Nuclear-Electronic Orbital Approach:Nuclear Electronic Orbital Approach:Electron-Proton Correlation,

Multicomponent Density FunctionalMulticomponent Density Functional Theory, and Tunneling Splittings

Sharon Hammes-SchifferPenn State University

Nuclear Quantum Effects Important

Zero point energy Hydrogen bonding Hydrogen tunnelingp gyVibrationally excited states

ET

AeApDp HPT

De ET

Proton-coupled electron transferHydrogen transfer in solution and enzymes

PT

Methods for Nuclear Quantum EffectsStandard Born-Oppenheimer approach

e n e n nelec nuctot ( , ) ( ; ) ( )Ψ Ψ Ψ≈r r r r rApproximate wavefunction :

n n ne

e eeleclec elec

n n nnuc el

elec

c n ce nu uc

( ; ) ( ) ( ; )

( ) ( ) ( )

H E

T E E

Ψ =

⎡ ⎤+ =⎦ Ψ Ψ⎣

Ψr r r r r

r r r

Electronic equation :

Nuclear equation :

Nuclear-electronic orbital (NEO) approach:treat specified protons QM on same level as electrons

c c cNEO NEO

e p e pNEO NEO( , ; ) ( ) ( , ; )H E=Ψ Ψr r r r r r r

e p c, , :r r r electrons, quantum protons, other nucleip :r transferring proton

c :r all other nuclei

Nuclear-Electronic Orbital (NEO) Method

• Solution of mixed nuclear-electronic time-independentSchrödinger equation with molecular orbital methods

Webb, Iordanov, and SHS, JCP 117, 4106 (2002)

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 expandedElectronic and nuclear MO s expanded in Gaussian basis sets

• Energy minimized with respect to all MO’s and centers f l b i f tiof nuclear basis functions

• Correlation among electrons and nuclei included with multiconfigurational, perturbation, and DFT approachesg , p , pp

• Provides structures, energies, minimum energy paths,and direct dynamics for chemical reactions

Advantages of NEO• Nuclear quantum effects incorporated during

electronic structure calculationsBorn Oppenheimer separation of• Born-Oppenheimer separation ofelectrons 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• Accuracy may be improved systematically• Related work

Tachikawa, Nakai, Shigeta, Gross, Jungwirth, Krylov, Sherrill, Valeev

Major challenges:• Electron-proton correlation is highly significantp g y g• Other modes may be strongly coupled to H motion

Nuclear-Electronic Hamiltoniane e c e

2e c eNEO e

1 12 | | | |

N N N NA

ii i A i ji A i j

H Z>

− ∇ − +− −

= ∑ ∑∑ ∑r r r r Electronic terms

p p pc2

p c ppp

1 12 | | | |

| | | |i i A i j

N N NNA

ii i A i ji A i j

i A i j

Zm ′

′ ′ ′ ′>′ ′

>

′

− ∇ + +− −∑ ∑∑∑ rr r r Nuclear terms

ep

e p

p | | | |

| |1

i i A i ji A i j

N N

i i i i

>

′ ′−−∑∑ r r

Nuclear-Electronic interaction term| |i i i i′ ′r r

e p c, ,N N N Number of electrons, quantum nuclei, and classical nucleie p c, ,i i′r r r Coordinates of electrons, quantum nuclei, and classical nuclei

NEO-HF (Hartree-Fock)• HF wavefunction

0, :Φ Φe p0 Slater determinantse p e e p p

tot 0 0( , ) ( ) ( )Ψ = Φ Φr r r r

• HF energye e p p e e p p( ) ( ) ( ) ( )= Φ Φ Φ Φr r r rE H

• Expand electronic, nuclear MO’s in Gaussian basis setss type p type d type Gaussian basis functions

0 0 NEO 0 0( ) ( ) ( ) ( )= Φ Φ Φ Φr r r rE H

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 protonsq q p

Fock operators depend on both Ce and Cp

Solve iteratively until self-consistencye e e e e

p p p p p

=

=

F S C εF S CC εC

Problem: Inadequate treatment of correlation

Include Correlation Effects• NEO-CI

peCI CI

e p e e p p( ) ( ) ( )NN

CΨ = Φ Φ∑∑r r r r

- Minimize energy with respect to CI coefficientsNEO MCSCF W bb I d SHS JCP 2002 P k S li SHS CP 2004

tot ' ''

( , ) ( ) ( )II I II I

CΨ = Φ Φ∑∑r r r r

• NEO-MCSCF Webb, Iordanov, SHS, JCP 2002; Pak, Swalina, SHS, CP 2004

- Minimize energy with respect to electronic andnuclear molecular orbitals and CI coefficientsnuclear molecular orbitals and CI coefficients

- Include all possible CI configurations from chosenelectronic and nuclear active spaces (NEO-CASSCF)

• NEO-MP2 Swalina, Pak, SHS, CPL 2005

- Use 2nd-order perturbation theory to calculate electron electron and electron proton correctionselectron-electron and electron-proton corrections

NEO-MP2 NEO-HF (2) (2)ee epE E E E= + +

Bihalides [XHX]−, X=F,ClS li d SHS JPC A 2005 P k Ch k b t SHS JPCA 2006Swalina and SHS, JPC A 2005; Pak, Chakraborty, SHS, JPCA 2006

R

RXX

RXX -0.3 -0.2 -0.1 0 0.1 0.2 0.3H coordinate (Angstroms)RH

• Single well at equilibrium X−X distance, anharmonic potential• Experimental and quantum 2D/4D grid data availablep q g• 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• 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)2D grid, MP2: Del Bene, Jordan, Spec. Acta. A (1999)4D VSCF and 2D VCI, MP2 and DFT : Pak, Chakraborty, SHS

H Vibrational Frequency: [He-H-He]+S li P k H S hiff J Ch Ph 2005Swalina, Pak, Hammes-Schiffer, J. Chem. Phys. 2005

Grid: NEO:• He nuclei classical (fixed)( )• H quantum mechanical• RHeHe = 1.86 Å• single well H potential• single well H potential

Method Splitting (cm-1)

NEO/2s2p2d(1) 4508

• NEO–full CI• H basis center positions

i i l [1 2 ] NEO/4s(2) 2229

NEO/4s4p(2) 1655

variational [1 or 2 centers]• NEO vibrational splitting improves with quality of basis set toward

NEO/4s4p4d(2) 1579

1D GRID 1400

q y1D grid value

Problem with Standard NEO ApproachesProblem• Nuclear wavefunctions too localized → impacts all properties• H vibrational frequencies much too large• H vibrational frequencies much too largePhysical explanation: X−H molecule: X classical, all e− and the H+ treated QM,• Grid calculation: electrons are explicitly correlated to H position

ETV

X{ }p← →r

• NEO-HF: proton feels average electronic wavefunction• NEO-MP2, NEO-CI, and NEO-CASSCF are not effective

Solution to Problem with Standard NEOSolution• 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 correlationGaussian-type geminal for electron-proton correlation depends on electron-proton distance rep

2⎡ ⎤ Improve description ETV2e pexpb γ⎡ ⎤− −⎢ ⎥⎣ ⎦

r r Improve descriptionof electron-proton cusp ψ

1V = −

• Quality of wavefunction at small r important

pep

e

Vr

= −

Quality of wavefunction at small rep important• Include explicit rep dependence via geminals in total wavefunction

Geminals for Electron-Proton Correlation

• Electron-electron dynamical correlationis often icing on the cakeis often icing on the cake- quantitatively important- repulsive e-e Coulomb interactionElectron proton correlation is the cake!• Electron-proton correlation is the cake!- qualitatively important- attractive e-p Coulomb interaction

ETV

• Use geminals only for electron-proton dynamical correlation• Use MP2 or DFT for electron-electron dynamical correlation

Geminals are computationally practical for electron-protoncorrelation because of small number of quantum protons

Electron-Proton Correlation: NEO-XCHFS li P k Ch k b t SHS JPCA 2006

( ) ( ) ( )p geme 2e p e e p p e p

gem1 1 1

, 1 expN NN

k k i ji j k

b γ⎧ ⎫⎪ ⎪⎡ ⎤Ψ = Φ Φ + − −⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭

∑∑∑r r r r r r

Swalina, Pak, Chakraborty, SHS, JPCA 2006

1 1 1i j k= = = ⎣ ⎦⎪ ⎪⎩ ⎭

• Gaussian-type geminals for electron-proton correlation• Only need to include ∼3 Gaussian geminalsOnly need to include 3 Gaussian geminals• Only requires 4-electron (and simpler) integrals• Maintains antisymmetry of overall wavefunction

ETV• Approaches correct limit (NEO-HF) as rep → ∞• bk and γk are constants pre-determined from models• Variational method: minimize total energy wrt molecular• 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

NEO-XCHF for Simple Model• Model system: e−, H+ in static field• X = +1 charge, infinite mass• e− and H+ treated QM

X •e and H treated QM

2

Ψ 1 k epγ rp p e ej j i i k

j i k

c φ c φ b e−⎛ ⎞⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠⎝ ⎠∑ ∑ ∑

cc-pVDZ electronic basis functions

H+ 5s/5p/4dbasis functions

Vibrational Frequencies (cm-1)

Isotope NEO-HF NEO-full CI NEO-XCHF VSCF

H 3621 2263 1649 1645D 2729 1707 1081 1177

NEO-XCHF improves frequencies significantly!NEO XCHF b tt th NEO f ll CI f l ti l l b i t

T 2302 1457 963 966

NEO-XCHF better than NEO-full CI for relatively large basis set

Frequencies calculated from vibrational splitting

NEO-XCHF for [He-H-He]+

He He

Chakraborty, Pak, SHS, JCP 2008

• He nuclei classical (fixed)

Electronic basis set: STO-2GNuclear basis set: 5s3 Gaussian geminals

• H quantum mechanical• RHeHe = 1.86 Å• single well H potential3 Gaussian geminals

Frequencies in cm−1 with H, D, T for central nucleus

single well H potential

Isotope NEO-HF NEO-XCHF 3D GridH 3759 1030 1107D 2738 725 783

G i l l d t lit ti i t f f

D 2738 725 783T 2274 558 639

• Geminals lead to qualitative improvement of frequencyFrequencies determined by fitting nuclear density along He-He axis to Gaussian

Status of NEO-XCHF• Proof of concept: provides accurate nuclear densities• Density matrix formulation Chakraborty and SHS JCP 2008Chakraborty and SHS, JCP 2008- express total energy in terms of densities and density matrix- straightforward extension to multiple quantum protons

d iti ld b bt i d f lti fi ti l NEO- densities could be obtained from multiconfigurational NEOwavefunction or other type of NEO wavefunction

• Speed up 3- and 4-particle integrals:ETV

p p p gresolution of identity, direct SCF, …

• Include electron-electron correlation with MP2, DFT,…

Advantages: rigorous, ab initio, can improve systematicallyDisadvantages: computationally expensive, difficult to include l t l t l ti i t tlelectron-electron correlation consistently

Multicomponent DFT• DFT with more than one type of quantum particle • Hohenberg-Kohn theorem for multicomponent systems:

f fthe total energy is a functional of 1-particle densitiese p e e p p

eC pCe p

univ[ , ] [ , ]E V d V d F ρρ ρρ ρ ρ= + +∫ ∫r r

• Kohn-Sham formalism

Coulomb interaction with classical nuclei

Kinetic energy and “classical” Coulomb

e p e e p p e peC pC ref

e e p pxc x

e pepcc

[ , ] [ , ]

[[ ][ ,] ]

E V d V d E

E EE ρ ρ

ρ ρ ρ ρ ρ ρ

ρ ρ

= + +

+ + +∫ ∫r r

Represent densities in terms of KS orbitals in Slater determinantsVariational method → 2 sets of equations (electronic, nuclear)

l d lf i t tl

xc x epcc

solved self-consistentlyParr, JCP1982; Kreibich, Gross, PRL 2001

Electron-Proton Density FunctionalDesired characteristics• Compatible with standard electronic functionals• Computationally fast• Computationally fast• Provide accurate nuclear densities

Cannot just reparametrize electronic functionals!

Strategy: gy• Define e-p functional as

e p e p 1 ep e p 1 e pepc ep ep

e p e p, ( , ) ( ) ( )E rd drd dρ ρ ρ ρ ρ− −⎡ ⎤ = −⎣ ⎦ ∫ ∫r rr rr rr r

• Use the geminal wavefunction to obtain expression for electron-proton pair density ρep(re,rp) in terms of one-particlel t d t d iti d

⎣ ⎦ ∫ ∫

electron and proton densities ρe(re) and ρp(rp)

Strategy for Developing e-p Functionale p

gem (1 )G+ ΦΨ Φ=

• Geminal ansatz defines map from auxiliary to geminal densities

ep e p, ,ρ ρ ρ e p, ,...ρ ρauxiliary densitiesgeminal densities

• Geminal ansatz defines map from auxiliary to geminal densities• Use this map to obtain a functional relationship between 1- and 2-particle geminal densities

e p

e

ep

e p

[ ][

, , ..., , ...]

ρ

ρ

ρ ρ

ρ ρ ep e p[ ],Fρ ρ ρ=?

p e p,[ , ...]ρρ ρ• Obtain geminal densities by integration of geminal wavefunction• Truncate expressions for geminal densities• Make a well-defined approximation that satisfies sum rules

Electron-Proton Functional

Contribution to total energy:e p e p 1 ep e p 1 e pE d d d d− −⎡ ⎤⎣ ⎦ ∫ ∫

Expression for ep pair density in terms of 1-particle densities:

e p e p 1 ep e p 1 e pepc ep ep,E d d dr rdρ ρ ρ ρ ρ⎡ ⎤ = −⎣ ⎦ ∫ ∫r r r r

ep e p e p e p 1 1e p ep

e p p 1 e 1e pp p e e

gN N

gN gNg

ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρρ ρ

− −

− −

⟨ ⟩

⟨ ⟩ ⟨ ⟩

= +

p p e ee p 1 1 e p 1 1

e p ep e p ep

e p e 1 e p p 1e e p p

1 1

g gggN N gN N

gN gN

ρ ρ ρ ρρ ρρ ρ ρ ρ

ρ ρ ρ ρ ρ ρ

− − − −

− −

⟨ ⟩ ⟨ ⟩

⟨ ⟩ ⟨ ⟩

⟨

+ ++ +

− ⟩ − ⟨ ⟩

N

Geminal function for electron-proton explicit correlation:

e e p pg gρ ρ ρ ρ ρ ρ⟨ ⟩ ⟨ ⟩

gem 2e p e p

1

( , ) expN

k kk

bg γ=

⎡ ⎤−⎢ ⎥⎣ ⎦= −∑r r r r

NEO-DFT for [He-H-He]+

He He

He nuclei classical, H nucleus quantumElectronic basis set: STO-2GNuclear basis set: 5s

Isotope NEO-HF NEO-XCHF 3D Grid NEO-DFT

Frequencies in cm−1 with H, D, and T for the central nucleus

Isotope NEO HF NEO XCHF 3D Grid NEO DFTH 3759 1030 1107 1072D 2738 725 783 770

• NEO-DFT agrees well with NEO-XCHF and grid method

T 2274 558 639 630

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

Status of NEO-DFT• Proof of concept: provides accurate nuclear densities• Same strategy could be used to design other functionals

A l th ti f th f ti l l ti ll• Analyze the properties of the functionals analytically• 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 ETV

Combine with QM/MM methods

Advantages: computationally efficient, includes electron-proton d l t l t l ti i t tland electron-electron correlation consistently

Disadvantages: same as for electronic DFT

Hydrogen Tunneling Splittings Example: malonaldehyde

O OH

O OH

OOH

CC

H

H HC C

CC

H

H HC

CC

H

HH

Experimental tunneling splitting: 21.6 cm-1

NEO-vibronic coupling theory: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• Dynamics described by vibronic Hamiltonian in approximatelydiabatic basis of two localized NEO states

Hazra, Skone, and SHS, JCP 2009

NEO-Vibronic Coupling Theory• Expand wavefunction in “diabatic” basis of two localized NEO wavefunctions

e p NEO e p( , , ) ( , ; ) ( )nn a a

a

φ χΨ =∑r r q r r q q

• Substitute into Schrödinger equation:

( ) ( ) ( ) ( )n n no a ab b n a

bT W Eχ χ χ+ =∑ q qq q

• Expand potential energy matrix elements in 2nd-order Taylor series

NEO NEONEO( ) ( ) ( )ab a bW d d Hφ φ= ∫ e p e p e pr r r ,r rq; ,r ;q q

C l l t li t t ith i l diff ti ti (fi it diff )1 1 ,

2

1

1 1(2 2

)q q q

i ijab ab i ab

N N Nia i abb

i i i ji jW E q q E q qE ω δ

= = =

+ + += ∑ ∑ ∑q

• Calculate coupling constants with numerical differentiation (finite difference)

• Expand in a basis of normal mode harmonic oscillator basis functions( )naχ q

• Solve for E0 and E1 to get the tunneling splitting E1−E0

Procedure for Malonaldehyde1. Calculate 18 normal mode coordinates and frequencies at TS2. Define reference geometry (q=0) as average reactant/product3 C l l t l li d t bit l t f t3. Calculate localized proton orbitals at reference geometry:

fit to grid-based hydrogen vibrational wavefunction

4 C l l t t ti l t i l t d i l4. Calculate potential energy matrix elements and numericalderivatives, keeping localized proton orbitals fixed

5. Solve vibronic Hamiltonian matrix equation to obtain E1-E0

Geometries and grid-based H wavefunctions: MP2/6-31G** levelLocalized diabatic states: NEO-HF level

Malonaldehyde ResultsCalculated 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

Status of NEO-Vibronic Coupling TheoryDisadvantagesNot as accurate as MCTDH and diffusion Monte Carlo methods*Advantagesg• Provides qualitatively accurate tunneling splittings• Relatively low computational cost• Enables calculations on larger molecules• Enables calculations on larger molecules• Identifies dominant modes coupled to H motion → reduceddimensionality calculations with more accurate methodsI l d di b ti ff t b t l t d t• Includes nonadiabatic effects between electrons and proton

• Accuracy can be improved systematically by improving diabaticbasis and including higher order coupling termsg g p g

Malonaldehyde: calculated accurate tunneling splitting andidentified dominant modes coupled to H motionidentified dominant modes coupled to H motion

* Meyer, Cederbaum, Manthe, Bowman, Carter, McCoy…

Conclusions• NEO approach is implemented in GAMESS• NEO-XCHF: includes explicit electron-proton correlation

fwith geminal functions; provides accurate nuclear densities• NEO-DFT: electron-proton functional designed from explicitlycorrelated wavefunction ansatz; treats electron-electron andcorrelated wavefunction ansatz; treats electron electron and electron-proton correlation consistently and provides accuratenuclear densities at low computational costNEO ib i li th t t l t d t f i H• NEO-vibronic coupling theory: treat electrons and transferring Hwith NEO and other nuclear modes with vibronic coupling theory;provides qualitatively accurate tunneling splittings atp q y g p grelatively low computational cost

Acknowledgments

Simon Webb, Tzvetelin Iordanov, Chet Swalina, Mike Pak,Jonathan Skone, Arindam Chakraborty, Anirban Hazra, Ben Auer

Funding: AFOSR, NIH, NSFGAMESS: Mark Gordon, Mike Schmidt