Derivative Couplings between Time-Dependent Density FunctionalTheory Excited States in the Random-Phase Approximation Based onPseudo-Wavefunctions: Behavior around Conical IntersectionsQi Ou, Ethan C. Alguire, and Joseph E. Subotnik*

Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States

ABSTRACT: In this paper, we present a formalism for derivative couplings betweentime-dependent density functional theory (TD-DFT) excited states within the random-phase approximation (RPA) using analytic gradient theory. Our formalism is based on apseudo-wavefunction approach in a companion paper (DOI 10.1021/jp505767b), andcan be checked against finite-difference overlaps. Our approach recovers the correctproperties of derivative couplings around a conical intersection (CI), which is a crucialprerequisite for any derivative coupling expression. As an example, we study the test caseof protonated formaldimine (CH2NH2

+).

I. INTRODUCTIONThe random-phase approximation (RPA) has been anextremely useful tool for calculating excited state propertiesand the oscillator strength of atoms and molecules since the1960s.1−5 More recently, the application of RPA in time-dependent density functional theory (TD-DFT/RPA) hasplayed an increasingly important role in the field of theo-retical chemistry for several reasons: (i) As a single-referenceab initio method, TD-DFT/RPA is computationally affordableand sometimes retains relatively high accuracy.6,7 (ii) TD-DFT/RPA is a size-consistent method which is able to give puresinglet and triplet states for closed-shell molecules.8 (iii) Incontrast to TD-DFT within the Tamm−Dancoff approximation(TDA), TD-DFT/RPA maintains the Thomas−Reiche−Kuhnsum rule of the oscillator strengths by taking into account the Bmatrix in the TD-DFT working equation;9 as such, TD-DFT/RPA gives improved results for transition moment calculations.8,10

For these reasons, despite its well-known triplet instability,8,11

TD-DFT/RPA is one of the most widely used approaches formodeling excited-state electronic structure.12−18

Recently, there has been a renewed interest in nonadiabaticdynamical transitions, as molecular photochemistry has gainedrenewed attention in the search for efficient solar energycapture. For these applications, the derivative coupling betweenexcited states as well as the derivative coupling between theground state and an excited state are necessary matrix elementsfor most theoretical approaches. To date, several researchershave investigated derivative couplings within TD-DFT andmost of them have focused on the ground−excited state deri-vative couplings. Historically, the derivative coupling matrixelements between TD-DFT ground and excited states were firstconstructed by Chernyak and Mukamel.19 A similar method forreal-time TD-DFT was developed by Baer and was applied to a

molecular system (H + H2).20 Extensive calculations of TD-

DFT ground−excited state derivative couplings were then pre-sented in both the plane-wave pseudopotential framework21−23

and the atomic orbital (AO) basis.24,25

The evaluation of TD-DFT derivative couplings is necessarilycomplicated by the fact that there are no rigorous staticTD-DFT wavefunctionsTD-DFT is based on time-dependentresponse theories as opposed to a quantum chemistry theorybased on a wavefunction ansatz. Despite this obstacle, however,TD-DFT derivative couplings should still meet certain bench-marks around a conical intersection (CI) point: (i) the magni-tude of the derivative couplings should become enormous; (ii)the derivative coupling vectors should lie in the branching plane;(iii) the path integral of the derivative coupling should giveBerry’s phase.26−28 Hu et al. have studied the behavior of theground−excited state derivative coupling around variousintersections,22,29−31 and they have found that the derivativecoupling (as given by their modified linear response schemewithin the TDA) recovers the correct Berry phase behavior.30,31

Looking forward, we are interested in the TD-DFT derivativecouplings between excited states. Unlike the case of ground−excited state crossings, TD-DFT recovers the correct dimen-sionality of a CI branching plane for the case of excited−excitedstate crossings.32,33 A method for calculating the derivativecoupling between TD-DFT excited states was proposed a fewyears ago by Tavernelli et al.34 based on Casida’s ansatz,9 butthe behavior around CIs was not fully investigated. To our

Special Issue: John R. Miller and Marshall D. Newton Festschrift

Received: June 10, 2014Revised: July 30, 2014

Article

pubs.acs.org/JPCB

© XXXX American Chemical Society A dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXX

knowledge, in practice, Tavernelli et al. have focused on theTDA exclusively, rather than full TD-DFT/RPA.34−37 From thederivations that follow in section II, one can show thatthe Tavernelli formalism does not agree with our result, nordoes the Tavernelli formalism satisfy the Chernyak−Mukamelequality. (More recently, while our article was under review, Liand Liu presented two abstract approaches for computingexcited state derivative couplings which are applicable to TD-DFT; the first is from a time-independent equation of motion(EOM) formalism, and the second is from time-dependent re-sponse theory.38 It can be shown that the final expression ob-tained in this work (eq 50) can be derived from Li and Liuusing their time-independent EOM formalism to calculateparticle−hole RPA derivate couplings.)In the following sections, our goal is to derive explicit expres-

sions for analytical derivative couplings between TD-DFT/RPAexcited states and analyze the behavior of the TD-DFT/RPAderivative couplings around a CI. Analogous to the configu-ration interaction singles (CIS) and TD-DFT/TDA derivativecouplings in our previous work,39,40 our theory for TD-DFT/RPA derivative couplings is based upon direct differentiation.For full response TD-DFT/RPA (rather than TD-DFT/TDA),we use the pseudo-wavefunctions proposed and evaluated forthe time-dependent Hartree−Fock (TD-HF) in ref 41. Tojustify our approach, first we will show in section V that ourderivative couplings behave correctly around a CI. To ourknowledge, the derivative couplings between TD-DFT/RPAexcited states around a CI have not yet been investigated(nor has the corresponding Berry phase). Second, we will showin section VI that our final answer satisfies the Chernyak−Mukamel equality near a crossing (again drawing on the resultsof ref 41); thus, our results are mostly consistent with responsetheory.An outline of this paper is as follows. In section II, our ana-

lytic gradient formalism for TD-DFT/RPA derivative couplingswill be presented. In section III, we compare our results versusthe finite-difference data. In section IV, using a simple model offour diabatic states, we analyze for TD-DFT/RPA (or TD-HF)derivative coupling vectors around a CI. In section V, we con-sider the specific case of protonated formaldimine and weexamine the CI between the first and second excited singletstates; we show that our derivative couplings lie in the correctbranching plane and recover the Berry phase exactly. Insection VI, we compare our derivative coupling expression withthe Chernyak−Mukamel equality and transition densities fromtime-dependent response theory.Unless otherwise specified, we use lowercase latin letters to

denote spin molecular orbitals (MO) (a, b, c, d for virtualorbitals, i, j, k, l, m for occupied orbitals, and p, q, r, s, w forarbitrary orbitals) and Greek letters (α, β, γ, δ, λ, σ, μ, ν) todenote AOs. The RPA excited states are denoted by Ψ (withuppercase latin indices I, J).

II. ANALYTIC DERIVATION FOR TD-DFT/RPADERIVATIVE COUPLINGS

A. TD-DFT/RPA Eigenvalue Equation. Before addressingderivative couplings between excited states, we first review theTD-DFT/RPA eigenvalue equations and establish the neces-sary notations. As discussed in ref 8, the non-Hermitian eigen-value TD-DFT/RPA equation can be written as

− −=⎜ ⎟⎛

⎝⎞⎠⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟E

A BB A

X

Y

X

Y

I

I I

I

I (1)

for excitation amplitudes XI and YI, and excitation energy EI ofTD-DFT/RPA excited state I. The orthogonality condition ofTD-DFT/RPA excitation amplitudes for any two excited statesI and J is

δ− = − =† † † †⎛⎝⎜⎜

⎞⎠⎟⎟X Y

X

YX X Y Y( )I I

J

JI J I J

IJ(2)

Multiplying (XJ† − YJ†) on both sides of eq 1 gives

δ

− − −= +

+ +=

† † † †

† †

⎜ ⎟⎛⎝

⎞⎠⎛⎝⎜⎜

⎞⎠⎟⎟

E

X YA B

B AX

YX AX Y AY

X BY Y BX

( ) (3)

(4)

J JI

IJ I J I

J I J I

IJ I

The matrix elements of A and B are given by

δ δ δ δ

= ⟨Φ | |Φ ⟩

= − + + Ω

A

F F E

(5)

(6)

iajb ia

jb

ij ab ab ij ij ab ajib

KS

DFT

= ⟨Φ | |Φ ⟩

= Ω

B (7)

(8)

iajb ijab

abij

KS DFT

where KS is the Kohn−Sham response operator. In second-quantized and antisymmetrized form (with physicists’ notationfor the two-electron integrals42), KS can be represented as

∑ ∑= + Ω

+ Ω +

† † †

† † † †

F a a a a a a

a a a a a a a a

[

( )]

pqpq p q

cdklclkd c k l d

cdkl c d l k l k c d

KS

(9)

where Fpq is the Fock matrix, and Ωpqsr is the two-electroneffective operator in DFT:

∑= + ΠF hpq pqm

pmqm(10)

ωΩ = Π +pqsr pqsr pqsr (11)

The diagonal entries of the Fock matrix Fpp ≡ εp are the usualKohn−Sham orbital energies, hpq

0 is the matrix element of thekinetic energy plus the external potential (eq 12), and gpq is thefirst derivative of the xc functional f xc (eq 13).

43 The sum of hpq0

and gpq gives the one-electron effective operator hpq (eq 14).

≡ ⟨ | | ⟩h p h qpq0 0

(12)

∫∑ ϕρ

ϕ≡∂

∂g

fr r

rrd ( )

( )( )pq

pqp

xcq

(13)

≡ +h h gpq pq pq0

(14)

Πpqsr is the Coulomb term plus whatever fraction of Hartree−Fock exchange is included in the DFT functional (cHF ineq 15), and ωpqsr is the second derivative of the xc functional(eq 16). The sum of Πpqsr and ωpqsr gives Ωpqsr (eq 11).

Π ≡ ⟨ | ⟩ − ⟨ | ⟩pq sr c pq rspqsr HF (15)

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dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXXB

∫∑

ω

ϕ ϕρ

ϕ ϕ

≡ ⟨ | ″ | ⟩

=∂

∂

pq f sr

fr r r

rr rd ( ) ( )

( )( ) ( )

pqsr xc

pqsrp q

xcr s

2

2(16)

With the definitions above, KS can be rewritten as

∑ ∑

∑ ω

ω

= + + Π

+ Π +

+ Π + +

†

† †

† † † †

h g a a

a a a a

a a a a a a a a

( )

[( )

( )( )]

pqpq pq

mpmqm p q

cdklclkd clkd c k l d

cdkl cdkl c d l k l k c d

KS0

(17)

B. The “Brute Force” Expression for TD-DFT/RPADerivative Couplings. In ref 41, we defined TD-HF pseudo-wavefunctions and we showed that the derivative couplingbetween ΨI and ΨJ obtained by direct differentiation can bewritten as

∑

∑

⟨Ψ|Ψ ⟩ = − ⟨Φ |Φ ⟩

+ −

X X Y Y

X X Y Y

( )

( )

I JQ

ijabiIa

jJb

iIa

jJb

ia

jb Q

iaiIa

iJa Q

iIa

iJa Q

[ ] [ ]

[ ] [ ]

(18)

We now treat TD-DFT/RPA as a natural extension ofTD-HF, and the goal of this paper is to evaluate eq 18 forTD-DFT/RPA derivative couplings. We begin with the secondterm in eq 18. The first step is to take the derivative on eachside of eqs 3 and 4 (for I ≠ J)

∑= + +

+ + +

+ + +

+ + +

X A X Y A Y X B Y

Y B X X A X Y A Y

X B Y Y B X X A X

Y A Y X B Y Y B X

0 {

}

ijabiIa

iajbQ

jJb

iIa

iajbQ

jJb

iIa

iajbQ

jJb

iIa

iajbQ

jJb

iIa Q

iajb jJb

iIa Q

iajb jJb

iIa Q

iajb jJb

iIa Q

iajb jJb

iIa

iajb jJb Q

iIa

iajb jJb Q

iIa

iajb jJb Q

iIa

iajb jJb Q

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ] [ ]

(19)

From eq 1, one finds that

∑ ∑+ =A X B Y E Xijab

iajb jIb

iajb jIb

iaI i

Ia

(20)

∑ ∑+ = −B X A Y E Yijab

iajb jIb

iajb jIb

iaI i

Ia

(21)

Inserting eqs 20 and 21 into eq 19, one has

∑

∑

−

=−

+ +

+

X X Y Y

E EX A X Y A Y X B Y

Y B X

( )

1{

}

iaiIa

iJa Q

iIa

iJa Q

J I ijabiIa

iajbQ

jJb

iIa

iajbQ

jJb

iIa

iajbQ

jJb

iIa

iajbQ

jJb

[ ] [ ]

[ ] [ ] [ ]

[ ](22)

Now, we replace the second term of eq 18 by the result ofeq 22 and the derivative coupling expression becomes

∑

∑

∑

∑

∑

⟨Ψ|Ψ ⟩ = − ⟨Φ |Φ ⟩

+−

+ ++

=−

+

+ +− −

− −

X X Y Y

E EX A X

Y A Y X B YY B X

E EX X Y Y A

X Y Y X BX X Y Y O

X X Y Y O

( )

1{

}

(23)

1{( )

( ) }( )

( )

(24)

I JQ

ijabiIa

jJb

iIa

jJb

ia

jb Q

J I ijabiIa

iajbQ

jJb

iIa

iajbQ

jJb

iIa

iajbQ

jJb

iIa

iajbQ

jJb

J I ijabiIa

jJb

iIa

jJb

iajbQ

iIa

jJb

iIa

jJb

iajbQ

iabiIa

iJb

iIa

iJb

baR Q

ijaiIa

jJa

iIa

jJa

jiR Q

[ ] [ ]

[ ]

[ ] [ ]

[ ]

[ ]

[ ]

[ ]

[ ]

where we define the “right” spin−orbital derivative overlap

∑ ∑ ∑

∑ ∑

μ ν ν

≡ ⟨ | ⟩

= ⟨ | | ⟩ + | ⟩

= +

μμ

νν

νν

μνμ μν ν

μνμ μν ν

O p q

C C C

C S C C S C

(25)

( )( ) (26)

(27)

pqR Q Q

p qQ Q

q

p qQ

pR Q

q

[ ] [ ]

[ ] [ ]

[ ] [ ]

Here Cμp denotes the MO coefficients; Sμν ≡ ⟨μ|ν⟩ is theatomic orbital overlap, and Sμν

R[Q] ≡ ⟨μ|ν[Q]⟩ is the right deriva-tive of the overlap. The detailed derivation of Opq

R[Q] can befound in ref 39, which leads to

∑

∑

= − − Θ

= − Θ

μνμ μν μν ν

μνμ μν ν

⎜ ⎟⎛⎝

⎞⎠O C S S C

C S C

12

(28)

(29)

pqR Q

pR Q Q

q pqQ

pA Q

q pqQ

[ ] [ ] [ ] [ ]

[ ] [ ]

where Θ is the occupied-to-virtual rotation angle matrix andSA[Q] is defined as SA[Q] ≡ SR[Q] − 1/2S

[Q].The derivatives of A and B in eq 24 can be written as

δ δ δ δ= Ω + − +A F F EiajbQ

ajibQ

ij abQ

ab ijQ

ij abQ[ ] [ ] [ ] [ ]

DFT[ ]

(30)

= ΩBiajbQ

abijQ[ ] [ ]

(31)

As we did in our recent paper on TD-DFT/TDA derivativecouplings,40 we make the following definitions for various typesof density matrices:

(1) The general density matrices:

∑=μν μ νP C Cm

m m(32)

∑ ∑ = = +μν μ ν μν μ νP C C P C Cp

p pa

a a(33)

Note that we may express the real-space density as ρ(r) =∑μν Pμνϕμ(r)ϕν(r)

(2) The RPA excitation-amplitude matrices:

∑=μν μ νR C X CXI

iaa i

Iai

(34)

∑=μν μ νR C Y CYI

iaa i

Iai

(35)

The Journal of Physical Chemistry B Article

dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXXC

(3) The generalized difference-density matrix:

∑

∑

= +

− +

μν μ ν

μ ν

D C X X Y Y C

C X X Y Y C

( )

( )

IJ

iaba i

IaiJb

iJa

iIb

b

ijai i

JajIa

iIa

jJa

j(36)

Using the expressions for Ω[Q] and F[Q] in refs 39 and 40,and plugging those expressions into Aiajb

[Q] and Biajb[Q], we find

this lengthy expression for the TD-DFT/RPA derivativecoupling:

∑ ∑

∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∑ ∑

∑

∑ ∑

∑ ∑

∑ ∑

∑

∑ ∑

∑

∑ ∑

∑ ∑

∑ ∑

∑

ε

ε

ε

ε

⟨Ψ|Ψ ⟩ =+−

Ω

− Ω + Ω + Ω

+ Ω + Ω Θ + Ω Θ

+ Ω Θ + Ω Θ

++−

×

+ Π −

+ −

Π + Π − Θ Π + Π

−+−

+ Π −

+ − Π

+ Π − Θ Π + Π

++−

Ω

− Ω + Ω + Ω

+ Ω + Ω Θ + Ω Θ

+ Ω Θ + Ω Θ

− −

+ −

μνλσμ ν μνλσ λ σ

αβα αβ β β β

β

μνμ μν ν

μνλσμ ν μνλσ λ σ

αβα αβ β

αβα αβ β

αβα αβ β

μνμ μν ν

μνλσμ ν μνλσ λ σ

αβα αβ β

αβα αβ β

αβα αβ β

μνλσμ ν μνλσ λ σ

αβα αβ β β β

β

μνμν μ ν

μ ν

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎧⎨⎩

⎡⎣⎤⎦

⎫⎬⎭

⎧⎨⎩

⎫⎬⎭

⎧⎨⎩

⎫⎬⎭

⎧⎨⎩

⎡⎣⎤⎦

⎫⎬⎭

X X Y Y

E EC C C C

C S C C C

C

X X Y YE E

C h C

C C C C C S C

C S C C S C

X X Y Y

E EC h C

C C C C C S C

C S C C S C

X Y X Y

E EC C C C

C S C C C

C

S X X Y Y C C

X X Y Y C C

( )

12

( )

12

(

)12

( ) ( )

( )

12

(

)12

(

) ( )

( )

12

[ ( )

( ) ]

I JQ

ijab

iIa

jJb

iIa

jJb

J Ia j

Qi b

ww

Qa wjib j awib i ajwb

b ajiwk

kjib akQ

cacib jc

Q

cajcb ic

Q

kajik bk

Q

iab

iIa

iJb

iIa

iJb

J Ia

Qb

ma m

Qb m a a

Qb

b bQ

amw

wQ

m

awbm ambwmc

cmQ

acbm ambc

ija

iIa

jJa

iIa

jJa

J Ii

Qj

mi m

Qj m i i

Qj

j jQ

imw

wQ

m iwjm

imjwmc

cmQ

icjm imjc

ijab

iIa

jJb

iIa

jJb

J Ia b

Qi j

ww

Qa wbij b awij i abwj

j abiwk

kbij akQ

cabic jc

Q

cabcj ic

Q

kakij bk

Q

A Q

iabiIa

iJb

iIa

iJb

b a

ijaiIa

jJa

iIa

jJa

j i

[ ] [ ]

[ ]

[ ] [ ]

[ ] [ ]

[ ]

[ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ]

[ ] [ ]

[ ] [ ]

[ ]

(37)

In eq 37, the terms in the first three braces come from A[Q],the last brace comes from B[Q], and the final line is contributedby OR[Q]. Note that the orbital rotations within the occupiedand virtual subspaces (Θab and Θij) disappear and are absent in

eq 37 (just as for CIS and TD-DFT/TDA39,40). These termsvanish because TD-DFT is invariant to one’s choice of occu-pied and virtual orbitals; as such, Θab and Θij cannot contributeto the final answer.In order to get our final expression into the AO basis, a few

more definitions are needed for the xc functional derivatives;these definitions are identical with those for TD-DFT/TDA.First, the first derivative of the xc functional g[Q] can be decom-posed in the AO representation as

≡ +μν μν μνg g gQ Q Y Q[ ] [ ] [ ](38)

where

∫ ∫

∫∑

ϕρ

ϕρ

ϕ ϕ

ϕ ϕρ

ϕ ϕ

≡∂

∂+

∂∂

+∂

∂

μν μ ν μ ν

λσμ ν λσ λ σ

gf f

fP

r rr

r rr

r r

r r rr

r r

d ( )( )

( ) d( )

( ( ) ( ))

d ( ) ( )( )

( ( ) ( ))

Q xc xc Q

xc Q

[ ] [ ]

2

2[ ]

(39)

∑ ω≡μνλσ

λσ μνλσg PY Q Q[ ] [ ]

(40)

Here, the first derivative ∫ [Q] represents differentiation withrespect to the Becke weights in the exchange-correlation nu-merical quadrature. The total one-electron-integral derivativefor TD-DFT can be written as

= +

= + +

≡ +

μν μν μν

μν μν μν

μν μν

h h g

h g g

h g

Q Q Q

Q Q Y Q

Q Y Q

[ ] 0[ ] [ ]

0[ ] [ ] [ ]

[ ] [ ](41)

Second, analogous definitions can be made for the two-electron-integral derivatives ω[Q]

ω ω ω= +μνλσ μνλσ μνλσQ Q Y Q[ ] [ ] [ ]

(42)

where

∫

∫

∫

∫∑

ω ϕ ϕρ

ϕ ϕ

ϕ ϕρ

ϕ ϕ

ϕ ϕρ

ϕ ϕ

ϕ ϕρ

ϕ ϕ

ϕ ϕ

≡∂

∂

+∂

∂

+∂

∂

+∂

∂

μνλσ μ ν λ σ

μ ν λ σ

μ ν λ σ

γδμ ν λ σ

γδ γ δ

f

f

f

f

P

r r rr

r r

r r rr

r r

r r rr

r r

r r rr

r r

r r

d ( ) ( )( )

( ) ( )

d ( ( ) ( ))( )

( ) ( )

d ( ) ( )( )

( ( ) ( ))

d ( ) ( )( )

( ) ( )

( ( ) ( ))

QQ

xc

Q xc

xc Q

xc

Q

[ ][ ] 2

2

[ ]2

2

2

2[ ]

3

3

[ ](43)

∑ω ≡ Ξμνλσγδ

γδ μνλσγδPY Q Q[ ] [ ]

(44)

and Ξμνλσγδ is the xc functional third derivative

∫ ϕ ϕρ

ϕ ϕ ϕ ϕΞ ≡∂

∂μνλσγδ μ ν λ σ γ δ

fr r r

rr r r rd ( ) ( )

( )( ) ( ) ( ) ( )xc

3

3

(45)

The Journal of Physical Chemistry B Article

dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXXD

Thus, the total two-electron-integral derivatives for TD-DFTcan be written as

ω

ω ω

ω

Ω = Π +

= Π + +

≡ Ω +

μνλσ μνλσ μνλσ

μνλσ μνλσ μνλσ

μνλσ μνλσ

Q Q Q

Q Q Y Q

Q Y Q

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ] [ ](46)

With the definitions made above, one can simplify the non-response and response terms of the TD-DFT/RPA derivativecouplings in a fashion completely analogous to the case inTD-DFT/TDA.40 In the end, a complete expression for theTD-DFT derivative coupling is

∑

∑

∑

∑

∑

∑

∑

∑ ∑

∑

⟨Ψ|Ψ ⟩ =−

+ + + +

× Ω + Π

− +

− + +

+ + + + +

× Ω − +

+ + + + +

+ Ω −

× + + + +

+ + + Ξ

+ + Ω

− Θ − −

× − −

μνμν μν

μνλσμλ σν μλ σν μλ νσ μλ νσ

μνλσ μλ σν μνλσ

αβμνμν μα βν νβ αβ

μναβγδμν μα νγ δβ νγ δβ δβ νγ

δβ νγ νγ βδ νγ βδ βδ νγ βδ νγ

αβγδμναβγδ

μν μα γν βδ γν βδ

βδ γν βδ γν γν δβ γν δβ δβ γν

δβ γν αβγδμναβγδλσ

μν μλ νσ

αγ δβ αγ δβ δβ αγ δβ αγ αγ βδ

αγ βδ βδ αγ βδ αγ αβγδλσ

μναβγδμν μα νδ γβ βγ αβγδ

μν

ν μ μνμν

ν μ μν

⎡⎣⎤⎦

E ED h

R R R R R R R R

D P

S P D D F

S P R R R R R R

R R R R R R R R R R

S P R R R R

R R R R R R R R R R

R R S P P

R R R R R R R R R R

R R R R R R

S P P D D

L X X Y Y

C C S X X Y Y C C S

1{

( )

12

( )

12 (

)12 (

)12

(

)12

( )

} ( )

( )

I JQ

J I

IJ Q

XI XJ YI YJ XI YJ YI XJ

Q IJ Q

Q IJ IJ

Q XI XJ YI YJ XI XJ

YI YJ XI YJ YI XJ XI YJ YI XJ

Q XI XJ YI YJ

XI XJ YI YJ XI YJ YI XJ XI YJ

YI XJ Q

XI XJ YI YJ XI XJ YI YJ XI YJ

YI XJ XI YJ YI XJ

Q IJ IJ

bibi bi

Q

iabiIa

iJb

iIa

iJb

a bA Q

ijaiIa

jJa

iIa

jJa

i jA Q

[ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ] [ ]

(47)

where

∑

∑

∑

∑

= + + + +

+ + + Ω

− + + +

+ + + + Ω

+ + + +

+ + + + Ξ

+ + Ω

μνλσμ λ νσ νσ νσ νσ σν

σν σν σν μνλσ

μνλσμ λ νσ νσ νσ νσ

σν σν σν σν μνλσ

μνλσγδγ δ μλ σν μλ σν σν μλ σν μλ

μλ νσ μλ νσ νσ μλ νσ μλ μνλσγδ

μνλσν σ μλ λμ μνλσ

L C C R X R Y X R Y R R Y

R X X R Y R

C C R X R Y X R Y R

R Y R X X R Y R

C C R R R R R R R R

R R R R R R R R

C C D D

(

)

(

)

(

)

( )

bid

b dXI

iJd YI

iJd

iId XJ

iId YJ XI

iJd

YIiJd

iId YJ

iId XJ

iXI Jb YI Jb Ib XJ Ib YJ

XI Jb YI Jb Ib YJ Ib XJ

b iXI XJ YI YJ XI XJ YI YJ

XI YJ YI XJ XI YJ YI XJ

b iIJ IJ

(48)

To evaluate the Θbi[Q] terms, one must solve the coupled-

perturbed Hartree−Fock (CPHF) equation44,45 by applying the“z-vector” method46

∑= ∂∂Θ ∂Θ

−⎛⎝⎜⎜

⎞⎠⎟⎟z L

Eaj

bibi

aj bi

21

(49)

Once the z-vector has been saved to disk, we construct themixed derivatives for each coordinate and our final expressionbecomes

∑

∑

∑

∑

∑

∑

∑

∑

∑

⟨Ψ|Ψ ⟩ =−

+ + +

+ Ω + Π

− +

− +

+ + + +

+ + Ω

− +

+ + + +

+ + Ω

− +

+ + + +

+ + Ξ

+ + Ω

− −

− −

μνμν μν

μνλσμλ σν μλ σν μλ νσ

μλ νσ μνλσ μλ σν μνλσ

αβμνμν μα βν νβ αβ

μναβγδμν μα νγ δβ νγ δβ

δβ νγ δβ νγ νγ βδ νγ βδ

βδ νγ βδ νγ αβγδ

μναβγδμν μα γν βδ γν βδ

βδ γν βδ γν γν δβ γν δβ

δβ γν δβ γν αβγδ

μναβγδλσμν μλ νσ αγ δβ αγ δβ

δβ αγ δβ αγ αγ βδ αγ βδ

βδ αγ βδ αγ αβγδλσ

μναβγδμν μα νδ γβ βγ αβγδ

μνν μ μν

μνν μ μν

⎡⎣⎤⎦

E ED h

R R R R R R

R R D P

S P D D F

S P R R R R

R R R R R R R R

R R R R

S P R R R R

R R R R R R R R

R R R R

S P P R R R R

R R R R R R R R

R R R R

S P P D D

X X Y Y C C S

X X Y Y C C S

1{

(

)

12

( )

12 (

)12 (

)12 (

)12

( ) }

( )

( )

I JQ

J I

IJ Q

XI XJ YI YJ XI YJ

YI XJ Q IJ Q

Q IJ IJ

Q XI XJ YI YJ

XI XJ YI YJ XI YJ YI XJ

XI YJ YI XJ

Q XI XJ YI YJ

XI XJ YI YJ XI YJ YI XJ

XI YJ YI XJ

Q XI XJ YI YJ

XI XJ YI YJ XI YJ YI XJ

XI YJ YI XJ

Q IJ IJ

iabiIa

iJb

iIa

iJb

a bA Q

ijaiIa

jJa

iIa

jJa

i jA Q

[ ] [ ]

[ ] [ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

(50)

where DIJ represents the relaxed difference density matrix

∑ ≡ − +

= − +

μν μν μ ν μ ν

μν μν νμ

D D z C C C C

D z z

( )

( )

IJ IJ

ajaj a j j a

IJ(51)

Before ending, we note that this formal expression (eq 50)can be easily transformed to the energy gradient when settingI = J. Also, eq 50 can be derived from a Hellmann−Feynmanapproach (just as for TD-HF in ref 41). See Appendix A.

III. COMPARISON WITH FINITE-DIFFERENCEIn order to check our implementation of eqs 50 and 51 forTD-DFT/RPA derivative couplings, in Table 1, we comparethe S1/S5 derivative coupling of formaldehyde calculated di-rectly from eq 50 with the finite-difference data given by theformula below

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dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXXE

⟨Ψ|Ψ ⟩ ≈

⟨Ψ |Ψ + Δ ⟩ − ⟨Ψ |Ψ − Δ ⟩Δ

Q Q Q Q Q Q

Q

( ) ( ) ( ) ( )

2

I JQ

I J I J

[ ]

(52)

where Q indexes the nuclear degree of freedom. To evaluate theoverlaps in eq 52, we employ the pseudo-wavefunction ansatzin ref 41 and calculate the finite-difference results according to

⟨Ψ|Ψ ⟩ ≈ ⟨ | + Δ ⟩ − ⟨ | − Δ ⟩Δ

− ⟨ | + Δ ⟩ − ⟨ | − Δ ⟩Δ

Q Q Q Q Q QQ

Q Q Q Q Q QQ

( ) ( ) ( ) ( )2

( ) ( ) ( ) ( )2

I JQ[ ]

(53)

where we define

∑≡ |Φ ⟩†X a aia

iIa

a i DFT(54)

∑≡ |Φ ⟩†Y a aia

iIa

a i DFT(55)

The quantum chemistry package Q-Chem47,48 was used forall calculations. Three TD-DFT xc functionals (B3LYP,49,50

ωB97, and ωB97X51) were tested using the 6-31G* basis set.As shown in Table 1, our numerical results from eq 50 matchthe finite-difference data with an error less than 10−4 for eachxc functional, which validates our implementation of theTD-DFT/RPA derivative couplings.

IV. GEOMETRIC PHASE AND BRANCHING PLANE FORRPA CONICAL INTERSECTIONS

The derivative couplings calculated above in eq 50 were derivedunder several assumptions, especially the necessity of small Yterms, which allowed us to ignore and/or manipulate severalterms in the derivation. To further justify our theory, we willnow evaluate our derivative couplings around a CI and examinewhether or not we recover the correct properties, which isan essential criterion for any derivative coupling approach.In particular, for the exact eigenstates of the Hamiltonian, the

derivative couplings must satisfy two criteria around a CI: (i)the derivative couplings must be in the branching plane; (ii) thederivative couplings must obey Berry’s phase.To date, almost all examinations of Berry’s phase and branch-

ing planes have focused on 2 × 2 Hermitian Hamiltonians.26,27

In our case, there are two slight complications because (i) theHamiltonian is not Hermitian and (ii) the Hamiltonian eigen-states come in pairs of positive and negative energy.Thus, whereas standard quantum chemistry considers a 2 × 2

Hermitian Hamiltonian for two diabatic states, we must nowconsider a 4 × 4 non-Hermitian Hamiltonian with four diabaticstates. In particular, for TD-DFT/RPA, the non-HermitianHamiltonian

=− −

⎜ ⎟⎛⎝

⎞⎠H

A BB A (56)

around a CI can be written as (to the first order)

ε

ε

ε

ε

=

− · · · ·

· + · · ·

− · − · − + · − ·

− · − · − · − − ·

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟H R

g R h R b R b R

h R g R b R b R

b R b R g R h R

b R b R h R g R

( )

11 12

21 22

11 12

21 22

(57)

Let the basis here be denoted as {|ΨI0⟩}, where I = 1, 2, 3, 4.

In this basis, g and h are the matrix elements of A around theCI point, ε is the (positive) energy of two TD-DFT/RPA statesat the CI point, and bij is a matrix representation of B. TheHamiltonian has been expanded to first order in R.We will now perform a series of linear transformations. Our

first step is to propose a branching plane. We propose that thebranching plane should be spanned by the g and h vectors. Letθ be the branching angle, θ = 0°, in the g direction. Second, werescale g and h to make the norm of the gradient differenceindistinguishable at every direction around the CI by takingg = ρ cos θ and h = ρ sin θ. Third, the A and −A matrix blockscan be diagonalized everywhere in the branching plane and theHamiltonian becomes

ρ θ

ε ρ ρ θ ρ θ

ε ρ ρ θ ρ θ

ρ θ ρ θ ε ρ

ρ θ ρ θ ε ρ

ρ ρ θ

=

−

+

− − − +

− − − −

≡ +

⊥

⊥ ⊥

⊥ ⊥

⊥ ⊥

⊥ ⊥

⊥

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

R

b R b R

b R b R

b R b R

b R b R

R

H

H B

( , , )

0 ( , , ) ( , , )

0 ( , , ) ( , , )

( , , ) ( , , ) 0

( , , ) ( , , ) 0

(58)

( ) ( , , ) (59)

11 12

21 22

11 12

21 22

0

Table 1. Derivative Couplings (in a0−1) between the S1 and S5 States of Formaldehyde (HCHO) as Computed by Finite-

Difference (FD) and Analytical Theory (DC)a

B3LYP ωB97 ωB97X

degree of freedom (Q) FD DC FD DC FD DC

Cy −0.88668 −0.88669 −0.84566 −0.84567 −0.85341 −0.85337Hx −0.05466 −0.05470 −0.04737 −0.04739 −0.04945 −0.04946Hy 0.10099 0.10099 0.09228 0.09224 0.09432 0.09433Oy 0.41617 0.41620 0.39715 0.39715 0.39984 0.39989

aNote that only the components larger than 10−4 a0−1 are listed here.

The Journal of Physical Chemistry B Article

dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXXF

where R⊥ denotes the coordinate perpendicular to the branch-ing plane. By doing the linear transformation, bij linearly de-pends on ρ and/or R⊥ and can be decomposed into two com-ponents: b ij = ρ·bij

0 + R⊥·bij⊥. Here bij

0 lies in the branching planeand bij

⊥ is perpendicular to the branching plane. In eq 59, H0contains only the diagonal elements (eq 60) and B containsonly the off-diagonal elements of H (eq 61).

ρ

ε ρε ρ

ε ρε ρ

=

−+

− +− −

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟H ( )

0 0 0

0 0 0

0 0 0

0 0 0

0

(60)

ρ θ

ρ θ ρ θ

ρ θ ρ θ

ρ θ ρ θ

ρ θ ρ θ

=

− −

− −

⊥

⊥ ⊥

⊥ ⊥

⊥ ⊥

⊥ ⊥

⎛

⎝

⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟

R

b R b R

b R b R

b R b R

b R b R

B( , , )

0 0 ( , , ) ( , , )

0 0 ( , , ) ( , , )

( , , ) ( , , ) 0 0

( , , ) ( , , ) 0 0

11 12

21 22

11 12

21 22

(61)

Note that the basis {|ΨI0⟩} for eqs 58−61 depends on θ:

θ θ θ|Ψ ⟩ = +

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟( ) cos

2

1000

sin2

0100

10

(62)

θ θ θ|Ψ ⟩ = −

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟( ) cos

2

0100

sin2

1000

20

(63)

θ θ θ|Ψ ⟩ = +

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟( ) cos

2

0010

sin2

0001

30

(64)

θ θ θ|Ψ ⟩ = −

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟( ) cos

2

0001

sin2

0010

40

(65)

It is crucial to keep in mind that, in changing coordinates, wehave performed only linear transformations.Because H is antisymmetric, we will need to construct

the “left” and “right” eigenfunctions, ⟨ΨIL| and ⟨ΨI

R|, of eq 56:

|Ψ ⟩ ≡ − |Ψ ⟩ ≡⎛⎝⎜⎜

⎞⎠⎟⎟X Y

X

Y( ),I

I II

I

IL R

(66)

where

⟨Ψ | = ⟨Ψ |EHI I IL L

(67)

|Ψ ⟩ = |Ψ ⟩EH I I IR R

(68)

We now assume that the elements of B (or B) are smallcompared with the diagonal elements of Athis is consistentwith the assumption that RPA is most accurate, the magnitudeof Y should be small.8 In particular, we assume that bij/ε ≪ 1.

Thus, we can estimate the energies (i.e., eigenvalues) of H.Using second order perturbation theory

ε ρε ρ ε

ε ρ

= − +|⟨Ψ | |Ψ ⟩|

−+

|⟨Ψ | |Ψ ⟩|

= − +

EB B

B

2( ) 2(69)

( ) (70)

130

10 2

40

10 2

2

ε ρε ε ρ

ε ρ

= + +|⟨Ψ | |Ψ ⟩|

+|⟨Ψ | |Ψ ⟩|

+

= + +

EB B

B

2 2( )(71)

( ) (72)

230

20 2

40

20 2

2

From this, we may conclude that the eigenvalues of H arethe same as the eigenvalues of H0 through ρ( )2 , ρ ⊥R( ), or

⊥R( )2(since B linearly depends on ρ and/or R⊥) and thus the

g−h plane is indeed the correct branching plane.Next, our second task is to consider the derivative couplings.

We are interested in the first two eigenstates with positiveenergy; these pseudo-wavefunctions can be expressed as (to thefirst order)

ε ρ ε|Ψ ⟩ = |Ψ ⟩ +

⟨Ψ | |Ψ ⟩−

|Ψ ⟩ +⟨Ψ | |Ψ ⟩

|Ψ ⟩B B

2( ) 21R

10 3

010

30 4

010

40

(73)

ε ε ρ|Ψ ⟩ = |Ψ ⟩ +

⟨Ψ | |Ψ ⟩|Ψ ⟩ +

⟨Ψ | |Ψ ⟩+

|Ψ ⟩B B2 2( )2

R20 3

020

30 4

020

40

(74)

ε ρ ε⟨Ψ | = ⟨Ψ | − ⟨Ψ |

⟨Ψ | |Ψ ⟩−

− ⟨Ψ |⟨Ψ | |Ψ ⟩B B

2( ) 21L

10

30 3

010

40 4

010

(75)

ε ε ρ⟨Ψ | = ⟨Ψ | − ⟨Ψ |

⟨Ψ | |Ψ ⟩− ⟨Ψ |

⟨Ψ | |Ψ ⟩+

B B2 2( )2

L20

30 3

020

40 4

020

(76)

where {|ΨI0⟩} are the eigenstates of H0. Because {|ΨI

0⟩} dependonly on θ, it is clear that our derivative couplings lie in thebranching plane.Finally, our third task is to compute the exact direction of the

derivative couplings. The derivative coupling between two RPAstates is

⟨Ψ | ∂∂

Ψ ⟩ = ⟨Ψ | ∂∂

Ψ ⟩ + + Q Q

B B( ) ( )1L

2R

10

20 2

(77)

where the latter two terms are the couplings contributed bythe B matrix. Note that the B( ) and B( )2 terms become in-finitesimal in the vicinity of a CI point so that only the firstterm in eq 77 gives a nonzero contribution to the derivativecouplings around a CI.

⟨Ψ | ∂∂

Ψ ⟩ ≈ ⟨Ψ | ∂∂

Ψ ⟩Q Q1

L2R

10

20

(78)

From eqs 62, 63, and 78, we may finally conclude that, just asfor CIS and TD-DFT/TDA, we can expect standard Berry phasebehavior between two RPA states for a loop around the CI:

∮ ∮ϕ π= ⟨Ψ | ∂∂

Ψ ⟩ ≈ ⟨Ψ | ∂∂

Ψ ⟩ =Q

QQ

Qd dC C

1L

2R

10

20

(79)

We now want to emphasize that our approach for derivativecouplings (in eq 18) must have the correct behavior aroundCIs. After all, we compute derivative couplings by looking at

The Journal of Physical Chemistry B Article

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the direct overlap of ⟨XiIaΦi

a[Q], −YiIaΦia[Q]| with |Xj

JbΦjb[Q],

YjJbΦj

b[Q]⟩. See, for instance, the finite-difference data in Table 1.Though there may be other approaches for derivativecouplings that recover the correct physics around a CI, directdifferentiation is always guaranteed to work. Thus, given thatthe TD-HF or TD-DFT/RPA derivative couplings shouldsatisfy a topology and Berry’s phase constraint around a CI,eq 79 is an excellent criterion by which we can judge thereasonability of TD-HF or TD-DFT/RPA derivative cou-plings.

V. APPLICATION TO PROTONATED FORMALDIMINETo convince the reader that our approach does recover thecorrect behavior around a CI in practice, we now provide a nu-merical example: we study the S1/S2 CI for protonated form-aldimine (CH2NH2

+), as previously examined by Tavernelli andco-workers.34−37 The minimum crossing point between S1 andS2 excited singlet states was determined using the penalty func-tion approach of Martinez et al.33 We consider both TD-HFand TD-DFT/RPA excited states and calculate their respectiveS1/S2 derivative coupling around the CI. All the calculationswere carried out with the 6-31G** basis set.After the minimum crossing point was located, the branching

plane of the S1/S2 CI of CH2NH2+ was obtained with the

approach described in ref 52. The g and h vectors are defined asfollows: (i) define D(θ) as the energy gradient difference in thebranching plane at angle θ around the CI point; (ii) g is definedas D(θmax) when θmax is the branching angle that maximizes∥D(θ)∥; (iii) h is defined as the gradient difference with theminimum norm, D(θmin). A pair of rescaled vectors, x and y, isthen generated to be used as the coordinates for a loop on thebranching plane (with the norm of the gradient differenceidentical in every direction):

=∥ ∥

xg

g1

(80)

=∥ ∥∥ ∥

yg

hh2 (81)

A total of 36 geometries around the CI point were chosen atthe distance r = 0.001 Å53 to perform the derivative couplingcalculation:

θ θ θ

θ

= + +

= ° ° ° °

R R x y( ) 0.001( cos sin ),

0 , 10 , 20 , ..., 350CI

(82)

where RCI is the coordinate of the CI point. For a detailedexplanation of the TD-HF derivative couplings, see ref 41. TheωB97X functional was used for our TD-DFT/RPA calculations.Previous studies by Tretiak and co-workers have shown thesignificant contribution of orbital response for semiempiricalexcited state−excited state derivative couplings.54 In order toassess the importance of such orbital response terms (Θ[Q])for RPA derivative couplings in this work, we computed theTD-HF and TD-DFT/RPA derivative couplings with and with-out response:

(1) Full-DC: complete derivative couplings given by eq 50.(2) NR-DC: derivative couplings given by eq 47 without the

Θ[Q] part.

A. Results. In Figure 1, we plot the derivative coupling vec-tors for each point on the loop around the CI. As Figure 1shows, for both TD-HF and TD-DFT/RPA, the full-DC vectors

lie rigorously on the branching plane for all geometries com-puted. By contrast, the NR-DC vectors do not reproduce thecorrect branching plane. The out-of-plane angle varies from 1to 15° for both TD-HF and TD-DFT/RPA. In Figure 2, we

project the derivative coupling vectors onto the branchingplane to get a clearer description of their behavior around theCI point. As can be seen from Figure 2, the full-DC vectors areperpendicular to the gradient difference direction and their

Figure 1. CH2NH2+ S1/S2 derivative coupling vectors on the circular

loop (r = 0.001 Å) around the CI point in the branching plane givenby TD-HF (left) and TD-DFT (right). 36 single-point calculationswere performed.

Figure 2. Projected derivative coupling vectors on the circular loop(r = 0.001 Å) around the CI point in the branching plane given byTD-HF (left) and TD-DFT (right).

The Journal of Physical Chemistry B Article

dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXXH

magnitudes are identical for each point on the loop. The NR-DC vectors, however, do not always point to the tangent direc-tion, nor do they show the same magnitude for different points.The results indicate the overwhelming necessity of taking theorbital response terms into consideration.Using θ = 90° as an example, we visualize the full-DC deri-

vative coupling vectors and the gradient difference vectors forboth methods in Figure 3. While the gradient difference vectors

at the given geometry point perpendicularly to the molecular(CH2NH2

+) plane, the derivative coupling vectors lie rigorouslyin the (CH2NH2

+) plane.Lastly, as demonstrated in section IV, it is important to check

the path integral of the derivative coupling around the CI pointto see if it gives the correct Berry phase. The path integral ofthe derivative coupling vectors is given by

∮ ∫ ∑θ θ θ θϕ δ= · = · ≈ ·π

=

r rd R R d d( ) d ( ) d ( )C

IJ IJi

IJ i i0

2

1

36

(83)

As shown in Table 2, both TD-HF and TD-DFT/RPAfull-DC vectors reproduced the expected Berry phase perfectly.

The projected NR-DC vectors do not satisfy eq 83 (though theerror is not enormous). Overall, our calculations highlight thereasonability of our analytic gradient theory for RPA derivative

couplings as well as the non-negligible role played by the orbitalresponse terms.

VI. THE CHERNYAK−MUKAMEL EXPRESSION ANDTHE TRANSITION DENSITY MATRIX ACCORDINGTO RESPONSE THEORY

Before ending, we emphasize that the theoretical approachin this paper (leading up to eq 18) is consistent with the Chernyak−Mukamel (CM) equality near a crossing. The CM equality thatthe derivative couplings should take the form

∑ ν γ=−E E

d1

IJJ I pq

pqQ

pqIJCM [ ]

(84)

where γpqIJ is the one-electron transition density matrix. Indeed,

for the case of TD-HF, ref 41 shows that our derivative cou-pling can be put in the form

∑=−

ΓE E

vd1

IJJ I pq

pqQ

pqIJCB [ ]

(85)

where ΓpqIJ is given by

∑

∑

∑

∑

Γ =

− + ∈

+ ∈

− + ∈∈

− + ∈∈

−

−

⎧

⎨

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

X X Y Y p q

X X Y Y p q

A B L pq

A B L qp

( ) for , occupied orbitals

( ) for , virtual orbitals

12

( ) for virtual orbitals,occupied orbitals

12

( ) for virtual orbitals,occupied orbitals

pqIJ

apJa

qIa

pIa

qJa

iiIp

iJq

iJp

iIq

biqpib bi

bipqib bi

1

1

(86)

and L is the Lagrangian for TD-HF orbital response. Near acrossing, ref 41 shows that Γij

IJ = γijIJ and Γaj

IJ + ΓjaIJ = γaj

(1),IJ + γja(2),IJ,

provided that the transition density matrix (γIJ) is computedwith time-dependent response theory.For the case of TD-DFT/RPA, one finds that the same corre-

spondence between our analytic derivative couplings and theCM expression holds. The only difference between the case ofTD-HF and the case of TD-DFT is the fact that our Lagrangianis now the TD-DFT Lagrangian. In particular, for the case ofTD-DFT, the γaj

(1),IJ and γja(2),IJ satisfy

γ

γ+ Δ

−= −⎜ ⎟

⎡⎣⎢⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟⎤⎦⎥⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟E

A BB A

II

L

L

00

IJ

IJ

(1),

(2),

(1)

(2)(87)

where

∑ ∑

∑

∑

= Ω + + Ω

+ + Ω − + Ω

+ + Ω + + Ξ

+ + Ξ

L D X X Y Y

X Y X Y X X Y Y

X Y X Y X X Y Y

X Y X Y

[( )

( ) ] [( )

( ) ] [( )

( ) ]

bipq

pbqi pqIJ

jcdiId

jJc

jIc

iJd

cbjd

iId

iJc

jIc

iJd

cdjbjlc

lIb

jJc

jIc

lJb

clji

jIb

lJc

jIc

lJb

cijljlcd

lId

jJc

jIc

lJd

cljdbi

lId

jJc

jIc

lJd

cdjlbi

(1)

(88)

Figure 3. Derivative coupling and gradient difference vectors forprotonated formaldimine at θ = 90° given by TD-HF (left) and TD-DFT (right).

Table 2. Circulations of Derivative Coupling Vectors aroundthe S1/S2 CI Point of Protonated Formaldiminea

magnitude (in units of π)

terms TD-HF TD-DFT

full-DC 0.99975 0.99984NR-DC (projected) 0.88373 0.93946

aNote that only the full-DC recovers the correct Berry phase for bothTD-HF and TD-DFT.

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dx.doi.org/10.1021/jp5057682 | J. Phys. Chem. B XXXX, XXX, XXX−XXXI

∑ ∑

∑

∑

= Ω + + Ω

+ + Ω − + Ω

+ + Ω + + Ξ

+ + Ξ

L D X X Y Y

Y X Y X X X Y Y

Y X Y X X X Y Y

Y X Y X

[( )

( ) ] [( )

( ) ] [( )

( ) ]

bipq

pbqi qpIJ

jcdjIc

iJd

iId

jJc

cbjd

jIc

iJd

iId

jJc

cdjbjlc

jIc

lJb

lIb

jJc

clji

jIc

lJb

lIb

jJc

cijljlcd

jIc

lJd

lId

jJc

cljdbi

jIc

lJd

lId

jJc

cdjlbi

(2)

(89)

and Lbi(1) + Lbi

(2) = Lbi. In the end, we conclude that, just as forthe case of TDHF,41 our TD-DFT/RPA derivative couplingsagree with the CM equality near an excited crossing (ΔE → 0)in the limit of an infinite atomic orbital basis.

VII. CONCLUSIONWe have developed an analytic gradient theory for the deriva-tive couplings between TD-DFT/RPA excited states based onpseudo-wavefunctions. Our solution matches the first (time-independent) formalism of Li and Liu38 but disagrees with theformalism of Tavernelli et al.34 Our final expressions have beenimplemented numerically and validated against finite-differencefor the S1/S5 derivative coupling of formaldehyde. For all threetypes of xc functionals (B3LYP, ωB97, and ωB97X) checked,the difference between our result and the finite-difference datais less than 10−4 a0

−1.To help justify our pseudo-wavefunction theory for both

TD-HF and TD-DFT/RPA, we first calculated the derivativecouplings around the S1/S2 CI of protonated formaldimine.The calculations showed that (i) the full-DC vectors recoverthe correct branching plane as well as the right orientationfor both methods and (ii) the full-DC vectors perfectlyreproduce the expected Berry phase behavior for a looparound the CI. We have also emphasized that orbital re-sponse is essential to recover these two properties. Second,we have shown that our final expression is consistent with theChernyak−Mukamel equality and time-dependent responsetheory near an excited crossing. Looking forward, it will beinteresting to see how the derivative couplings presented herematch with analytical work based exclusively on responsetheory, i.e., the second formalism presented by Li and Liu.38

In the meanwhile, our derivative couplings should find imme-diate use in nonadiabatic calculations and studies ofelectronic relaxation.

■ APPENDIX A: HELLMANN−FEYNMAN DERIVATIVECOUPLINGS FOR TD-DFT/RPA

As illustrated in ref 41., derivative couplings for TD-HF canalso be expressed in the form of a Hellmann−Feynmanexpression:

∑=−

+ ⟨Φ | |Φ ⟩

+ + ⟨Φ | |Φ ⟩

E EX X Y Y A

X Y Y X B

d1

{( )

[( ) }

IJQ

J I ijabiIa

jJb

iIa

jJb

ia Q

jb

iIa

jJb

iIa

jJb

ijab Q

TDHF[ ] [ ]

[ ]HF (A1)

Here A and B are defined as

= A H1 1 (A2)

= B H2 0 (A3)

where H is the TD-HF Hamiltonian and n are configurationinteraction projection operators:

= |Φ ⟩⟨Φ |0 HF HF (A4)

∑= |Φ ⟩⟨Φ |ia

ia

ia

1(A5)

∑= |Φ ⟩⟨Φ |14 ijab

ijab

ijab

2(A6)

Note that TD-DFT/RPA is a natural extension of TD-HFand therefore one can construct Hellmann−Feynman deriva-tive couplings for TD-DFT/RPA by simply replacing H ineqs A2 and A3 with the Kohn−Sham response operator KS(eq 17) and using the Kohn−Sham orbitals to construct theprojection operators.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: subotnik@sas.upenn.edu.

NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSJ.E.S. thanks Filipp Furche for very stimulating conversa-tions and his suggestion to compare our results versus theChernyak−Mukamel equality. This work was supported byNSF CAREER Grant CHE-1150851; J.E.S. also acknowledgesan Alfred P. Sloan Research Fellowship and a David and LucilePackard Fellowship.

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