermochemical recipes including G3(MP2) and T1. Quantum chemistry calculations in Spartan are powered by Q-Chem.

Computational chemical methods in solid-state physics follow the same approach as they do for molecules, but with two differences. First, the translational symmetry of the solid has to be utilised, and second, it is possible to use completely delocalised basis functions such as plane waves as an alternative to the molecular atom-centered basis functions. The electronic structure of a crystal is in general described by a band structure, which defines the energies of electron orbitals for each point in the Brillouin zone. Ab initio and semi-empirical calculations yield orbital energies, therefore they can be applied to band structure calculations. Since it is time-consuming to calculate the energy for a molecule, it is even more time-consuming to calculate them for the entire list of points in the Brillouin zone.

Semi-empirical quantum chemistry methods are based on the Hartree–Fock formalism, but make many approximations and obtain some parameters from empirical data. They are very important in computational chemistry for treating large molecules where the full Hartree–Fock method without the approximations is too expensive. The use of empirical parameters appears to allow some inclusion of electron correlation effects into the methods.

The fragment molecular orbital method (FMO) is a computational method that can be used to calculate very large molecular systems with thousands of atoms using ab initio quantum-chemical wave functions.

Quantum chemistry composite methods are computational chemistry methods that aim for high accuracy by combining the results of several calculations. They combine methods with a high level of theory and a small basis set with methods that employ lower levels of theory with larger basis sets. They are commonly used to calculate thermodynamic quantities such as enthalpies of formation, atomization energies, ionization energies and electron affinities. They aim for chemical accuracy which is usually defined as within 1 kcal/mol of the experimental value. The first systematic model chemistry of this type with broad applicability was called Gaussian-1 (G1) introduced by John Pople. This was quickly replaced by the Gaussian-2 (G2) which has been used extensively. The Gaussian-3 (G3) was introduced later.

<span class="mw-page-title-main">CP2K</span>

CP2K is a freely available (GPL) quantum chemistry and solid state physics program package, written in Fortran 2008, to perform atomistic simulations of solid state, liquid, molecular, periodic, material, crystal, and biological systems. It provides a general framework for different methods: density functional theory (DFT) using a mixed Gaussian and plane waves approach (GPW) via LDA, GGA, MP2, or RPA levels of theory, classical pair and many-body potentials, semi-empirical and tight-binding Hamiltonians, as well as Quantum Mechanics/Molecular Mechanics (QM/MM) hybrid schemes relying on the Gaussian Expansion of the Electrostatic Potential (GEEP). The Gaussian and Augmented Plane Waves method (GAPW) as an extension of the GPW method allows for all-electron calculations. CP2K can do simulations of molecular dynamics, metadynamics, Monte Carlo, Ehrenfest dynamics, vibrational analysis, core level spectroscopy, energy minimization, and transition state optimization using NEB or dimer method.

Martin Schütz was a Swiss theoretical chemist and quantum chemist.

References

  1. 1 2 Levine, Ira N. (1991). Quantum Chemistry. Englewood Cliffs, New jersey: Prentice Hall. pp. 455–544. ISBN   978-0-205-12770-2.
  2. ↑ Parr, Robert G. "History of Quantum Chemistry".
  3. ↑ Parr, Robert G.; Craig D. P.; 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. Bibcode:1950JChPh..18.1561P. 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. ↑ Leach, Dr Andrew (January 30, 2001). Molecular Modelling: Principles and Applications (2 ed.). Harlow: Prentice Hall. ISBN   9780582382107.
  6. ↑ Friesner, Richard A. (May 10, 2005). "Ab initio quantum chemistry: Methodology and applications". Proceedings of the National Academy of Sciences of the United States of America. 102 (19): 6648–6653. Bibcode:2005PNAS..102.6648F. doi: 10.1073/pnas.0408036102 . ISSN   0027-8424. PMC   1100737 . PMID   15870212.
  7. ↑ Jensen, Frank (2007). Introduction to Computational Chemistry. Chichester, England: John Wiley and Sons. pp. 80–81. ISBN   978-0-470-01187-4.
  8. 1 2 3 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. Bibcode:2003JChPh.118.8149W. doi:10.1063/1.1564816.
  9. ↑ 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. Bibcode:1987JChPh..86..914S. doi:10.1063/1.452293.
  10. 1 2 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. Bibcode:1999JChPh.111.5691S. doi:10.1063/1.479957.
  11. ↑ Hanaor, Dorian A. H. (2012). "Ab initio study of phase stability in doped TiO2". Computational Mechanics. 50 (2): 185–194. arXiv: 1210.7555 . Bibcode:2012CompM..50..185H. doi:10.1007/s00466-012-0728-4. S2CID   95958719.
  12. ↑ Alireza Mashaghi et al., Hydration strongly affects the molecular and electronic structure of membrane phospholipids. J. Chem. Phys. 136, 114709 (2012) http://jcp.aip.org/resource/1/jcpsa6/v136/i11/p114709_s1 Archived 2016-05-15 at the Portuguese Web Archive
  13. ↑ Mischa Bonn et al., Interfacial Water Facilitates Energy Transfer by Inducing Extended Vibrations in Membrane Lipids, J Phys Chem, 2012 http://pubs.acs.org/doi/abs/10.1021/jp302478a
  14. 1 2 Ma, Xiaoyao (January 5, 2016). "Sign Learning Kink-based (SiLK) Quantum Monte Carlo for molecular systems". The Journal of Chemical Physics. 144 (1): 014101. arXiv: 1512.05455 . Bibcode:2016JChPh.144a4101M. doi:10.1063/1.4939145. PMC   6910603 . PMID   26747795.
  15. ↑ Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd. pp. 153–189. ISBN   978-0-471-48552-0.
  16. 1 2 Cramer, Christopher J. (2002). Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd. pp. 191–232. ISBN   978-0-471-48552-0.
  17. ↑ Jensen, Frank (2007). Introduction to Computational Chemistry. Chichester, England: John Wiley and Sons. pp. 98–149. ISBN   978-0-470-01187-4.
  18. ↑ Binkley, J. S. (1983). "Theoretical studies of the relative stability of C2H2 of Si2H2". Journal of the American Chemical Society. 106 (3): 603–609. doi:10.1021/ja00315a024.
  19. ↑ Lischka, H.; H-J Kohler (1983). "Ab initio investigation on the lowest singlet and triplet state of Si2H2". Journal of the American Chemical Society. 105 (22): 6646–6649. doi:10.1021/ja00360a016.
  20. ↑ Colegrove, B. T.; Schaefer, Henry F. III (1990). "Disilyne (Si2H2) revisited". Journal of Physical Chemistry. 94 (14): 5593–5602. doi:10.1021/j100377a036.
  21. ↑ Grev, R. S.; Schaefer, Henry F. III (1992). "The remarkable monobridged structure of Si2H2". Journal of Chemical Physics. 97 (11): 7990. Bibcode:1992JChPh..97.7990G. doi:10.1063/1.463422.
  22. ↑ 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.
  23. ↑ Stephens, J. C.; Bolton, E. E.; Schaefer H. F. III; Andrews, L. (1997). "Quantum mechanical frequencies and matrix assignments to Al2H2". Journal of Chemical Physics. 107 (1): 119–223. Bibcode:1997JChPh.107..119S. doi:10.1063/1.474608.
  24. ↑ 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. Bibcode:1993CPL...203..195P. doi:10.1016/0009-2614(93)85386-3.
  25. ↑ DeLeeuw, B. J.; Grev, R. S.; 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. Bibcode:1992JChEd..69..441D. doi:10.1021/ed069p441.
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