Diagrammatic theories of anharmonic molecular vibrations

Hermes, Matthew R.; Hirata, So

doi:10.1080/0144235X.2014.1001220

Diagrammatic theories of anharmonic molecular vibrations

Journal Article · Wed Apr 01 00:00:00 EDT 2015 · International Reviews in Physical Chemistry

DOI:https://doi.org/10.1080/0144235X.2014.1001220· OSTI ID:3021787

Hermes, Matthew R. ^[1]; Hirata, So ^[1]

University of Illinois at Urbana-Champaign, IL (United States)

In this review, we summarise recent developments in our laboratory in the field of many-body quantum-mechanical calculations of the anharmonic vibrational structure of molecules. Our size-extensive vibrational self-consistent field (XVSCF) and size-extensive second-order many-body perturbation (XVMP2) methods are, unlike their parent methods (VSCF and VMP2), defined in diagram- matic formulations of the energies and Dyson self-energies, leading to manifestly size-consistent expressions for zero-point energies and anharmonic vibrational frequencies calculable with much greater efficiency. The effective one-mode potentials of XVSCF are quadratic and hence the Schrödinger equation for each mode can be solved analytically, unlike VSCF, where a basis-set expansion of wave functions on more complex one-mode potentials need to be performed; VSCF potentials and their minima (anharmonic geometry) are shown to reduce to the quadratic potentials and their minima (also given analytically) of XVSCF in the thermodynamic limit. By self-consistently solving the Dyson equation with frequency-dependent self-energies, XVMP2 has the ability to calculate anharmonic frequencies of fundamentals as well as combinations and overtones in the presence of strong anharmonic resonance without a multireference or quasi-degenerate formulation, which tends to be non-size-consistent. To eliminate the computational bottleneck of XVSCF and XVMP2, which is the high-rank force-constant evaluation, we have developed alternative algorithms in which the diagrammatic equations are recast as a small number of high-dimensional integrals and then evaluated stochastically using a Metropolis Monte Carlo (MC) method. These MC-XVSCF and MC-XVMP2 methods not only remove the need for force-constant evaluation or storage, but also take into account force constants of up to infinite order according to their importance. Furthermore, they are a new branch of quantum Monte Carlo which can calculate frequencies (excitation energies) directly without fixed-node errors.

View Accepted Manuscript (DOE)

Research Organization:: University of Illinois at Urbana-Champaign, IL (United States)

Sponsoring Organization:: USDOE Office of Science (SC), Basic Energy Sciences (BES). Chemical Sciences, Geosciences & Biosciences Division (CSGB)

Grant/Contract Number:: SC0006028

OSTI ID:: 3021787

Journal Information:: International Reviews in Physical Chemistry, Journal Name: International Reviews in Physical Chemistry Journal Issue: 1 Vol. 34; ISSN 1366-591X; ISSN 0144-235X

Publisher:: Informa UK LimitedCopyright Statement

Country of Publication:: United States

Language:: English

References (82)

Quantum Monte Carlo for Ab Initio calculations of energy-relevant materials Wagner, Lucas K. International Journal of Quantum Chemistry, Vol. 114, Issue 2 https://doi.org/10.1002/qua.24526	journal	August 2013
Thermodynamic limit and size-consistent design Hirata, So Theoretical Chemistry Accounts, Vol. 129, Issue 6 https://doi.org/10.1007/s00214-011-0954-4	journal	June 2011
Phonons from a many-body viewpoint Mattuck, R. D. Annals of Physics, Vol. 27, Issue 2 https://doi.org/10.1016/0003-4916(64)90106-X	journal	April 1964
Threshold energy dependence of intramolecular vibrational redistribution (IVR) rates of isolated polyatomic molecules Kuzmin, M. V.; Stuchebrukhov, A. A. Chemical Physics Letters, Vol. 119, Issue 6 https://doi.org/10.1016/0009-2614(85)85388-4	journal	September 1985
Green's functions and the diagrammatic technique in the theory of gas-phase electron diffraction on vibrationally excited polyatomic molecules. Part I. Stuchebrukhov, A. A. Journal of Molecular Structure, Vol. 178 https://doi.org/10.1016/0022-2860(88)85024-5	journal	August 1988
From concepts to algorithms for the treatment of large amplitude internal motions and unimolecular reactions Barone, Vincenzo; Minichino, Camilla Journal of Molecular Structure: THEOCHEM, Vol. 330, Issue 1-3 https://doi.org/10.1016/0166-1280(94)03862-F	journal	January 1995
Toward an Exact One–Electron Picture of Chemical Bonding Ortiz, J. V. Advances in Quantum Chemistry https://doi.org/10.1016/S0065-3276(08)60454-2	book	January 1999
Many – Body Methods in Chemistry and Physics Shavitt, Isaiah; Bartlett, Rodney J. https://doi.org/10.1017/CBO9780511596834	book	January 2009
The self-consistent-field approach to polyatomic vibrations Bowman, Joel M. Accounts of Chemical Research, Vol. 19, Issue 7 https://doi.org/10.1021/ar00127a002	journal	June 1986
Quantum Monte Carlo and Related Approaches Austin, Brian M.; Zubarev, Dmitry Yu.; Lester, William A. Chemical Reviews, Vol. 112, Issue 1 https://doi.org/10.1021/cr2001564	journal	December 2011
Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems Neuhauser, Daniel; Rabani, Eran; Baer, Roi Journal of Chemical Theory and Computation, Vol. 9, Issue 1 https://doi.org/10.1021/ct300946j	journal	December 2012
Convergence Acceleration of Parallel Monte Carlo Second-Order Many-Body Perturbation Calculations Using Redundant Walkers Willow, Soohaeng Yoo; Hermes, Matthew R.; Kim, Kwang S. Journal of Chemical Theory and Computation, Vol. 9, Issue 10 https://doi.org/10.1021/ct400557z	journal	September 2013
New Formulation and Implementation of Vibrational Self-Consistent Field Theory Hansen, Mikkel B.; Sparta, Manuel; Seidler, Peter Journal of Chemical Theory and Computation, Vol. 6, Issue 1 https://doi.org/10.1021/ct9004454	journal	November 2009
Excited vibrational states of polyatomic molecules: the semiclassical self-consistent field approach Ratner, Mark A.; Gerber, R. B. The Journal of Physical Chemistry, Vol. 90, Issue 1 https://doi.org/10.1021/j100273a008	journal	January 1986
IR spectrum of highly vibrationally excited osmium tetraoxide in the region of strong nonlinear interaction of vibrational modes Boyarkin, O. V.; Ionov, S. I.; Stuchebrukhov, A. A. The Journal of Physical Chemistry, Vol. 94, Issue 4 https://doi.org/10.1021/j100367a018	journal	February 1990
Application of SCF-SI theory to vibrational motion in polyatomic molecules Bowman, Joel M.; Christoffel, Kurt.; Tobin, Frank. The Journal of Physical Chemistry, Vol. 83, Issue 8 https://doi.org/10.1021/j100471a005	journal	April 1979
First-Order Dyson Coordinates and Geometry Hermes, Matthew R.; Hirata, So The Journal of Physical Chemistry A, Vol. 117, Issue 32 https://doi.org/10.1021/jp4008834	journal	May 2013
Vibrational Analysis of an Ice Ih Model from 0 to 4000 cm ^–1 Using the Ab Initio WHBB Potential Energy Surface Liu, Hanchao; Wang, Yimin; Bowman, Joel M. The Journal of Physical Chemistry B, Vol. 117, Issue 34 https://doi.org/10.1021/jp405865c	journal	August 2013
Second-Order Many-Body Perturbation Theory: An Eternal Frontier Hirata, So; He, Xiao; Hermes, Matthew R. The Journal of Physical Chemistry A, Vol. 118, Issue 4 https://doi.org/10.1021/jp410587b	journal	December 2013
Quantum Calculations of Intramolecular IR Spectra of Ice Models Using Ab Initio Potential and Dipole Moment Surfaces Liu, Hanchao; Wang, Yimin; Bowman, Joel M. The Journal of Physical Chemistry Letters, Vol. 3, Issue 24 https://doi.org/10.1021/jz3016777	journal	November 2012
Expeditious Stochastic Calculation of Random-Phase Approximation Energies for Thousands of Electrons in Three Dimensions Neuhauser, Daniel; Rabani, Eran; Baer, Roi The Journal of Physical Chemistry Letters, Vol. 4, Issue 7 https://doi.org/10.1021/jz3021606	journal	March 2013
A Guided Stochastic Energy-Domain Formulation of the Second Order Møller–Plesset Perturbation Theory Ge, Qinghui; Gao, Yi; Baer, Roi The Journal of Physical Chemistry Letters, Vol. 5, Issue 1 https://doi.org/10.1021/jz402206m	journal	December 2013
Vibrational quasi-degenerate perturbation theory: applications to fermi resonance in CO2, H2CO, and C6H6 Yagi, Kiyoshi; Hirata, So; Hirao, Kimihiko Physical Chemistry Chemical Physics, Vol. 10, Issue 13 https://doi.org/10.1039/b719093j	journal	January 2008
Experimental Energy Levels of the Water Molecule Tennyson, Jonathan; Zobov, Nikolai F.; Williamson, Ross Journal of Physical and Chemical Reference Data, Vol. 30, Issue 3 https://doi.org/10.1063/1.1364517	journal	May 2001
Møller–Plesset perturbation theory for vibrational wave functions Christiansen, Ove The Journal of Chemical Physics, Vol. 119, Issue 12 https://doi.org/10.1063/1.1601593	journal	September 2003
A second quantization formulation of multimode dynamics Christiansen, Ove The Journal of Chemical Physics, Vol. 120, Issue 5 https://doi.org/10.1063/1.1637578	journal	February 2004
Vibrational coupled cluster theory Christiansen, Ove The Journal of Chemical Physics, Vol. 120, Issue 5 https://doi.org/10.1063/1.1637579	journal	February 2004
Vibrational zero-point energies and thermodynamic functions beyond the harmonic approximation Barone, Vincenzo The Journal of Chemical Physics, Vol. 120, Issue 7 https://doi.org/10.1063/1.1637580	journal	February 2004
Equation of State Calculations by Fast Computing Machines Metropolis, Nicholas; Rosenbluth, Arianna W.; Rosenbluth, Marshall N. The Journal of Chemical Physics, Vol. 21, Issue 6 https://doi.org/10.1063/1.1699114	journal	June 1953
Anharmonic vibrational properties by a fully automated second-order perturbative approach Barone, Vincenzo The Journal of Chemical Physics, Vol. 122, Issue 1 https://doi.org/10.1063/1.1824881	journal	January 2005
Efficient configuration selection scheme for vibrational second-order perturbation theory Yagi, Kiyoshi; Hirata, So; Hirao, Kimihiko The Journal of Chemical Physics, Vol. 127, Issue 3 https://doi.org/10.1063/1.2748774	journal	July 2007
Vibrational coupled cluster theory with full two-mode and approximate three-mode couplings: The VCC[2pt3] model Seidler, Peter; Matito, Eduard; Christiansen, Ove The Journal of Chemical Physics, Vol. 131, Issue 3 https://doi.org/10.1063/1.3158946	journal	July 2009
Automatic derivation and evaluation of vibrational coupled cluster theory equations Seidler, Peter; Christiansen, Ove The Journal of Chemical Physics, Vol. 131, Issue 23 https://doi.org/10.1063/1.3272796	journal	December 2009
First-principles theories for anharmonic lattice vibrations Hirata, So; Keçeli, Murat; Yagi, Kiyoshi The Journal of Chemical Physics, Vol. 133, Issue 3 https://doi.org/10.1063/1.3462237	journal	July 2010
First-principles calculations on anharmonic vibrational frequencies of polyethylene and polyacetylene in the Γ approximation Keçeli, Murat; Hirata, So; Yagi, Kiyoshi The Journal of Chemical Physics, Vol. 133, Issue 3 https://doi.org/10.1063/1.3462238	journal	July 2010
Finite-temperature infrared spectroscopy of polycyclic aromatic hydrocarbon molecules. II. Principal mode analysis and self-consistent phonons Calvo, F.; Parneix, P.; Van-Oanh, N. -T. The Journal of Chemical Physics, Vol. 133, Issue 7 https://doi.org/10.1063/1.3465554	journal	August 2010
Vibrational coupled cluster response theory: A general implementation Seidler, Peter; Sparta, Manuel; Christiansen, Ove The Journal of Chemical Physics, Vol. 134, Issue 5 https://doi.org/10.1063/1.3536499	journal	February 2011
Computation of expectation values from vibrational coupled-cluster at the two-mode coupling level Zoccante, Alberto; Seidler, Peter; Christiansen, Ove The Journal of Chemical Physics, Vol. 134, Issue 15 https://doi.org/10.1063/1.3560027	journal	April 2011
Size-extensive vibrational self-consistent field method Keçeli, Murat; Hirata, So The Journal of Chemical Physics, Vol. 135, Issue 13 https://doi.org/10.1063/1.3644895	journal	October 2011
A random‐walk simulation of the Schrödinger equation: H ⁺ ₃ Anderson, James B. The Journal of Chemical Physics, Vol. 63, Issue 4 https://doi.org/10.1063/1.431514	journal	August 1975
Self‐consistent field energies and wavefunctions for coupled oscillators Bowman, Joel M. The Journal of Chemical Physics, Vol. 68, Issue 2 https://doi.org/10.1063/1.435782	journal	January 1978
Dynamical and optical properties of the ethylene crystal: Self‐consistent phonon calculations using an ‘‘ a b i n i t i o ’’ intermolecular potential Luty, T.; van der Avoird, A.; Berns, R. M. The Journal of Chemical Physics, Vol. 75, Issue 3 https://doi.org/10.1063/1.442152	journal	August 1981
Anharmonic vibrational properties of CH ₂ F ₂ : A comparison of theory and experiment Amos, R. D.; Handy, N. C.; Green, W. H. The Journal of Chemical Physics, Vol. 95, Issue 11 https://doi.org/10.1063/1.461259	journal	December 1991
A general discrete variable method to calculate vibrational energy levels of three‐ and four‐atom molecules Bramley, Matthew J.; Carrington, Tucker The Journal of Chemical Physics, Vol. 99, Issue 11 https://doi.org/10.1063/1.465576	journal	December 1993
The anharmonic force field of ethylene, C ₂ H ₄ , by means of accurate ab initio calculations Martin, Jan M. L.; Lee, Timothy J.; Taylor, Peter R. The Journal of Chemical Physics, Vol. 103, Issue 7 https://doi.org/10.1063/1.469681	journal	August 1995
Investigation of the full configuration interaction quantum Monte Carlo method using homogeneous electron gas models Shepherd, James J.; Booth, George H.; Alavi, Ali The Journal of Chemical Physics, Vol. 136, Issue 24 https://doi.org/10.1063/1.4720076	journal	June 2012
Mo/ller–Plesset perturbation theory applied to vibrational problems Norris, Lawrence S.; Ratner, Mark A.; Roitberg, Adrian E. The Journal of Chemical Physics, Vol. 105, Issue 24 https://doi.org/10.1063/1.472922	journal	December 1996
Size-extensive vibrational self-consistent field methods with anharmonic geometry corrections Hermes, Matthew R.; Keçeli, Murat; Hirata, So The Journal of Chemical Physics, Vol. 136, Issue 23 https://doi.org/10.1063/1.4729602	journal	June 2012
Stochastic evaluation of second-order many-body perturbation energies Willow, Soohaeng Yoo; Kim, Kwang S.; Hirata, So The Journal of Chemical Physics, Vol. 137, Issue 20 https://doi.org/10.1063/1.4768697	journal	November 2012
Self-consistent phonons revisited. II. A general and efficient method for computing free energies and vibrational spectra of molecules and clusters Brown, Sandra E.; Georgescu, Ionuţ; Mandelshtam, Vladimir A. The Journal of Chemical Physics, Vol. 138, Issue 4 https://doi.org/10.1063/1.4788977	journal	January 2013
Stochastic evaluation of second-order Dyson self-energies Willow, Soohaeng Yoo; Kim, Kwang S.; Hirata, So The Journal of Chemical Physics, Vol. 138, Issue 16 https://doi.org/10.1063/1.4801862	journal	April 2013
Stochastic determination of effective Hamiltonian for the full configuration interaction solution of quasi-degenerate electronic states Ten-no, Seiichiro The Journal of Chemical Physics, Vol. 138, Issue 16 https://doi.org/10.1063/1.4802766	journal	April 2013
Second-order many-body perturbation expansions of vibrational Dyson self-energies Hermes, Matthew R.; Hirata, So The Journal of Chemical Physics, Vol. 139, Issue 3 https://doi.org/10.1063/1.4813123	journal	July 2013
On the ground state calculation of a many-body system using a self-consistent basis and quasi-Monte Carlo: An application to water hexamer Georgescu, Ionuţ; Jitomirskaya, Svetlana; Mandelshtam, Vladimir A. The Journal of Chemical Physics, Vol. 139, Issue 20 https://doi.org/10.1063/1.4829836	journal	November 2013
Stochastic, real-space, imaginary-time evaluation of third-order Feynman–Goldstone diagrams Willow, Soohaeng Yoo; Hirata, So The Journal of Chemical Physics, Vol. 140, Issue 2 https://doi.org/10.1063/1.4861561	journal	January 2014
Communication: Stochastic evaluation of explicitly correlated second-order many-body perturbation energy Willow, Soohaeng Yoo; Zhang, Jinmei; Valeev, Edward F. The Journal of Chemical Physics, Vol. 140, Issue 3 https://doi.org/10.1063/1.4862255	journal	January 2014
Stochastic many-body perturbation theory for anharmonic molecular vibrations Hermes, Matthew R.; Hirata, So The Journal of Chemical Physics, Vol. 141, Issue 8 https://doi.org/10.1063/1.4892614	journal	August 2014
The lattice dynamics of an anharmonic crystal Cowley, R. A. Advances in Physics, Vol. 12, Issue 48 https://doi.org/10.1080/00018736300101333	journal	October 1963
Treatment of Fermi resonance effects on transition moments in vibrational perturbation theory Vázquez, Juana; Stanton, John F. Molecular Physics, Vol. 105, Issue 1 https://doi.org/10.1080/00268970601135784	journal	January 2007
Calculated stretching overtone levels and Darling–Dennison resonances in water: a triumph of simple theoretical approaches Matthews, Devin A.; Vázquez, Juana; Stanton, John F. Molecular Physics, Vol. 105, Issue 19-22 https://doi.org/10.1080/00268970701618424	journal	October 2007
Variational quantum approaches for computing vibrational energies of polyatomic molecules Bowman, Joel M.; Carrington, Tucker; Meyer, Hans-Dieter Molecular Physics, Vol. 106, Issue 16-18 https://doi.org/10.1080/00268970802258609	journal	August 2008
The electron propagator Öhrn, Yngve; Ortiz, J. V. Molecular Physics, Vol. 108, Issue 21-23 https://doi.org/10.1080/00268976.2010.498390	journal	November 2010
On the relationship of normal modes to local modes in molecular vibrations Mills, I. M.; Robiette, A. G. Molecular Physics, Vol. 56, Issue 4 https://doi.org/10.1080/00268978500102691	journal	November 1985
Diffusion Monte Carlo approaches for investigating the structure and vibrational spectra of fluxional systems McCoy, Anne B. International Reviews in Physical Chemistry, Vol. 25, Issue 1-2 https://doi.org/10.1080/01442350600679347	journal	January 2006
The use of a model in anharmonic lattice dynamics Hooton, D. J. Philosophical Magazine, Vol. 3, Issue 25 https://doi.org/10.1080/14786435808243224	journal	January 1958
Self-consistent phonon spectrum of h.c.p. H₂and D₂ Klein, M. L.; Koehler, T. R. Journal of Physics C: Solid State Physics, Vol. 3, Issue 5 https://doi.org/10.1088/0022-3719/3/5/007	journal	May 1970
Theoretical methods for small-molecule ro-vibrational spectroscopy Lodi, Lorenzo; Tennyson, Jonathan Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 43, Issue 13 https://doi.org/10.1088/0953-4075/43/13/133001	journal	June 2010
Continuum variational and diffusion quantum Monte Carlo calculations Needs, R. J.; Towler, M. D.; Drummond, N. D. Journal of Physics: Condensed Matter, Vol. 22, Issue 2 https://doi.org/10.1088/0953-8984/22/2/023201	journal	December 2009
Derivation of the Brueckner many-body theory Goldstone, Jeffrey Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, Vol. 239, Issue 1217, p. 267-279 https://doi.org/10.1098/rspa.1957.0037	journal	February 1957
Scattering of Neutrons by an Anharmonic Crystal Maradudin, A. A.; Fein, A. E. Physical Review, Vol. 128, Issue 6 https://doi.org/10.1103/PhysRev.128.2589	journal	December 1962
Properties of Crystalline Argon and Neon in the Self-Consistent Phonon Approximation Gillis, N. S.; Werthamer, N. R.; Koehler, T. R. Physical Review, Vol. 165, Issue 3 https://doi.org/10.1103/PhysRev.165.951	journal	January 1968
Helmholtz free energy of an anharmonic crystal to O(λ4). III. Equation of state for the Lennard-Jones solid Shukla, R. C.; Cowley, E. R. Physical Review B, Vol. 31, Issue 1 https://doi.org/10.1103/PhysRevB.31.372	journal	January 1985
Anharmonic self-energies of phonons in silicon Narasimhan, Shobhana; Vanderbilt, David Physical Review B, Vol. 43, Issue 5 https://doi.org/10.1103/PhysRevB.43.4541	journal	February 1991
Fast coupled-cluster singles and doubles for extended systems: Application to the anharmonic vibrational frequencies of polyethylene in theΓapproximation Keçeli, Murat; Hirata, So Physical Review B, Vol. 82, Issue 11 https://doi.org/10.1103/PhysRevB.82.115107	journal	September 2010
Anharmonic free energies and phonon dispersions from the stochastic self-consistent harmonic approximation: Application to platinum and palladium hydrides Errea, Ion; Calandra, Matteo; Mauri, Francesco Physical Review B, Vol. 89, Issue 6 https://doi.org/10.1103/PhysRevB.89.064302	journal	February 2014
Self-Averaging Stochastic Kohn-Sham Density-Functional Theory Baer, Roi; Neuhauser, Daniel; Rabani, Eran Physical Review Letters, Vol. 111, Issue 10 https://doi.org/10.1103/PhysRevLett.111.106402	journal	September 2013
Theory of the Self-Consistent Harmonic Approximation with Application to Solid Neon Koehler, Thomas R. Physical Review Letters, Vol. 17, Issue 2 https://doi.org/10.1103/PhysRevLett.17.89	journal	July 1966
An Improved Self-Consistent Phonon Approximation Goldman, V.; Horton, G.; Klein, M. Physical Review Letters, Vol. 21, Issue 22 https://doi.org/10.1103/PhysRevLett.21.1527	journal	November 1968
Theoretical Temperature-Dependent Phonon Spectra of Solid Neon Koehler, T. R. Physical Review Letters, Vol. 22, Issue 15 https://doi.org/10.1103/PhysRevLett.22.777	journal	April 1969
Stochastic Perturbation Theory: A Low-Scaling Approach to Correlated Electronic Energies Thom, Alex J. W.; Alavi, Ali Physical Review Letters, Vol. 99, Issue 14 https://doi.org/10.1103/PhysRevLett.99.143001	journal	October 2007
Quantum Monte Carlo simulations of solids Foulkes, W. M. C.; Mitas, L.; Needs, R. J. Reviews of Modern Physics, Vol. 73, Issue 1 https://doi.org/10.1103/RevModPhys.73.33	journal	January 2001
Extensivity of Energy and Electronic and Vibrational Structure Methods for Crystals Hirata, So; Keçeli, Murat; Ohnishi, Yu-ya Annual Review of Physical Chemistry, Vol. 63, Issue 1 https://doi.org/10.1146/annurev-physchem-032511-143718	journal	May 2012

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