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Accurate wavefunctions for multi-sorted quantum systems
Michael Pak University of Illinois at Urbana-Champaign
Multi-sorted quantum systems
• Quantum Chemistry:
• Multi-sorted Quantum Systems:
H = −12me
∇i2
i
Ne
∑ −ZA
| rie − RA
c |A
Nc
∑i
Ne
∑ +1
| rie − r j
e |i> j
Ne
∑
H = −12me
∇i2
i
Ne
∑ −ZA
| rie − RA
c |A
Nc
∑i
Ne
∑ +1
| rie − r j
e |i> j
Ne
∑
−12mp
∇i '2
i '
Np
∑ −ZA
| ri 'p − RA
c |A
Nc
∑i '
Np
∑ +1
| ri 'p − r j '
p |i '> j '
Np
∑
−1
| rie − r $i
p |i
Ne
∑$i
Np
∑
“Electronic” terms
“Nuclear” terms
“Nuclear-Electronic” interaction term
Multi-sorted quantum systems
me ~ mp Type I: positronic chemistry, muonized atoms and molecules
me < < mp Type II: “Chemically” non-adiabatic molecular systems
e: electrons p: positron/muon/etc. Np= 1
e: electrons p: quantum nuclei Np= 1
e p n
quantum classical
Ψ re | R( )→Ψ re,Rp | R( )
• Mean-field (HF) wavefunction: HF energy: • Configuration interaction (CI) / MCSCF
CI/MCSCF energy: • Perturbation theory (MP2) Use 2nd-order perturbation theory to calculate electron-electron and electron-proton corrections
“Imitating” Quantum Chemistry: mean-field based approaches
0, :Φ Φe p0 Slater determinantsΨHF (re ,rp ) =Φ0
e (re )Φ0p (rp )
EHF = Φ0e (re )Φ0
p (rp ) H Φ0e (re )Φ0
p (rp )
ΨCI (re ,rp ) = CII 'Φ Ie (re )Φ I '
p (rp )I '
NCIp
∑I
NCIe
∑
EMP2 = EHF + Eee(2) + Eep
(2)
ECI = ΨCI (re ,rp ) H ΨCI (re ,rp )
Testing electron-positron wavefunctions
• Positron binding energies - no binding predicted for most systems: e.g., Li, Be, Na, Mg atoms, LiH molecule etc. - accounting for the Ps dissociation channel is problematic
• Positron annihilation rate - direct measure of quality of e/e+ wavefunctions -
• Good reference points: - experimental: e+ binding energies and annihilation rates - computational: high accuracy results for small few-electron systems (QMC,SVM, SVM+ECG,,etc.)
Ψ(r1, r2, rp ) = A12 Cijklmnr1ir2
jrpkr1p
l r2 pm r12
ne(−α12
v r12−α1pv r1p−α2 p
v r2 p )v∑
i, j,k,l,m,n∑
Ψ(r1, r2, rp ) = A12 exp(a1r1 + c1r1
2
1+ b1r1+a2r2 + c2r2
2
1+ b2r2+aprp + cprp
2
1+ bprp+a12r12 + c12r12
2
1+ b12r12+a1pr1p + c1pr1p
2
1+ b1pr1p+a2 pr2 p + c2 pr2 p
2
1+ b2 pr2 p)
Ψ (r1e ,r2
e ,rp ) → ρ ep r( ) = Ψ (r1e ,r2
e ,rp ) | δ rie − r( )
i=1
N
∑ δ r p − r( ) |Ψ (r1e ,r2
e ,rp ) → λ 2γ =C ρ ep r( )dr∫
Bound Positrons – Annihilation Rate
>Ψ−Ψ<>=Ψ−Ψ<= ∑∑==
|)(||)(ˆ|41
20
1
20
ee N
i
pei
N
i
pei
Sep crScr rrrr δπδπλ
r0 =α! /mec
0)(ˆ)(ˆ Ω=Ψ ++ qapbi
0)(ˆ)(ˆ Ω−+=Ψ ++ kqpckcf
crH if202 4)0,0(ˆ πσ γ →ΨΨ=
))]3()2()3()2()(1()[,,(ˆ),,( 32123321 αββαα −=Ψ=Ψ rrrfAxxxi
032321321321 )],(ˆ),(ˆ),(ˆ),(ˆ)[,(ˆ),,( Ω↑↓−↓↑↑=Ψ +++++∫ papapapapbpppfdpdpdpi
)]()'()['(),,('ˆ2/1ˆ)( 213321321232 pppppppppfdpdpdpdpAH if −−−=ΨΨ= ∫∑ δδχσχλχ
γ
021 ),(ˆ),(ˆ),'(ˆ)'(' Ω−+↑=Ψ ∫ +++ ekqpcekcpapdpf χ
∫ ===
rrrrrdr
21),(2/1 212 ρσλ γ
q
p
∑=
ʹ′ʹ′=N
iii
HF Scr1
11202πλ
rrrrr dS jijijiij ∫ ʹ′ʹ′ʹ′ʹ′ = )()()()( φφφφ
∑ʹ′
ʹ′ʹ′ʹ′ʹ′ −−+
ʹ′ʹ′−=
rarrar
rra
Srra
cr 11
20
)1( 14
εεεεπλ
λFCI = 2πr02c
CI !I CI !JI !J∑
I !I∑ SIiIi !I p !Jp
i=1
N
∑ +
2 CI !I CJ !J SIiJ j !I p !JpI !I <J !J∑
#
$
%%%%
&
'
(((( 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
[6s]
[6s1p
][6s
2p]
[6s3p
]
[6s2p
1d]
[6s3p
1d]
Basis Setи
(ns-
1)
HF
FCI
MP2
HF
MP2
FCI
6s 6s1p 6s2p1d 6s3p1d 0.2972 0.2976 0.2978 0.2977 0.3490 0.4018 0.5287 0.5376 0.3723 0.6304 0.8662 0.8993
HFλ2MPλ
FCIλ
PsH Annihilation Rate
Dynamical correlation • Electron-electron dynamical correlation: - defined as difference between exact and mean-field answers E.g. for the energy: - mathematically is (mostly) due to deviation of from when - typically is ‘icing on the cake’: e-e Coulomb interaction is repulsive… • Electron-proton correlation is the cake! - qualitatively important: e-p Coulomb interaction is attractive
Ecorr = Ψexact H Ψexact − ΨHF H ΨHF
Ψexact ΨHF | rie − r j
e | → 0
ΨXCHF xe ,xp( ) =Φe xe( )Φp xp( ) 1+ g rie ,r j
p( )j=1
Np
∑i=1
Ne
∑$%&
'&
()&
*&
• Gaussian-type geminals for electron-positron correlation • bk and γk are constants pre-determined from models • Variational method: minimize total energy wrt molecular orbital coefficients → Modified Hartree-Fock equations, solve iteratively to self-consistency
g rie ,r j
p( ) = bk exp −γ k rie − r j
p 2"#$
%&'k=1
Ngem
∑Gaussian-type geminals:
Explicitly correlated wavefunctions
ΨRXCHF-fe xe ,x p( ) =χ p x p( )N !
χ1e x1
e( )g r1e ,r p( ) χ2e x1
e( ) ! χNe x1
e( )χ1e x2
e( )g r2e ,r p( ) χ2e x2
e( ) ! χNe x2
e( ). ! " !
χ1e xN
e( )g rNe ,r p( ) χ2e xN
e( ) ! χNe xN
e( )
Method E λ2γ(ns-1)
HF -0.664337 0.3189
XCHF-1G -0.693216 0.6883
XCHF-2G -0.705863 1.1402
XCHF-3G -0.712496 2.0122
XCHF-4G* 0.716097 2.0443
FCI -0.758965 0.8993
SVM -0.789198 2.4714
PsH Annihilation Rate from NEO XCHF
Basis sets: HF. XCHF: 6s FCI: 6s2p1d
2
, 01
( , ) | ( ) ( , )e
e p
N
ep e p x x x e i p e p ei
x x dx dλ ρ δ= ==
= − Ψ =∑∫ ∫ r r r r r:
1 2 31
( , ) ( , , ) ( , , , )e e eN N N
i p i j p i j k pi i j i j k
x x x x x x x x x= ≠ ≠ ≠
Ψ Ω Ψ + Ψ Ω Ψ + Ψ Ω Ψ∑ ∑ ∑2
1( , ) 2i p ip ip ip ip ipx x g gδ δ δΩ = + +
3( , , , )i j k p ip jp kpx x x x g g δΩ =
Ω2 (xi ,x j ,xp ) = 2gipδ jp + gip2δ jp
+2gipg jpδ jp
( ) ( ) ( ) ( )gem 2
, ,1 1 1
, 1 exppe N NN
e p e e p p k k e i p ji j k
b γ= = =
⎧ ⎫⎪ ⎪⎡ ⎤Ψ =Ψ Ψ + − −⎨ ⎬⎢ ⎥⎣ ⎦⎪ ⎪⎩ ⎭∑∑∑r r r r r r
Two-Photon Annihilation Rates
• HF very inaccurate • XCHF better but still inaccurate • RXCHF-ne/ae perform similarly and agree well with highly accurate ECG/SVM results • Electronic and positronic densities are also in decent agreement • RXCHF is 235 (25) times faster than XCHF for e+Li (LiPs)
aMitroy, Phys. Rev. A 70, 024502 (2004). bRyzhikh, Mitroy, Varga, J. Phys. B 31, 3965 (1998).
cStrasburger, J. Chem. Phys. 111, 10555 (1999).
e+Li LiPs e+LiH NEO-HF 3.3556×10-4 0.0512 0.03 XCHF 1.7361 0.566 0.62
RXCHF-ne 1.6759 1.940 1.08 RXCHF-ae 1.6657 1.940 1.11 SVM/ECG 1.7512a 2.107b 1.26c
Units: ns-1
Conclusions: Electron-positron wavefunctions
• No usable method for computing accurate wavefunctions is available: - highly accurate methods of ECG+SVM type are not practical for systems with more than 5-10 electrons. ((NEO-XCHF/RXCHF is perhaps in the same class)) - methods easily applied to larger positronic molecules (HF, MP2, CI) are utterly unreliable even for systems with 3 electrons or less. - trivial example: experimentally, e+ binds to propane. No known theoretical method can predict this binding… • The field is wide open, and in need of new ideas…
• PET applications
Multi-sorted quantum systems: electron-nuclear wavefunctions
• 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 - retain at least two classical nuclei • Easy access to ‘exact’ answer is available for benchmarking electron-
nuclear orbital methods p :r quantum proton
c :r all other nuclei
Electron-proton wavefunctions: how accurate is BO approximation?
Ψktot re,rp | R( ) = ciµk Ψi
elec re | rp,R( )Ψiµprot rp | R( )
i,µ∑
Hiµ, jν = δijδµνεµi( ) R( ) −
!2
mpΨiµprot | dij
ep( ) ⋅ ∇rp | Ψ jνprot
p−!2
2mpΨiµprot | gij
ep( ) | Ψ jνprot
p
dijep( ) = Ψi
elec | ∇rp | Ψ jelec
egijep( ) = Ψi
elec | ∇rp2 | Ψ j
elece
• Expand exact solution in double-adiabatic basis
• Need to evaluate non-adiabatic coupling terms:
Phenoxyl-phenol: electronically non-adiabatic
Benzyl-toluene: electronically adiabatic
C000 = .9973
C000 = .9997
-2 -1 1 2
0.2
0.4
0.6
0.8
1.0
1.2
HF
Electron-proton wavefunctions: what is ‘accurate’?
exact
• Isotope Effects on Geometries
X HX DX TX
F 0.9186 0.9134 0.9110
Cl 1.2884 1.2815 1.2785
Br 1.4235 1.4165 1.4134
Bond lengths in Å
• NEO-HF → As mass increases, - bond length decreases - magnitude of negative partial charge on X decreases • Same trends observed for H2O, NH3, H3O+
• Proton density distribution:
typically, HF frequency is off by 200% or more
He He
Nuclear Quantum Effects and H-bonding • Large amplitude, anharmonic zero-point motion
Prism Cage
H5O2+ Hydrogen fluoride
• Clusters: increase H-bond donor–acceptor distances - (H2O)n=2-6 → QDMC: Clary, CR 2000 - (HF)2 → Experiment: Klemperer, JCP 1984 • Liquids: decrease H-bond donor–acceptor distances - H2O, HF → PICPMD: Klein, JACS 2003, PRL 2004
• Zero point energy effects: – bending-type modes → increase D–A distance – stretching modes → decrease D–A distance
Proton-Containing Systems with RXCHF
• 14-electron system • Explicitly correlate two electronic
orbitals to proton (CH bond orbital) Frequency (cm-1)
HF 5077 NEO-RXCHF-ne 3604 NEO-RXCHF-ae 3476 3D grid 3544 black: grid (adiabatic)
blue: NEO-HF red: RXCHF-ne green: RXCHF-ae
N ≡ C – H
F – H – F-
Frequency (cm-1) NEO-HF 4614 NEO-RXCHF-ne 3348 NEO-RXCHF-ae 2616 3D grid 2639
• 20-electron system • Explicitly correlate four electronic
orbitals to proton (FH bond orbitals)
Problems with explicitly correlated methods for BO systems
ϕi (r){ }i∈ℜ ∈ L2[r] ϕ(r,R)∈ L2[r⊗ R]
0.5 1.0 1.5 2.0 2.5 3.0
0.5
1.0
1.5
electron density: BO(solid line) vs. RXCHF (dotted line)
),( 21 rrΨφ(r1)χ (r2 ) | r1 − r2 |>> 0
Ce−γ |r1−r2| | r1 − r2 |→ 0Φe xe( )Φp xp( ) 1+ g ri
e ,r jp( )
j=1
Np
∑i=1
Ne
∑#$%
&%
'(%
)%
HR[x]= T[x]+ V[x]+1
x − RΨR (x)
Description of Bilobal Wavefunctions
HF: variational solution is always localized In practice, so is CI/CASSCF Delocalization: at least FCI in a very large basis set
Pak and SHS, PRL 2004
D A-H D- A H
• TS for H transfer reactions corresponds to equal H probability near donor and acceptor
Model system studies (analytical proof)
Conclusions • Electron-positron correlation is too large to make HF a viable reference. • Post-HF (CI,MP2) methods fail to produce reasonably accurate electron-positron wavefunctions. • Explicitly correlated methods do somewhat better, but are very expensive.
• New approaches are needed to describe positron behavior in large molecules.
• BO ground state electron-nuclear wavefunctions are almost exact even for chemically non-adiabatic systems.
• Orbital-based methods fail to reproduce the shape and size of exact nuclear density. This is true, to some extent, even for explicitly-correlated approaches.
• This basic failure probably implies that the functional dependence of electronic wavefunction on nuclear coordinate is described incorrectly.
• Producing bilobal proton wavefunctions (in a double-well potential) with nuclear-electron orbital methods is essentially an unsolved problem.
Acknowledgments Sharon Hammes-Schiffer (PSU, UIUC) Simon Webb Andrès Reyes Arindam Chakraborty Kurt Brorsen Tzvetelin Iordanov Chet Swalina Jonathan Skone Andrew Sirjoosingh Tanner Culpitt Funding: AFOSR, NSF NEO method is incorporated into GAMESS
Extra: Topology of nodal surfaces
• Diffusion MC (DMC) depends on the fixed-node approximation. • Topology of the nodal surface of multi-sorted wavefunctions responds to the level of correlation in an unknown way. • Nodal surface of an N+M-particle system is an embedded manifold (in R3N+3M). • Alternatively, after a polynomial approximation, it is an algebraic
variety over R.
• We need a computable pathway from the embedding function to some suitable topological characteristic (e.g. Betti numbers for some cohomology theory) of the surface.
• Example: de Rham cohomology groups for (a complement to) an affine variety over C can be computed from the polynomial coefficients of the variety.
Toshinori Oaku, Nobuki Takayama, Journal of Pure and Applied Algebra, 139 (1999) 201
Multicomponent DFT
Parr, JCP1982
Hohenberg-Kohn theorem has been proven for multi-component systems: functional exists for electronic and nuclear 1-particle densities
Kreibich, Gross, PRL 2001
)(),( rr Ne ρρ )(rVext
)(rVext),...,(),( 1 Ne RRr Γρ
Strategy for Developing e-p Functional
• Geminal ansatz defines map from auxiliary to geminal densities • Use this map to obtain a functional relationship between 1- and 2-particle geminal densities
Ψgem = (1+G)ΦeΦp!
ep e p, ,ρ ρ ρ !ρ e , !ρ p ,...auxiliary densities geminal densities
ρ ep[ !ρ e , !ρ p ,...]ρ e[ !ρ e , !ρ p ,...]ρ p[ !ρ e , !ρ p ,...]
ρ ep = F[ρ e ,ρ p ]
• Obtain geminal densities by integration of geminal wavefunction • Truncate expressions for geminal densities • Make a well-defined approximation that satisfies sum rules
?
Chakraborty, Pak, SHS, PRL 2008
Electron-Proton Functional
gem 2e p e p
1( , ) exp
N
k kkbg γ
=
⎡ ⎤−⎢ ⎥⎣ ⎦= −∑r r r rGeminal function:
Expression for ep pair density in terms of 1-particle densities:
Contribution to total energy:
e p e p 1 ep e p 1 e pepc ep ep,E d d dr d rρ ρ ρ ρ ρ− −⎡ ⎤ = − +⎣ ⎦ ∫ ∫r r r r
Motivating work: Colle and Salvetti, 1975, developed an electronic functional
e p e p 1 1 p
ep e p e p e p 1 1e p ep
e p e 1 e p p 1e e
ee p p e
1 1 e pe p e
p
p
p
1
gN N
g N N g gN N g
gN gN
ρ ρ ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ
ρ ρ
− −
− −
− −
− −
= +
+ 〈 〉 〈 〉+
+ 〈
〈 〉
〈 〈− 〉 − 〉
〉
( ) egem
p1 G+ ΦΨ Φ=
NEO-DFT for [He-H-He]+
He nuclei classical, H nucleus quantum Electronic basis set: STO-2G Nuclear basis set: 5s
• NEO-DFT agrees well with NEO-XCHF and grid method • NEO-DFT ~1400 times faster than NEO-XCHF for this system
He He
Frequencies determined by fitting nuclear density along He-He axis to Gaussian
Isotope NEO-HF NEO-XCHF 3D Grid NEO-DFT H 3759 1030 1107 1072 D 2738 725 783 770 T 2274 558 639 630
NEO-HF
NEO-DFT
3D Grid
Frequencies in cm−1 with H, D, and T for the central nucleus
XCHF vs. RXCHF: Physical Assumptions • XCHF correlates all electrons with positron using same geminal parameters → geminal functions are used to account for interactions other than short-ranged electron-positron dynamical correlation → wavefunction is not optimized to accurately describe this specific interaction but rather is more globally optimized • RXCHF correlates only one electron with positron using geminal parameters optimized for a single e-/e+ interaction → geminal functions are used to account mainly for the short-ranged electron-positron dynamical correlation → wavefunction produces highly accurate description of this interaction, although overall wavefunction is not optimal
NEO-DFT for [He-H-He]+
Nuclear basis set: 1s exponent optimized variationally 2 Gaussian-type geminals, scaled with single scaling factor to fit H cc-pVDZ result
• NEO-DFT provides accurate nuclear densities at low cost! • Electron-electron and electron-proton correlation predominantly additive
He He
Isotope NEO-HF cc-pVDZ
NEO-DFT cc-pVDZ
3D Grid cc-pVDZ
NEO-DFT cc-pVTZ
3D Grid cc-pVTZ
H 3098 1191 1191 1103 1111 D 2284 820 801 782 740 T 1903 660 633 646 581
Sirjoosingh, Pak, SHS, JCTC 2011; JCP 2012
Frequencies in cm−1 with H, D, T for central nucleus
• He nuclei classical (fixed) • H quantum mechanical
PCET: Multiconfigurational RXCHF• Two-configuration SCF approach for O-H bond on left or right • Only explicitly correlate 2 electronic orbitals to proton orbital (O-H bond) • Restricted basis set: only expand explicitly correlated orbitals in terms of valence basis functions on donor, acceptor, and H • Includes static and dynamical electron-proton correlation • Substantial computational savings à allows applications to larger systems
I I RXCHF I RXCHF1 1 2 2
II II RXCHF II RXCHF1 1 2 2
C CC C
Ψ = Ψ + Ψ
Ψ = Ψ + Ψ
State 1 State 2
Electronic and Positronic Densities
RXCHF-ne and RXCHF-ae perform similarly and agree well with highly accurate SVM/ECG results
LiPs e+LiH
electronic density positronic density solid: RXCHF-ae dashed: SVM/ECG
SVM data (dashed lines): Ryzhikh, Mitroy, Varga, J. Phys. B 31, 3965 (1998). ECG data (dashed lines): Strasburger, J. Chem. Phys. 111, 10555 (1999).
e+Li
Multicomponent DFT
Strategy for designing electron-proton functionals: • Define e-p functional as
• Use an explicitly correlated wavefunction to obtain expression for electron-proton pair density ρ ep(re,rp) in terms of one-particle electron and proton densities ρ e(re) and ρ p(rp) Advantages: provides accurate nuclear densities, consistent treatment of el-el & el-proton correlation Disadvantages: inherent DFT approximations, still expensive
e p e p 1 ep e p 1 e pe
e p e ppc ep ep, ( , ) ( ) ( )rE d rd d dρ ρ ρ ρ ρ− −⎡ ⎤ = − +⎣ ⎦ ∫ ∫r rr rr rr r
Chakraborty, Pak, SHS, PRL 2008; Sirjoosingh, Pak, SHS, JCTC 2011; JCP 2012
e pe
e p e p e pext ref
e pe pcxc pxc
[ , ] [ , ] [ , ]
[ ,[ ] [ ] ]
E E EE E E
ρ ρ ρ ρ
ρ
ρ ρ
ρ ρ ρ
= +
+ + +
Parr, JCP1982; Kreibich, Gross, PRL 2001; Chakraborty, Pak, SHS, JCP 2009
kinetic energy of noninteracting system, “classical” Coulomb
Electron-Proton Functional
gem 2e p e p
1( , ) exp
N
k kkbg γ
=
⎡ ⎤−⎢ ⎥⎣ ⎦= −∑r r r rGeminal function:
Expression for ep pair density in terms of 1-particle densities:
Contribution to total energy:
e p e p 1 ep e p 1 e pepc ep ep,E d d dr d rρ ρ ρ ρ ρ− −⎡ ⎤ = − +⎣ ⎦ ∫ ∫r r r r
Satisfies sum rules:
( ) ( )e e 1 ep e pp p
,Nρ ρ−=r r r ( ) ( )p p 1 ep e pe e
,Nρ ρ−=r r r
Motivating work: Colle and Salvetti, 1975, developed an electronic functional
e p e p 1 1 p
ep e p e p e p 1 1e p ep
e p e 1 e p p 1e e
ee p p e
1 1 e pe p e
p
p
p
1
gN N
g N N g gN N g
gN gN
ρ ρ ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ
ρ ρ
− −
− −
− −
− −
= +
+ 〈 〉 〈 〉+
+ 〈
〈 〉
〈 〈− 〉 − 〉
〉( ) e
gemp1 G+ ΦΨ Φ=
Born-Oppenheimer Approximation
• Schrödinger equation for electrons and protons • Fix proton position and solve electronic part • For each electronic state, solve proton part • Born-Oppenheimer approximation
Tp +Te +V re,rp,R( )( )Ψktot re,rp;R( ) = Ek R( )Ψk
tot re,rp;R( )
Te +V re,rp,R( )( )Ψielec re;rp,R( ) = εi rp,R( )Ψi
elec re;rp,R( )
Tp + εi rp,R( )( )Ψiµprot rp;R( ) = εµi( ) R( )Ψiµ
prot rp;R( )
Ψktot re,rp;R( ) ≈ Ψi
elec re;rp,R( )Ψiµprot rp;R( )
re,rp,R: electrons, protons, and other nuclei e p n
quantum classical
Non-Born-Oppenheimer Effects
• Expand exact solution in double-adiabatic basis
• Express Hamiltonian in matrix form
• Nonadiabatic couplings provide a measure of nonadiabaticity
Ψktot re,rp;R( ) = ciµk Ψi
elec re;rp,R( )Ψiµprot rp;R( )
i,µ∑
Hiµ, jν = δijδµνεµi( ) R( ) −
!2
mpΨiµprot | dij
ep( ) ⋅ ∇rp | Ψ jνprot
p−!2
2mpΨiµprot | gij
ep( ) | Ψ jνprot
p
dijep( ) = Ψi
elec | ∇rp | Ψ jelec
egijep( ) = Ψi
elec | ∇rp2 | Ψ j
elece
e p n
quantum classical
ep nad
Nuclear Quantum Effects
Zero point energy Vibrationally excited states
Hydrogen bonding Hydrogen tunneling
Proton-coupled electron transfer in solution, proteins, electrochemistry
ET
PT
Nuclear Quantum vs. Non-Adiabatic Effects
)()|(),( RRrRr ΧΦ=Ψ
),(ˆˆˆ),(ˆ RrVHHRrH rR ++=
),(),(),(ˆ RrERrRrH Ψ=Ψ
0),(ˆ ≈Φ RrTR
)|()()|()],(ˆˆ[ RrRWRrRrVHr Φ=Φ+
)()()](ˆˆ[ RERRWHR Χ=Χ+
0),(ˆ ≈Φ RrTR0),(ˆ1
≠Φ RrTR
)|,()()|,()],(ˆ),(ˆˆˆ[ 1111RRrRWRRrRrVRrVHH Rr Φ=Φ+++
)()()](ˆˆ[ RERRWHR Χ=Χ+
)()|,(),( 1 RRRrRr ΧΦ=Ψ