Electron-Proton Correlation, Theory, and Tunneling Splittings. Sharon Hammes-Schiffer Penn State University

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1 Nuclear-Electronic Orbital Approach: Electron-Proton Correlation, Multicomponent Density Functional Theory, and Tunneling Splittings Sharon Hammes-Schiffer Penn State University

2 Nuclear Quantum Effects Important Zero point energy Hydrogen bonding Hydrogen tunneling Vibrationally excited states D e D p ET H PT A p A e ET Proton-coupled electron transfer Hydrogen transfer in solution and enzymes PT

3 Methods for Nuclear Quantum Effects Standard Born-Oppenheimer approach Approximate wavefunction : Ψ ( r e, r n ) Ψ ( r e ; r n ) Ψ ( r n ) tot elec nuc e n n e n Electronic equation : HelecΨ elec r r = Eelec r Ψ elec r r Nuclear equation : ( ; ) ( ) ( ; ) n n n Tnuc + Eel ec ( r ) Ψ nuc( r ) = E Ψ nuc( r ) Nuclear-electronic orbital (NEO) approach: treat specified protons QM on same level as electrons e p c c e p c H Ψ ( r, r ; r ) = E ( r ) Ψ ( r, r ; r ) NEO NEO e p c NEO NEO r, r, r : electrons, quantum protons, other nuclei r p : transferring proton r c : all other nuclei

4 Nuclear-Electronic Orbital (NEO) Method Webb, Iordanov, and SHS, JCP 117, 4106 (2002) Solution of mixed nuclear-electronic time-independent Schrödinger equation with molecular orbital methods Treat specified nuclei quantum mechanically on same level as electrons - treat only key H nuclei QM - at least two classical nuclei Electronic and nuclear MO s expanded in Gaussian basis sets Energy minimized with respect to all MO s and centers of nuclear basis functions Correlation among electrons and nuclei included with multiconfigurational, perturbation, and DFT approaches Provides structures, energies, minimum energy paths, and direct dynamics for chemical reactions

5 Advantages of NEO Nuclear quantum effects incorporated during electronic structure calculations Born-Oppenheimer separation of electrons and quantum nuclei is avoided Excited vibrational-electronic states are provided Nonadiabatic effects may be included in dynamics Computationally practical Accuracy may be improved systematically Related work Tachikawa, Nakai, Shigeta, Gross, Jungwirth, Krylov, Sherrill, Valeev Major challenges: Electron-proton correlation is highly significant Other modes may be strongly coupled to H motion

6 Nuclear-Electronic Hamiltonian H NEO Ne Ne Nc Ne 1 Z r r r r Electronic terms = 2 A i e c e e i i A i A i > j i j N N N 2 Z A i p c p p i i A i A i > j i r j p p N c p m r r r Nuclear terms p N p N e 1 r r Nuclear-Electronic interaction term e p i i r i r i Ne, Np, Nc Number of electrons, quantum nuclei, and classical nuclei r, r, r Coordinates of electrons, quantum nuclei, and classical nuclei e p c i i

7 HF wavefunction NEO-HF (Hartree-Fock) Ψ ( r, r ) =Φ ( r ) Φ ( r ) e p e e p p tot 0 0 Φ e p 0, Φ0 : Slater determinants HF energy E = Φ ( r ) Φ ( r ) H Φ ( r ) Φ ( r ) e e p p e e p p 0 0 NEO 0 0 Expand electronic, nuclear MO s in Gaussian basis sets s-type, p-type, d-type, Gaussian basis functions Minimize energy with respect to electronic and nuclear MO s HF-Roothaan equations for electrons and quantum protons e e F C e e e = S C ε Fock operators depend on both C e and C p Solve iteratively until self-consistency F C = S C ε p p p p p Problem: Inadequate treatment of correlation

8 NEO-CI Include Correlation Effects e CI p CI N N e p e e p p tot ( r, r ) CII ' I ( r ) I '( r ) I I' Ψ = Φ Φ - Minimize energy with respect to CI coefficients NEO-MCSCF Webb, Iordanov, SHS, JCP 2002; Pak, Swalina, SHS, CP Minimize energy with respect to electronic and nuclear molecular orbitals and CI coefficients - Include all possible CI configurations from chosen electronic and nuclear active spaces (NEO-CASSCF) NEO-MP2 Swalina, Pak, SHS, CPL Use 2 nd -order perturbation theory to calculate electron-electron and electron-proton corrections E = E + E + E NEO-MP2 NEO-HF (2) (2) ee ep

9 Bihalides [XHX], X=F,Cl Swalina and SHS, JPC A 2005; Pak, Chakraborty, SHS, JPCA 2006 R XX R XX R H H coordinate (Angstroms) Single well at equilibrium X X distance, anharmonic potential Experimental and quantum 2D/4D grid data available NEO-MP2 X X distances agree well with 4D VSCF and 2D VCI X X frequencies in reasonable agreement with experiment/grid NEO-MP2 computationally much faster than grid methods Experiment: Kawaguchi, Hirota JCP (1987); Kawaguchi JCP (1988) 2D grid, MP2: Del Bene, Jordan, Spec. Acta. A (1999) 4D VSCF and 2D VCI, MP2 and DFT : Pak, Chakraborty, SHS

10 H Vibrational Frequency: [He-H-He] + Swalina, Pak, Hammes-Schiffer, J. Chem. Phys He nuclei classical (fixed) H quantum mechanical R HeHe = 1.86 Å single well H potential Grid: NEO: NEO full CI H basis center positions variational [1 or 2 centers] NEO vibrational splitting improves with quality of basis set toward 1D grid value Method Splitting (cm -1 ) NEO/2s2p2d(1) 4508 NEO/4s(2) 2229 NEO/4s4p(2) 1655 NEO/4s4p4d(2) D GRID 1400

11 Problem with Standard NEO Approaches Problem Nuclear wavefunctions too localized impacts all properties H vibrational frequencies much too large Physical explanation: X H molecule: X classical, all e and the H + treated QM Grid calculation: electrons are explicitly correlated to H position V ET X { r p } NEO-HF: proton feels average electronic wavefunction NEO-MP2, NEO-CI, and NEO-CASSCF are not effective

12 Solution to Problem with Standard NEO Solution Explicitly-correlated Hartree-Fock with Gaussian-type geminals geminal: basis function that depends on two coordinates rather than a single coordinate Gaussian-type geminal for electron-proton correlation depends on electron-proton distance r ep Improve description V of electron-proton cusp ψ 1 Vep = r 2 e p Improve description ET bexp γ r r ep Quality of wavefunction at small r ep important Include explicit r ep dependence via geminals in total wavefunction

13 Geminals for Electron-Proton Correlation Electron-electron dynamical correlation is often icing on the cake - quantitatively important - repulsive e-e Coulomb interaction Electron-proton correlation is the cake! - qualitatively important - attractive e-p Coulomb interaction V ET Use geminals only for electron-proton dynamical correlation Use MP2 or DFT for electron-electron dynamical correlation Geminals are computationally practical for electron-proton correlation because of small number of quantum protons

14 Electron-Proton Correlation: NEO-XCHF Swalina, Pak, Chakraborty, SHS, JPCA 2006 N Np Ngem e ( e p) e( e) p( p) e p 2 Ψ gem r, r =Φ r Φ r 1+ bk exp γ k ri rj i= 1 j= 1 k= 1 Gaussian-type geminals for electron-proton correlation Only need to include 33 Gaussian geminals Only requires 4-electron (and simpler) integrals Maintains antisymmetry of overall wavefunction ET Approaches correct limit (NEO-HF) as r ep V b k and γ k are constants pre-determined from models Variational method: minimize total energy wrt molecular orbital coefficients Modified Hartree-Fock equations, solve iteratively to self-consistency Related work in electronic structure theory: Szalewicz, Taylor, Manby, Adamowicz, Valeev, Mazziotti, Schaefer, Kutzelnigg, Klopper, Noga, Rassolov

15 NEO-XCHF for Simple Model Model system: e, H + in static field X = +1 charge, infinite mass e and H + treated QM 2 γ r p p e e k ep Ψ = c jφj ciφi 1+ be k j i k X cc-pvdz electronic basis functions H + 5s/5p/4d basis functions Vibrational Frequencies (cm-1) Isotope NEO-HF NEO-full CI NEO-XCHF VSCF H D T NEO-XCHF improves frequencies significantly! NEO-XCHF better than NEO-full CI for relatively l large basis set Frequencies calculated from vibrational splitting

16 NEO-XCHF for [He-H-He] + Chakraborty, Pak, SHS, JCP 2008 He He Electronic basis set: STO-2G Nuclear basis set: 5s 3 Gaussian geminals He nuclei classical (fixed) H quantum mechanical R HeHe = 1.86 Å single well H potential Frequencies in cm 1 with H, D, T for central nucleus Isotope NEO-HF NEO-XCHF 3D Grid H D T Geminals lead to qualitative ti improvement of frequency Frequencies determined by fitting nuclear density along He-He axis to Gaussian

17 Status of NEO-XCHF Proof of concept: provides accurate nuclear densities Density matrix formulation Chakraborty and SHS, JCP express total energy in terms of densities and density matrix - straightforward extension to multiple quantum protons - densities could be obtained from multiconfigurational NEO wavefunction or other type of NEO wavefunction Speed up 3- and 4-particle integrals: ET resolution of identity, direct SCF, V Include electron-electron correlation with MP2, DFT, Advantages: rigorous, ab initio, can improve systematically Disadvantages: computationally expensive, difficult to include electron-electron l t correlation consistently tl

18 Multicomponent DFT DFT with more than one type of quantum particle Hohenberg-Kohn theorem for multicomponent systems: the total energy is a functional of 1-particle densities E = V d + V d + F ρ e p e e p p e p [ ρ, ρ ] ecρ r pcρ r univ[, ρ ] Coulomb interaction with classical nuclei Kohn-Sham formalism Kinetic energy and classical Coulomb E[ ρ, ρ ] = V ρ dr + V ρ dr + E [ ρ, ρ ] e p e e p p e p ec pc ref + E + + e e p p e p xc[ ρ ] E x c[ ρ ] E epc[ ρ, ρ ] Represent densities in terms of KS orbitals in Slater determinants Variational method 2 sets of equations (electronic, nuclear) solved self-consistently tl Parr, JCP1982; Kreibich, Gross, PRL 2001

19 Electron-Proton Density Functional Desired characteristics Compatible with standard electronic functionals Computationally fast Provide accurate nuclear densities Cannot just reparametrize electronic functionals! Strategy: Define e-p functional as E ρ, ρ = dr dr r ρ ( r, r ) dr dr r ρ ( r ) ρ ( r ) e p e p 1 ep e p e p 1 e e p p epc ep ep Use the geminal wavefunction to obtain expression for electron-proton pair density ρ ep (r e,r p ) in terms of one-particle electron and proton densities ρ e (r e ) and ρ p (r p )

20 Strategy for Developing e-p Functional ep e p ρ, ρ, ρ geminal densities Ψgem = (1 + G) Φ Geminal ansatz defines map from auxiliary to geminal densities Use this map to obtain a functional relationship between 1- and 2-particle geminal densities ρ ep e p [ ρ, ρ,...] e ρ [ ρ e p, ρ,...] p e p ρ [ ρ, ρ,...]? e Φ p ρ e p, ρ,... auxiliary densities ep e p ρ = F[ ρ, ρ ] Obtain geminal densities by integration of geminal wavefunction Truncate expressions for geminal densities Make a well-defined approximation that satisfies sum rules

21 Electron-Proton Functional Contribution to total energy: E ρ, ρ = dr dr r ρ dr dr r ρ ρ e p e p 1 ep e p 1 e p epc ep ep Expression for ep pair density in terms of 1-particle densities: ρ = ρ ρ + ρ ρ ρ ρ gn N ep e p e p e p 1 1 e p ep e p ρρ g ρρ ρgn ρ gn + + e p 1 1 e p ρρgn N 1+ ρρgn N e p p 1 e 1 p p e e e p ep e p ep ρρ ρgn ρρ ρ gn e p e 1 e p p 1 e e p p Geminal function for electron-proton explicit correlation: Ngem e p e p = b k γ k k = 1 g( r, r ) exp r r 2

22 NEO-DFT for [He-H-He] + He He He nuclei classical, H nucleus quantum Electronic basis set: STO-2G Nuclear basis set: 5s Frequencies in cm 1 with H, D, and T for the central nucleus Isotope NEO-HF NEO-XCHF 3D Grid NEO-DFT H D T NEO-DFT agrees well with NEO-XCHF and grid method NEO-DFT ~1400 times faster than NEO-XCHF for this system Frequencies determined by fitting nuclear density along He-He axis to Gaussian

23 Status of NEO-DFT Proof of concept: provides accurate nuclear densities Same strategy could be used to design other functionals Analyze the properties of the functionals analytically ll Test the functionals by comparison to NEO-XCHF Both NEO-XCHF and NEO-DFT interfaced to GAMESS Include electron-electron correlation with standard electronic functionals such as B3LYP Combine with QM/MM methods ET V Advantages: computationally efficient, includes electron-proton and electron-electron l t correlation consistently tl Disadvantages: same as for electronic DFT

24 Hydrogen Tunneling Splittings Example: malonaldehyde H H H O O O O O O H C C C H H C C C H H C C C H H H H Experimental tunneling splitting: 21.6 cm -1 NEO-vibronic coupling theory: Transferring hydrogen and electrons treated with NEO Other nuclear modes treated with vibronic coupling theory Dynamics described by vibronic Hamiltonian in approximately diabatic basis of two localized NEO states Hazra, Skone, and SHS, JCP 2009

25 NEO-Vibronic Coupling Theory Expand wavefunction in diabatic basis of two localized NEO wavefunctions Ψ = φ χ e p NEO e p n n( r, r, q) a ( r, r ; q) a( q) a Substitute into Schrödinger equation: n n n Toχa( q) + Wab( q) χb( q) = Enχa( q) b W = d d H e p NEO e p NEO e p ab( q) r r φa ( r,r ; q) NEOφb ( r,r ; q) Expand potential energy matrix elements in 2 nd -order Taylor series Nq Nq Nq i 1 i 2 1 ij ab( q) = Eab + ab i + ωδab i + ab i= 1 2 i= 1 2 i, j= 1 W E q q E qq Calculate l coupling constants t with numerical differentiation (finite it difference) ( ) n Expand χa q in a basis of normal mode harmonic oscillator basis functions i j Solve for E 0 and E 1 to get the tunneling splitting E 1 E 0

26 Procedure for Malonaldehyde 1. Calculate 18 normal mode coordinates and frequencies at TS 2. Define reference geometry (q=0) as average reactant/product 3. Calculate l localized li proton orbitals at reference geometry: fit to grid-based hydrogen vibrational wavefunction 4. Calculate l potential ti energy matrix elements and numerical derivatives, keeping localized proton orbitals fixed 5. Solve vibronic Hamiltonian matrix equation to obtain E 1 -E 0 Geometries and grid-based H wavefunctions: MP2/6-31G** level Localized diabatic states: NEO-HF level

27 Malonaldehyde Results Calculated tunneling splitting: 24.5 cm -1 (experimental: 21.6 cm -1 ) Identified dominant modes coupled to H motion: Symmetric mode: proton donor-acceptor motion Antisymmetric mode: backbone double bond rearrangement Blue solid: diabatic Red dashed: adiabatic

28 Status of NEO-Vibronic Coupling Theory Disadvantages Not as accurate as MCTDH and diffusion Monte Carlo methods* Advantages Provides qualitatively accurate tunneling splittings Relatively low computational cost Enables calculations on larger molecules Identifies dominant modes coupled to H motion reduced dimensionality calculations with more accurate methods Includes nonadiabatic effects between electrons and proton Accuracy can be improved systematically by improving diabatic basis and including higher order coupling terms Malonaldehyde: calculated accurate tunneling splitting and identified dominant modes coupled to H motion * Meyer, Cederbaum, Manthe, Bowman, Carter, McCoy

29 Conclusions NEO approach is implemented in GAMESS NEO-XCHF: includes explicit electron-proton correlation with geminal functions; provides accurate nuclear densities NEO-DFT: electron-proton functional designed from explicitly correlated wavefunction ansatz; treats electron-electron and electron-proton correlation consistently and provides accurate nuclear densities at low computational cost NEO-vibronic i coupling theory: treat t electrons and transferring H with NEO and other nuclear modes with vibronic coupling theory; provides qualitatively accurate tunneling splittings at relatively low computational cost

30 Acknowledgments Simon Webb, Tzvetelin Iordanov, Chet Swalina, Mike Pak, Jonathan Skone, Arindam Chakraborty, Anirban Hazra, Ben Auer Funding: AFOSR, NIH, NSF GAMESS: Mark Gordon, Mike Schmidt