Derivative couplings between TDDFT excited states obtained by direct differentiation inthe Tamm-Dancoff approximationQi Ou, Shervin Fatehi, Ethan Alguire, Yihan Shao, and Joseph E. Subotnik Citation: The Journal of Chemical Physics 141, 024114 (2014); doi: 10.1063/1.4887256 View online: http://dx.doi.org/10.1063/1.4887256 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Two-component hybrid time-dependent density functional theory within the Tamm-Dancoff approximation J. Chem. Phys. 142, 034116 (2015); 10.1063/1.4905829 Publisher's Note: “Derivative couplings between TDDFT excited states obtained by direct differentiation in theTamm-Dancoff approximation” [J. Chem. Phys.141, 024114 (2014)] J. Chem. Phys. 141, 069903 (2014); 10.1063/1.4891539 Valence excitation energies of alkenes, carbonyl compounds, and azabenzenes by time-dependent densityfunctional theory: Linear response of the ground state compared to collinear and noncollinear spin-flip TDDFTwith the Tamm-Dancoff approximation J. Chem. Phys. 138, 134111 (2013); 10.1063/1.4798402 Analytical Hessian of electronic excited states in time-dependent density functional theory with Tamm-Dancoffapproximation J. Chem. Phys. 135, 014113 (2011); 10.1063/1.3605504 Nonadiabatic coupling vectors for excited states within time-dependent density functional theory in theTamm–Dancoff approximation and beyond J. Chem. Phys. 133, 194104 (2010); 10.1063/1.3503765

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THE JOURNAL OF CHEMICAL PHYSICS 141, 024114 (2014)

Derivative couplings between TDDFT excited states obtained by directdifferentiation in the Tamm-Dancoff approximation

Qi Ou,1 Shervin Fatehi,1,a) Ethan Alguire,1 Yihan Shao,2 and Joseph E. Subotnik1,b)

1Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA2Q-Chem, Inc., 6601 Owens Drive, Suite 105, Pleasanton, California 94588, USA

(Received 17 February 2014; accepted 25 June 2014; published online 14 July 2014;publisher error corrected 25 July 2014)

Working within the Tamm-Dancoff approximation, we calculate the derivative couplings betweentime-dependent density-functional theory excited states by assuming that the Kohn-Sham superpo-sition of singly excited determinants represents a true electronic wavefunction. All Pulay terms areincluded in our derivative coupling expression. The reasonability of our approach can be establishedby noting that, for closely separated electronic states in the infinite basis limit, our final expres-sion agrees exactly with the Chernyak-Mukamel expression (with transition densities from responsetheory). Finally, we also validate our approach empirically by analyzing the behavior of the deriva-tive couplings around the T1/T2 conical intersection of benzaldehyde. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4887256]

I. INTRODUCTION

A. Derivative couplings: Essential matrix elementsfor modeling nonadiabatic processes

Within the Born-Oppenheimer (BO) approximation,1, 2

one attempts to separate electronic and nuclear motion by di-agonalizing the electronic Hamiltonian and generating adi-abatic electronic states which are (hopefully) minimallycoupled. Nevertheless, the BO approximation often breaksdown,3–5 and nonadiabatic dynamical transitions between dif-ferent electronic states—including charge transfer, electronicquenching, and spin-forbidden reactions6—are ubiquitous inchemistry.3, 4 When the BO approximation fails, the adiabaticelectronic states are coupled together (to first order) by the so-called derivative couplings 〈�I|∇Q|�J〉, which promote elec-tronic transitions as mediated by the motion of a given nucleusQ.7, 8 Applying the logic of the Hellmann-Feynman theorem,one can always express the derivative coupling as9

dIJ (R) = 〈�I |∇RH |�J 〉EJ − EI

. (1)

Calculating derivative couplings is absolutely essential formodeling nonadiabatic processes.

B. Previous computational studies and motivation

Theoretical studies of derivative couplings date backto the early work of Lengsfield and Yarkony.10–12 Origi-nally, these authors focused on multi-reference configuration-interaction (MR-CI) wavefunction theory and developed thenecessary computational formalism.8, 13, 14 MR-CI is an espe-cially attractive theory because it generates ground and ex-

a)Current address: Department of Chemistry and Henry Eyring Center forTheoretical Chemistry, University of Utah, 315 South 1400 East, Salt LakeCity, Utah 84103, USA.

b)Electronic mail: subotnik@sas.upenn.edu

cited states in a balanced framework, which cannot be ob-tained by single-reference methods15, 16 (though see Ref. 17).Thus, one can study electronic relaxation from an excitedstate to the ground state via MR-CI. Today, analytic gradi-ents and derivative couplings are readily available for MR-CI calculations,18–23 and many MR-CI applications have beenconducted.14, 24–28 However, the large computational cost isone of the unavoidable downsides of using MR-CI, and as aresult, calculations are limited to relatively small systems (orat least to systems with small active spaces).29–31

Because of the large cost of MR-CI, many modern stud-ies have focused on single-reference approaches which canevaluate excited states and derivative couplings in a moreaffordable way,32–39 although almost certainly with less ac-curacy. In particular, several decades after the formulationof the Runge-Gross theorem,40 time-dependent density func-tional theory (TDDFT) appears to be the current methodof choice for modeling electronic relaxation in photoexcitedorganics.41–43 (DFT orbitals have been shown to resembleDyson orbitals.44) Unfortunately, calculating derivative cou-plings is complicated for TDDFT because the Kohn-Shamwavefunction is not a true wavefunction. To circumvent thisobstacle, the Chernyak-Mukamel approach is to notice that,for an exactly stationary state, the derivative coupling can beexpressed as a function of the transition density (γ IJ) betweentwo states and their relative energy gap32, 45, 46 (with detaileddiscussion given in Appendix C),

dCMIJ = 1

EJ − EI

∑pq

v[x]pq γ IJ

pq . (2)

Furthermore, a meaningful transition density can be com-puted from linear response TDDFT,47 and Send andFurche48, 49 have already used this fact to calculate exactTDDFT derivative couplings from ground to excited state.

Recently, Tavernelli et al.33–36 have derived expressionsfor the derivative couplings between TDDFT excited states

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024114-2 Ou et al. J. Chem. Phys. 141, 024114 (2014)

that partially reduce to the Chernyak-Mukamel expression inthe limit of excited state wavefunctions that exactly solve theSchrödinger equation, as have Hu et al.37–39 (Ref. 38 includ-ing the Pulay terms50). Derivative couplings between TDDFTexcited states are likely more meaningful than those betweenground state and excited state because, in the former case,one recovers the correct dimensionality of a conical intersec-tion branching plane, whereas this is not always true in thelatter case.15, 16 Nevertheless, for the Tamm-Dancoff approx-imation (TDA) states, the Tavernelli formalism constructs aderivative coupling by differentiating only the Kohn-ShamFock matrix and omits the full coupling that is induced by thesecond derivative of the exchange-correlation (xc) function-als. Thus, for example, whereas the third derivative of the xcfunctional appears in TDA analytic gradient theory,51 it is notpresented in the Tavernelli formalism for the derivative cou-pling. As such, the Tavernelli formalism cannot exactly agreewith the Chernyak-Mukamel expression.84

Following up on our earlier work on CIS derivativecouplings,52 the goal of this article is to reexamine the deriva-tive couplings between TDDFT excited states (including allPulay and response terms) by using the simplest possible ap-proach: direct differentiation. More specifically, we will treatthe TDDFT/TDA Kohn-Sham wavefunction as if it were atrue electronic wavefunction and, through direct differentia-tion, we will derive an analytical expression for the derivativecouplings between two Tamm-Dancoff TDDFT states. Ourrationality is as follows: even though the Kohn-Sham linear-response wavefunction is not rigorously meaningful, the truederivative couplings between TDA excited states must obeycertain conditions around a conical intersection. For instance,(i) the derivative coupling should be orthogonal to the differ-ence gradient in properly scaled coordinates; (ii) it should di-verge at a conical intersection; (iii) its integral should be π fora loop encircling the conical intersection. For all of these rea-sons, it is not unreasonable to consider a derivative couplingdIJ from direct differentiation where the behavior around aconical intersection is guaranteed to be correct (and the for-mal expression can be easily transformed to the energy gradi-ent when I = J). Moreover, we will show in Appendix C that,near a conical intersection, our computed derivative couplingsmatch the exact derivative coupling expression in Eq. (2) withthe transition density calculated according to response theory(in the limit of an infinite basis).

C. Outline

An outline of this article is as follows: all of our theorywill be presented in Sec. II. Our analytical expressions will bea little messy because, as is common for molecular systems,we will work in a basis of atomic orbitals (AOs) – and byincluding all Pulay terms, we will quantitatively investigatethe effect of the basis set on the final answer. In Sec. III, weshow that our results match finite-difference data for lithiumhydride using three different xc functionals (B3LYP, ωB97,and ωB97X). In Sec. IV, we will apply our new formalism tothe T1/T2 conical intersection of benzaldehyde, which showsthe substantial effects of Pulay terms, response terms, andsecond derivative of the xc functionals. In Sec. V, we con-

clude. In Appendix C, a comparison of our approach with theChernyak-Mukamel formula is given.

D. Notation

Here, we summarize the notation in this work as follows:spin molecular orbitals (MO) are denoted by lowercase latinletters (a, b, c, d for virtual orbitals, i, j, k, l, m for occupiedorbitals, p, q, r, s, w for arbitrary orbitals). AOs are denotedby Greek letters (α, β, γ , δ, λ, σ , μ, ν). The TDA excitedstates are denoted by � (with uppercase latin indices I, J).

II. ANALYTIC DERIVATION FOR TDDFT/TDADERIVATIVE COUPLINGS

A. Single-reference excited states within theTamm-Dancoff approximation

The derivation of TDDFT/TDA derivative couplings pre-sented here will parallel the derivation given in Ref. 52 forCIS derivative couplings. TDDFT/TDA Kohn-Sham eigen-states have the form

|�I 〉 =∑ia

t Iai

∣∣�ai

⟩(3)

=∑ia

t Iai a

†aai |�DFT〉. (4)

TDDFT/TDA states are linear combinations of singly ex-cited determinants {|�a

i 〉}, which differ from the DFT groundstate determinant |�DFT〉 by the replacement of a spin orbitalfrom the occupied subspace (of size O) with a spin orbitalfrom the virtual subspace (of size V ). The procedure for de-termining the TDDFT/TDA amplitudes (tai ) requires diago-nalizing the Kohn-Sham linear-response tensor A,

Aiajb = ⟨�a

i

∣∣OKS

∣∣�bj

⟩, (5)

that appears in the eigenvalue equation

AT = ETDAT, (6)

where T is the tensor of amplitudes for all states. The resultingTDDFT/TDA energy is given by

∑ijab t Ia

i AiajbtJbj = EIδIJ .

The Kohn-Sham operator OKS can be written in second-quantized and antisymmetrized form (with physicists notationfor the two-electron-integrals53)

OKS =∑pq

Fpqa†paq

+∑cdkl

[ clkda†caka

†l ad + cdkl(a

†ca

†dalak + a

†l a

†kacad )],

(7)

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

Fpq = hpq +∑m

�pmqm, (8)

pqsr = �pqsr + ωpqsr . (9)

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024114-3 Ou et al. J. Chem. Phys. 141, 024114 (2014)

The diagonal entries of the Fock matrix Fpp ≡ εp are theusual Kohn-Sham orbital energies, if we define h0

pq as the ma-trix element of the kinetic energy plus the external potential(Eq. (10)) and gpq is the first derivative of the xc functionalfxc (Eq. (11)),54 then the sum of h0

pq and gpq gives the one-electron effective operator hpq (Eq. (12)),

h0pq ≡ 〈p|h0|q〉, (10)

gpq ≡∑pq

∫drφp(r)

∂fxc

∂ρ(r)φq(r), (11)

hpq ≡ h0pq + gpq. (12)

�pqsr is the Coulomb term plus whatever fraction of Hartree-Fock exchange is included in the DFT functional (cHF inEq. (13)), and ωpqsr is the second derivative of the xc func-tional (Eq. (14)). The sum of �pqsr and ωpqsr gives pqsr(Eq. (9)),

�pqsr ≡ 〈pq|sr〉 − cHF〈pq|rs〉, (13)

ωpqsr ≡ 〈pq|f ′′xc|sr〉

=∑pqsr

∫drφp(r)φq(r)

∂2fxc

∂ρ(r)2φr (r)φs(r). (14)

With the definitions above, OKS can be rewritten as

OKS =∑pq

(h0

pq + gpq +∑m

�pmqm

)a†paq

+∑cdkl

[(�clkd + ωclkd )a†caka

†l ad

+ (�cdkl + ωcdkl)(a†ca

†dalak + a

†l a

†kacad )]. (15)

Note that the cdkl term in OKS will not contribute to thesingle-single coupling as stated in TDA. Inserting these ex-pressions into Eq. (5), one has

Aiajb = ajib + δijFab − δabFij + δij δabEDFT. (16)

B. The “brute force” expression

Before we start this section, it is helpful to define severaldensity matrices for future use. These definitions are consis-tent with those in Ref. 52.

1. The general density matrices

Pμν =∑m

CμmCνm, (17)

Pμν =∑

p

CμpCνp = Pμν +∑

a

CμaCνa. (18)

Here, C is the matrix of MO coefficients. Notethat we may express the real-space density as ρ(r)= Pμνφμ(r)φν(r).

2. The TDA excitation-amplitude matrix, also called thetransition density matrix

RIμν =

∑ia

CμatIai Cνi . (19)

3. The generalized difference-density matrix55

DIJμν =

∑iab

CμatIai tJb

i Cνb −∑ija

CμitJai t Ia

j Cνj . (20)

We will now construct derivative couplings by a “bruteforce” approach. For the Hellmann-Feynman analogue, seeAppendix B. The “brute force” expression for the derivativecoupling can be obtained by direct differentiation (with a su-perscript [x] denoting a gradient in the x direction),⟨�I

∣∣�[x]J

⟩ = ∑ijab

t Iai

⟨�a

i

∣∣ (tJbj

∣∣�bj

⟩)[x](21)

=∑ijab

t Iai t

Jb[x]j

⟨�a

i

∣∣�bj

⟩+∑ijab

t Iai tJb

j

⟨�a

i

∣∣�b[x]j

⟩(22)

=∑ia

t Iai t

Ja[x]i +

∑ijab

t Iai tJb

j

⟨�a

i

∣∣�b[x]j

⟩(23)

= 1

EJ − EI

∑ijab

t Iai A

[x]iajbt

Jbj −

∑iab

t Iai tJb

i OR[x]ba

−∑ija

t Iai tJa

j OR[x]ji , (24)

where we define the “right” spin-orbital derivative overlap,

OR[x]pq ≡ 〈p|q[x]〉. (25)

In the MO representation,

OR[x]bi = 〈b|i[x]〉 (26)

=(∑

μ

Cμb 〈μ|)(∑

ν

|ν〉C[x]νi +

∑ν

|ν[x]〉Cνi

)(27)

=∑μν

CμbSμνC[x]νi +

∑μν

CμbSR[x]μν Cνi, (28)

where the atomic orbital overlap is

Sμν ≡ 〈μ|ν〉 . (29)

Analogous to the OR[x]bi , we define the right derivative of

the overlap as follows:

S[x]μν ≡ 〈μ[x]|ν〉 + 〈μ|ν[x]〉 (30)

≡ SL[x]μν + S

R[x]μν . (31)

At this point, we need to calculate the tensor A and its deriva-tives. Formally, from Eq. (16),

A[x]iajb =

[x]ajib + δijF

[x]ab − δabF

[x]ij + δij δabE

[x]DFT. (32)

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024114-4 Ou et al. J. Chem. Phys. 141, 024114 (2014)

Here, all derivative terms are important except for E[x]DFT,

which cannot contribute to the final expression by orthogonal-ity of TDDFT/TDA excited states. The following expressionis then obtained by plugging Eq. (32) into Eq. (24):

⟨�I

∣∣�[x]J

⟩ = 1

EJ − EI

∑ijab

t Iai tJb

j [x]ajib

+∑iab

t Iai tJb

i

(1

EJ − EI

F[x]ab − O

R[x]ba

)

−∑ija

t Iai tJa

j

(1

EJ − EI

F[x]ij + O

R[x]ji

). (33)

Finally, we now use the formal derivations in Ref. 52 to ac-quire the expressions for F[x], �[x], and OR[x]. Expanding theseterms in our expression for A

[x]iajb, we find a long and compli-

cated coupling expression,

⟨�I

∣∣�[x]J

⟩ = 1

EJ − EI

∑ijab

t Iai tJb

j

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩∑μνλσ

CμaCνj [x]μνλσCλiCσb

−1

2

∑αβw

CαwS[x]αβ

⎡⎢⎢⎢⎢⎣

Cβa wjib

+Cβj awib

+Cβi ajwb

+Cβb ajiw

⎤⎥⎥⎥⎥⎦

+∑

k

kjib�[x]ak +

∑c

acib�[x]jc

+∑

c

ajcb�[x]ic +

∑k

ajik�[x]bk

⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭

+∑iab

t Iai tJb

i

⎧⎨⎩ 1

EJ − EI

⎡⎣∑

μν

Cμah[x]μνCνb

+∑

μνλσm

CμaCνm�[x]μνλσCλbCσm

−1

2

⎛⎝εa

∑αβ

CαaS[x]αβ Cβb + εb

∑αβ

CαbS[x]αβ Cβa

⎞⎠

−1

2

∑αβmw

CαwS[x]αβ Cβm

(�awbm + �ambw

)

−∑mc

�[x]cm(�acbm+�ambc)

]−∑μν

CμbCνaSA[x]μν

}

−∑ija

t Iai tJa

j

{1

EJ − EI

[∑μν

Cμih[x]μνCνj

+∑

μνλσm

CμiCνm�[x]μνλσCλjCσm

−1

2

⎛⎝εi

∑αβ

CαiS[x]αβ Cβj + εj

∑αβ

CαjS[x]αβ Cβi

⎞⎠

−1

2

∑αβmw

CαwS[x]αβ Cβm(�iwjm + �imjw)

−∑mc

�[x]cm(�icjm+�imjc)

]+∑μν

CμjCνiSA[x]μν

}, (34)

where SA[x]μν is defined as S

R[x]μν − 1

2S[x]μν . Henceforward, all

terms in Eq. (34) with a factor of S[x] will be called “Pulay”terms.50 Note that the orbital rotations between occupied andvirtual subspaces (�ab and �ij) disappear and are absent inEq. (34) (just as for CIS).52 Compared with the correspondingCIS expression (Eq. (A21) in Ref. 52), Eq. (34) differs by in-cluding the gradient of the first derivative of the xc functionalin the derivative of the Fock matrix F[x] (g[x]), an extra termin �[x] which is the gradient of the second derivative of the xcfunctional (ω[x]), and the fraction of Hartree-Fock exchangeterm (cHF) present in the DFT functional (�[x]).

C. Exact TDDFT/TDA derivative couplings:The non-response component

Equation (34) is quite lengthy and, for the most part, theright hand side is almost identical with the corresponding CISderivative couplings (Eq. (A21) in Ref. 52). Thus, at this junc-ture, let us focus only on those terms which are unique to theTDDFT expressions. To begin, we need several new defini-tions. First, note that the gradient of the first derivative of thexc functional g[x] can be decomposed in the AO representationas

g[x]μν ≡ g

[x]μν + g

Y[x]μν , (35)

where54

g[x]μν ≡

∫ [x]

drφμ(r)∂fxc

∂ρ(r)φν(r)

+∫

dr∂fxc

∂ρ(r)(φμ(r)φν(r))[x]

+∑λσ

∫drφμ(r)φν(r)

∂2fxc

∂ρ(r)2Pλσ (φλ(r)φσ (r))[x],

(36)

gY[x]μν ≡

∑λσ

P[x]λσ ωμνλσ . (37)

The integral∫

[x] shown in the first term of Eq. (36) rep-resents differentiation with respect to the Becke weights inthe quadrature for the exchange-correlation functional. Notethat g

[x]μν enters in h

[x]μν , and the total one-electron-integral

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024114-5 Ou et al. J. Chem. Phys. 141, 024114 (2014)

derivative for TDDFT can be written as

h[x]μν = h

0[x]μν + g

[x]μν

= h0[x]μν + g

[x]μν + g

Y[x]μν

≡ h[x]μν + g

Y[x]μν . (38)

Recall that h0[x]μν is the derivative of the kinetic energy

plus the external potential, which is exactly the same as inthe CIS expression. We label the sum of h

0[x]μν and g

[x]μν as h

[x]μν

to denote the non-response part of the one-electron-integralderivative.

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

ω[x]μνλσ = ω

[x]μνλσ + ω

Y[x]μνλσ , (39)

where

ω[x]μνλσ ≡

∫ [x]

drφμ(r)φν(r)∂2fxc

∂ρ(r)2φλ(r)φσ (r)

+∫

dr(φμ(r)φν(r))[x] ∂2fxc

∂ρ(r)2φλ(r)φσ (r)

+∫

drφμ(r)φν(r)∂2fxc

∂ρ(r)2(φλ(r)φσ (r))[x]

+∑γ δ

∫drφμ(r)φν(r)

∂3fxc

∂ρ(r)3φλ(r)φσ (r)

×Pγδ(φγ (r)φδ(r))[x], (40)

ωY[x]μνλσ ≡

∑γ δ

P[x]γ δ �μνλσγ δ, (41)

and �μνλσγ δ is the xc functional third derivative,

�μνλσγ δ ≡∫

drφμ(r)φν(r)∂3fxc

∂ρ(r)3φλ(r)φσ (r)φγ (r)φδ(r).

(42)

Thus, the total two-electron-integral derivatives forTDDFT can be written as

[x]μνλσ = �

[x]μνλσ + ω

[x]μνλσ

= �[x]μνλσ + ω

[x]μνλσ + ω

Y[x]μνλσ

≡ [x]μνλσ + ω

Y[x]μνλσ . (43)

With the definitions made in Sec. II B and the expressionsabove, Eq. (34) can be rewritten as⟨

�I

∣∣�[x]J

⟩

= 1

EJ − EI

{∑μν

DIJμν

(h

[x]μν + g

Y[x]μν

)

+∑μνλσ

(RI

μλRJσν

(

[x]μνλσ + ω

Y[x]μνλσ

)+ DIJμλPσν�

[x]μνλσ

)

−1

2

∑αβμν

S[x]μν Pμα

(DIJ

βν + DIJνβ

)Fαβ

−1

2

∑μναβγ δ

S[x]μν Pμα

(RI

νγ RJδβ + RI

δβRJνγ

) αβγ δ

−1

2

∑μναβγ δ

S[x]μν Pμα

(RI

γνRJβδ + RI

βδRJγ ν

) αβγ δ

+1

2

∑μναβγ δ

S[x]μν PμαPνδ

(DIJ

γβ + DIJβγ

)�αβγ δ

−∑bi

Ybi�[x]bi

}

−∑μνiab

CνatIai tJb

i CμbSA[x]μν −

∑μνija

Cνi tIai tJa

j CμjSA[x]μν ,

(44)

where

Ybi =∑

μνλσd

CμbCλd

(RI

νσ tJdi + t Id

i RJνσ

) μνλσ

+∑μνλσ

CνbCσi

(DIJ

μλ + DIJλμ

)�μνλσ

+∑

μνλσ�

Cμ�Cσi

(RI

νλtJb� + t Ib

� RJνλ

) μνλσ . (45)

D. Response terms in TDDFT/TDA derivativecouplings

Finally, all that remains to do is to treat the so-called “re-sponse” terms, gY[x] and ωY[x] in Eq. (44). Combining gY[x]

and ωY[x] with their multiplying coefficients in Eq. (45), onehas ∑

μν

DIJμνg

Y[x]μν =

∑μνλσ

DIJμνP

[x]λσ ωμνλσ , (46)

∑μνλσ

RIμλR

Jσνω

Y[x]μνλσ =

∑μνγ δλσ

RIμλR

JσνP

[x]γ δ �μνλσγ δ. (47)

Now, using standard analytic gradient theory summarizedin Ref. 52, one can always write (a derivation is provided inAppendix A),

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P[x]νσ = −1

2

∑αβ

(PναPσβ + PσαPνβ)S[x]αβ −

∑ib

(CνbCσi + CνiCσb)�[x]bi . (48)

Combining Eqs. (46)–(48), one finds

∑μν

DIJμνg

Y[x]μν = −

∑μνλσ

DIJμν

[12

∑αβ(PλαPσβ + PσαPλβ)S[x]

αβ

+∑bi(CλbCσi + CλiCσb)�[x]

bi

]ωμνλσ , (49)

∑μνλσ

RIμλR

Jσνω

Y[x]μνλσ = −

∑μνγ δλσ

RIμλR

Jσν

[12

∑αβ(Pγ αPδβ + PδαPγβ)S[x]

αβ

+∑bi(CνbCσi + CνiCσb)�[x]

bi

]�μνλσγ δ, (50)

and after relabeling the indices, Eqs. (49) and (50) become∑μν

DIJμνg

Y[x]μν = 1

2

∑μναβγ δ

S[x]αβ PαμPβσ

(DIJ

λν + DIJνλ

)ωμνλσ

−∑

μνλσbi

CνbCσi

(DIJ

μλ + DIJλμ

)ωμνλσ�

[x]bi , (51)

∑μνλσ

RIμλR

Jσνω

Y[x]μνλσ = −1

2

∑μναβγ δλσ

S[x]αβ Pαγ Pβδ

(RI

μλRJσν + RI

σνRJμλ

)�μνλσγ δ

−∑

μνλσγ δbi

CγbCδi

(RI

μλRJσν + RI

σνRJμλ

)�μνλσγ δ�

[x]bi . (52)

Finally, after a bit of tedious algebra and simplification,one arrives at the final expression,⟨

�I

∣∣�[x]J

⟩= 1

EJ − EI

{∑μν

DIJμν h

[x]μν

+∑μνλσ

(RI

μλRJσν

[x]μνλσ + DIJ

μλPσν�[x]μνλσ

)

−1

2

∑αβμν

S[x]μν Pμα

(DIJ

βν + DIJνβ

)Fαβ

−1

2

∑μναβγ δ

S[x]μν Pμα

(RI

νγ RJδβ + RI

δβRJνγ

) αβγ δ

−1

2

∑μναβγ δ

S[x]μν Pμα

(RI

γνRJβδ + RI

βδRJγ ν

) αβγ δ

−1

2

∑μναβγ δλσ

S[x]μν PμλPνσ

(RI

αγ RJδβ + RI

δβRJαγ

)�αβγ δλσ

+1

2

∑μναβγ δ

S[x]μν PμαPνδ

(DIJ

γβ + DIJβγ

) αβγ δ

−∑bi

Ybi�[x]bi

}

−∑μνiab

CνatIai tJb

i CμbSA[x]μν −

∑μνija

Cνi tIai tJa

j CμjSA[x]μν ,

(53)

where

Ybi =∑

μνλσd

CμbCλd

(RI

νσ tJdi + t Id

i RJνσ

) μνλσ

+∑μνλσ

CνbCσi

(DIJ

μλ + DIJλμ

) μνλσ

+∑

μνλσ�

Cμ�Cσi

(RI

νλtJb� + t Ib

� RJνλ

) μνλσ

+∑

μνλσγ δ

CγbCδi

(RI

μλRJσν + RI

σνRJμλ

)�μνλσγ δ.

(54)

Note that the complete orbital-response Lagrangian Y inthe TDDFT expression (Eq. (54)) differs from that in CIS(Eq. (60b) in Ref. 52) in three ways. First, there is only afraction of Hartree-Fock exchange (cHF) in the two-electronintegral. Second, the second derivative of the xc functionalappears, i.e., there is a ω term in � (compared with � forCIS). Third, the third derivative of the xc functional � alsoappears (which has no CIS counterpart).56

E. Coupled-perturbed Hartree-Fock equation (CPHF)

As a practical matter, in order to compute the � deriva-tive in Eq. (53), one needs to solve the CPHF equation,57, 58

�[x]bi = −

∑ja

(∂2E

∂�aj∂�bi

)−1

M[x]aj , (55)

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024114-7 Ou et al. J. Chem. Phys. 141, 024114 (2014)

where M[x] is the matrix that contains all the mixed deriva-tives,

M[x]aj =

∑αβ

∂2E

∂�aj∂Sαβ

S[x]αβ +

∑αβ

∂2E

∂�aj∂hαβ

h[x]αβ (56)

+∑αβγ δ

∂2E

∂�aj∂�αβγ δ

�[x]αβγ δ.

As is standard in analytic gradient methods, we use the“z-vector” method developed by Handy and Schaefer,59 andwe iteratively construct

zaj =∑ib

Ybi

(∂2E

∂�aj∂�bi

)−1

. (57)

With this z-vector saved to disk, we compute M[x]aj for

each coordinate x so that∑

bi Ybi�[x]bi can be obtained. Thus,

Eqs. (53) and (54) become⟨�I

∣∣�[x]J

⟩= 1

EJ − EI

{∑μν

DIJμν h

[x]μν

+∑μνλσ

(RI

μλRJσν

[x]μνλσ + DIJ

μλPσν�[x]μνλσ

)

−1

2

∑αβμν

S[x]μν Pμα

(DIJ

βν + DIJνβ

)Fαβ

−1

2

∑μναβγ δ

S[x]μν Pμα

(RI

νγ RJδβ + RI

δβRJνγ

) αβγ δ

−1

2

∑μναβγ δ

S[x]μν Pμα

(RI

γνRJβδ + RI

βδRJγ ν

) αβγ δ

−1

2

∑μναβγ δλσ

S[x]μν PμλPνσ

(RI

αγ RJδβ + RI

δβRJαγ

)�αβγ δλσ

+1

2

∑μναβγ δ

S[x]μν PμαPνδ

(DIJ

γβ + DIJβγ

) αβγ δ

}

−⎛⎝∑

μνiab

CνatIai tJb

i Cμb +∑μνija

Cνi tIai tJa

j Cμj

⎞⎠ S

A[x]μν ,

(58)

where DIJ represents the relaxed difference density matrix,

DIJμν ≡ DIJ

μν −∑aj

zaj (CμaCνj + CμjCνa)

= DIJμν − (zμν + zνμ). (59)

Equations (54), (57)–(59) are a complete recipe forderivative couplings that is easy to evaluate. In summary,the differences between TDDFT and CIS expressions are thepresences of (i) different one-electron-integral derivative forTDDFT h[x] (compared with h0[x] in CIS), (ii) �

[x]and �

(compared with �[x] and � in CIS), which include the sec-ond xc functional derivative and specific fraction of Hartree-Fock exchange term involved in TDDFT functionals, and(iii) �, the third xc functional derivative (which has no CIScounterpart).

III. COMPARISON WITH FINITE-DIFFERENCE

In order to verify the equations above and also check ournumerical implementations of TDDFT/TDA derivative cou-plings, we calculated the magnitude of the derivative cou-plings between the S1 and S4 states of lithium hydride (LiH),and compared the results with finite-difference method. Thestandard central-difference formula yields an expression forthe derivative coupling as

⟨�I

∣∣�[x]J

⟩ ≈ 〈�I (x)|�J (x + �x)〉 − 〈�I (x)|�J (x − �x)〉2�x

.

(60)

The ab initio quantum chemistry package Q-Chem60, 61

was employed for the calculations. Three different function-als (B3LYP,62, 63 ωB97, and ωB97X64) were tested using6-31G* basis set (CIS results are also listed for comparison).As shown in Table I, our analytical approach matches thefinite-difference data with an error ∼10−4 a−1

0 . When Pulayterms (S[x]) are neglected, the resulting derivative couplingschange significantly.

IV. APPLICATION TO BENZALDEHYDE

As a prototypical aromatic carbonyl compound, ben-zaldehyde has drawn significant attention in both experimen-tal and theoretical studies because of its unique spectroscopicproperties.65–74 A great deal of theory and experiment has fo-cused on the two lowest triplet states of benzaldehyde in orderto explain the mechanism of the molecule’s highly phospho-rescent radiation.65, 66, 71, 73–77 The interstate mixing betweenthe T1(n-π*) and T2(π -π*) states of benzaldehyde was es-timated as long ago as the 1970s.68, 78, 79 Since the deriva-tive coupling is essential for understanding nonadiabatic dy-namics and radiationless transitions in general, we will also

TABLE I. Derivative couplings between the S1 and S4 states of LiH as com-puted by finite difference (FD), analytical theory (full-DC), and DC withoutPulay terms (NP).

〈�1|�[Q]4 〉(a−1

0 )

xc functional Atom moved (Q) FD Full-DC NP

B3LYP H − 0.00564 − 0.00570 − 0.12611Li 0.15804 0.15810 0.12611

ωB97 H − 0.04049 − 0.04053 − 0.06985Li 0.15150 0.15154 0.06985

ωB97X H − 0.05952 − 0.05955 − 0.07422Li 0.16457 0.16460 0.07422

CIS H − 0.05765 − 0.05766 − 0.04389Li 0.17475 0.17476 0.04389

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024114-8 Ou et al. J. Chem. Phys. 141, 024114 (2014)

investigate here the T1/T2 derivative coupling of benzalde-hyde with our analytic gradient method.

In a previous paper,80 we showed that a conical in-tersection point can be located between the first and thesecond triplet states of benzaldehyde according to theTDDFT/ωB97X functional. Now, with a working code forcalculating derivative couplings, we revisit geometries aroundthe conical intersection point on the branching plane and cal-culate the derivative coupling with and without Pulay termsfor each geometry. Consistent with the previous work, theωB97X functional and 6-31G** basis set are used for all cal-culations.

In Ref. 80, we defined raw g and h vectors and locatedthe branching plane for benzaldehyde as follows:

i. Find a CI point, RCI .ii. Displace each of the 3N Cartesian coordinates in posi-

tive and negative directions, and at every displaced pointperform (6N) gradient calculations for the adiabaticenergies

D±i = 1

2

(∇Ead2 ( RCI ± �R · ei)

−∇Ead1 ( RCI ± �R · ei)

), i = 1, . . . , 3N. (61)

iii. Notice that all 3N gradients actually lie in a single 2Dplane – the g − h branching plane. At this point, we mustmake a non-unique choice of g and h that corresponds toa unique diabatic basis.

iv. In our calculations, we check the energy difference gra-dient D along a circle that is centered at RCI in theg − h plane. If θ is the angle of rotation around RCI ,we have already computed D(θ ) in step ii. We define gas D(θmax) when θmax is chosen as the angle that maxi-mizes ‖ D(θ )‖.

v. Finally, we check for the angle that minimizes ‖ D(θ )‖.By construction, D(θmin) must be perpendicular to g andcan be defined as h.

In this article, we rescale g and h vectors so that the normof the gradient difference is identical at every point on theloop,

x = 1

‖g‖g, (62)

y = ‖g‖‖h‖2

h. (63)

Note that x and y are perpendicular (just like g and h). Inwhat follows, we will investigate the derivative couplings ina loop around the CI point, chosen as x cos θ + y sin θ forθ = 0◦, 10◦, 20◦, . . . , 350◦ at the distance r = 0.001 Å. Thus,with normalized coordinates x and y, we construct a loop de-fined by the Cartesian coordinates

R(θ ) = RCI + 0.001(x cos θ + y sin θ ). (64)

Note that 0.001 is in units of Angstroms. Five differenttypes of derivative couplings below are calculated:

i. Full-DC: complete derivative couplings given byEq. (58).

ii. ETF-DC: corrected derivative couplings given byEq. (58) when setting SA[x] = 0 or SR[x] = 1

2 S[x]. As wasillustrated in Refs. 52 and 81, this replacement is equiv-alent to including perturbative electron-translation fac-tors (ETF) in the derivative coupling and restores trans-lational invariance.

iii. NP: derivative couplings without all Pulay terms inEq. (58), i.e., SR[x], S[x] → 0.

iv. Rel-DH: derivative couplings given by using aHellmann-Feynman expression with the Fock operatorin place of the Hamiltonian. In other words, we collectall terms that contain DIJ in Eq. (58) (including orbitalresponse terms).

v. DH: derivative couplings given by using a Hellmann-Feynman expression with the Fock operator in place ofthe Hamiltonian. In other words, we collect all terms thatcontain DIJ in Eq. (53) (no orbital response terms).

A. Results

For a physical picture of the nondiabatic motion in ben-zaldehyde, the full-DC, NP, rel-DH, and DH vectors are visu-alized by the quiver plots shown in Fig. 1. Note the ETF-DC

(a) (b)

(c) (d)

FIG. 1. Derivative coupling vectors for benzaldehyde at θ = 30◦ (a) usingfull analytic gradient theory (full-DC, Eq. (58)), (b) neglecting all Pulay terms(NP), (c) using only the terms that contain DIJ in Eq. (58) (rel-DH), (d) usingthe terms that contain DIJ in Eq. (53) (DH).

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024114-9 Ou et al. J. Chem. Phys. 141, 024114 (2014)

x

y

z

x

y

z

(a) (b)

x

y

z

x

y

z

(c) (d)

FIG. 2. Full-DC, NP, rel-DH, and DH derivative coupling vectors on the circular loop (r = 0.001 Å) in the branching plane. 36 single-point calculations wereperformed. (a) full-DC, (b) NP, (c) rel-DH, (d) DH.

plot is omitted (since it looks almost exactly the same as full-DC). In Fig. 2, we plot the derivative couplings on the looparound the conical intersection. As Fig. 2 shows, only the full-DC and ETF-DC vectors (which are roughly identical) lie rig-orously on the same physically correct branching plane. Theother vectors, NP, rel-DH, and DH, are somewhat out of theplane. The out-of-plane angle varies from 2◦ to 10◦ for rel-DH vectors and from 15◦ to 20◦ for DH and NP vectors. InTable II, we list the out-of-plane angles for the rel-DH, DH,and NP vectors at θ = 30◦. To further assess the behavioraround the conical intersection, we project these derivativecoupling vectors on the branching plane in order to make ameaningful evaluation of their performances around the con-ical intersection. The full-DC and ETF-DC vectors as wellas the rel-DH projections on the branching plane are tangentto the loop, i.e., perpendicular to the gradient difference. Bycontrast, even after projection, the other methods (NP and DHprojection) do not yield that correct orientation for the deriva-

TABLE II. Out-of-plane angles for the rel-DH, DH, and NP vectors atθ = 30◦.

Terms Out-of-plane angles (deg)

rel-DH 5.642DH 16.831NP 15.898

tive couplings around the conical intersection point. To provethis point, for θ = 30◦, in Table III, we list the exact magni-tudes of the derivative couplings and their angles relative tothe gradient difference.

Another means to check the reasonability of our deriva-tive coupling is to calculate the phase factor around the con-ical intersection. It is well-known that Berry’s phase will notdisappear for a closed path surrounding a conical intersec-tion. Mathematically, for a loop C in the branching plane, the

TABLE III. Magnitudes of derivative coupling vectors between T1 andT2 states of benzaldehyde and their angles relative to the gradient differ-ence at θ = 30◦. Full-DC: The complete derivative couplings given by Eq.(58). ETF-DC: Corrected derivative couplings given by Eq. (58) when set-ting SR[x] = 1

2 S[x]. rel-DH: All terms that contain DIJ in Eq. (58). DH: Allterms that contain DIJ in Eq. (53). NP: Derivative couplings without all Pu-lay terms. Note that only the full-DC, ETF-DC, and projected rel-DH vectorsare effectively orthogonal to the energy gradient difference in properly scaledcoordinates.

Terms Magnitudes (a−10 ) Angles (deg)

full-DC 261.05361 90.234ETF-DC 260.45083 90.228rel-DH (unprojected) 263.67060 90.034rel-DH (projected) 262.39583 90.034DH (projected) 180.52366 74.510NP (projected) 362.58573 96.113

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024114-10 Ou et al. J. Chem. Phys. 141, 024114 (2014)

TABLE IV. Circulations of derivative couplings vectors around the T1/T2conical intersection point of benzaldehyde. Note that only the full-DC, ETF-DC, and the projected rel-DH vectors recover the correct Berry’s phase.

Terms Magnitudes (in units of π )

full-DC 0.99939ETF-DC 0.99938rel-DH (projected) 1.00137DH (projected) 0.77513NP (projected) 1.46068

circulation of the derivative coupling must be π ,

φ =∮

C

dIJ (R) · dR =∫ 2π

0rdIJ (θ ) · dθ = π, (65)

where dIJ refers to the derivative coupling between the corre-sponding states. With a finite number of points (i.e., 36) takenon the loop as shown in Fig. 2, we calculated the sum of thederivative coupling vectors dotted into the direction of eachδθ , i.e., the tangential direction,

φ ≈36∑i=1

rdIJ (θ i) · δθ i . (66)

Results for different variations of derivative couplingvectors are shown in Table IV. The result for the full-DC(or ETF-DC), which is very close to π , perfectly reproducesthe geometric phase factor and further justifies our analyti-cal theory for the derivative coupling. Interestingly, note thatthe projected rel-DH vectors also recover the exact Berry’sphase. Given the fact that the out-of-plane angles for rel-DHvectors are relatively small (less than 10◦), it may be true thatthe rel-DH is a decent approximation to the exact derivativecoupling around a conical intersection point. Further investi-gation is needed. Finally, we observe that NP and DH approx-imations do not come close to satisfying

∮CdIJ(R)dR = π . In

particular, referring to Table III, one can see that the magni-tude of the projected NP vector is twice as large as the oneof the full-DC vector, and the magnitude of the projected DHvector is significantly smaller; this explains the incorrect be-havior of the NP and DH circulations in Table IV. Overall,our calculation highlights the facts that, (i) in a finite atomic-orbital basis, Pulay terms are non-negligible for the correctderivative coupling in the vicinity of a conical intersection;(ii) the orbital responses also need to be taken into account inorder to yield the exact properties of the derivative coupling;(iii) only the full-DC recovers both the correct branch-ing plane and Berry’s phase behavior around a conicalintersection.

V. CONCLUSION

In this work, we calculated derivative couplings betweenTDDFT/TDA states via analytic gradient theory by assumingthat we can treat the Kohn-Sham excited state wavefunctionas if it were a true wavefunction, and then we implementedthe resulting equations numerically. With all Pulay terms in-cluded, our theory has been numerically validated against

the finite-difference data for lithium hydride, with an errorless than 10−4 a−1

0 for three types of xc functionals (B3LYP,ωB97, and ωB97X).

As an application, we investigated benzaldehyde and westudied the T1/T2 conical intersection point located in Ref. 80.The considerable differences between the NP, DH, and rel-DH derivative couplings and the full-DC result emphasizesthe qualitative significance of Pulay terms as well as the or-bital responses. Only the full-DC and ETF-DC vectors lie inthe branching plane and are perpendicular to the energy gradi-ent difference. Furthermore, the full-DC and ETF-DC vectorsfor benzaldehyde computed by our analytical method also sat-isfy the expected Berry’s phase behavior for a loop around theconical intersection point. Finally, in Appendix C, we showthat our derivative couplings agree with the exact derivativecouplings according to the Chernyak-Mukamel expression inEq. (2) (with the transition density matrix calculated accord-ing to response theory). Altogether, these results strongly sug-gest that our TDDFT/TDA derivative coupling are quite rea-sonable. Given the current popularity of TDDFT and the mod-ern interest in photochemistry and photoexcited nonadiabaticdynamics, we believe this computational formalism will bevery useful in the future.

ACKNOWLEDGMENTS

We thank Filipp Furche for extremely helpful conver-sations and his suggestion to include Appendix C. Thiswork was supported by National Science Foundation (NSF)CAREER Grant No. CHE-1150851. J.E.S. also acknowledgesan Alfred P. Sloan Research Fellowship and a David and Lu-cile Packard Fellowship.

APPENDIX A: DERIVATION OF THE DENSITY MATRIXDERIVATIVE P[x]

To justify Eq. (48) above, note that besides one- andtwo-electron-integrals, the only independent variables inquantum chemistry are S (the overlap) and � (the orbitalrotations).52, 82 Therefore, for any matrix P,

P[x]μν =

∑αβ

∂Pμν

∂Sαβ

S[x]αβ +

∑aj

∂Pμν

∂�aj

�[x]aj . (A1)

Now, the definition of the one-electron ground state den-sity matrix is

Pμν =∑m

CμmCνm. (A2)

To find these partial derivatives, one simply differentiatesP with respect to S and � to find

∂Pμν

∂�aj

=∑m

∂Cμm

∂�aj

Cνm +∑m

Cμm

∂Cνm

∂�aj

, (A3)

∂Pμν

∂Sαβ

=∑m

∂Cμm

∂Sαβ

Cνm +∑m

Cμm

∂Cνm

∂Sαβ

. (A4)

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024114-11 Ou et al. J. Chem. Phys. 141, 024114 (2014)

Combining Eqs. (A1)–(A4) and Eq. (63) in Ref. 52 intothe above expressions, one gets the final expression for P[x],

P[x]μν = −1

2

∑αβ

(PμαPνβ + PναPμβ)S[x]αβ

−∑aj

(CμaCνj + CμjCνa)�[x]aj . (A5)

APPENDIX B: HELLMANN-FEYNMANDERIVATIVE COUPLINGS

In the text above, we formed derivative couplings froma “brute force” expression whereby we differentiate the ketdirectly. We will now show that such an approach has aclear Hellmann-Feynman analogue. By applying the logic ofthe Hellmann-Feynman theorem, the TDDFT/TDA derivativecoupling should be equivalent to

dIJ (R) = 〈�I |∇RHKS |�J 〉EJ − EI

(B1)

=∑ijab

t Iai tJb

j

⟨�a

i

∣∣H[x]KS |�b

j

⟩EJ − EI

, (B2)

where HKS = QOKSQ. Here, OKS is defined in Eq. (7) andQ is the projector onto the singles manifold. Recall that thematrix element of the Kohn-Sham linear-response tensor A isgiven by Eq. (5), so that

HKS =∑ijab

∣∣�ai

⟩Aiajb

⟨�b

j

∣∣. (B3)

Inserting Eq. (B3) to Eq. (B2), one has

dIJ (R) = 1

EJ − EI

∑ijab

{t Iai tJb

j A[x]iajb + ⟨

�I

∣∣�a[x]i

⟩Aiajbt

Jbi

+ t Iai Aiajb

⟨�

b[x]j

∣∣�J

⟩}(B4)

= 1

EJ − EI

⎧⎨⎩∑ijab

t Iai tJb

j A[x]iajb

+EJ

∑jkbc

t Ick tJb

j

⟨�c

k

∣∣�b[x]j

⟩− EI

∑ilad

t Iai tJd

l

⟨�d

l

∣∣�a[x]i

⟩⎫⎬⎭ .

(B5)

By relabeling the indices, one can combine the like termsand reach the following equation:

dIJ (R) =∑ijab

t Iai tJb

j

[1

EJ − EI

A[x]iajb + ⟨

�ai |�b[x]

j

⟩](B6)

which is exactly the same expression as Eq. (24). Therefore,our direct differentiation method for derivative couplings hasan obvious Hellmann-Feynman analogue.

APPENDIX C: THE CHERNYAK-MUKAMELEXPRESSION AND THE TRANSITION DENSITYMATRIX ACCORDING TO RESPONSE THEORY

In the limit of two exact eigenstates of the Hamiltonian,|�I〉 and |�J〉, the derivative coupling takes a very simpleform known as the Chernyak-Mukamel formula (Eq. (C1)),

dCMIJ = 1

EJ − EI

∑pq

v[x]pq γ IJ

pq , (C1)

where γ IJpq is the one-electron transition density matrix. Here,

v[x]pq = ∑

μν Cμpv[x]μν cνq , where v

[x]μν is the derivative of the

nuclear-electronic potential in the AO basis. Equation (C1)can be derived easily from the exact Hellmann-Feynmanexpression,

dexactIJ =

⟨�exact

I

∣∣∇RH exact∣∣�exact

J

⟩EJ − EI

(C2)

using the fact that only vμν in the Hamiltonian depends onnuclear coordinates in the limit of an infinite basis. Accordingto time-dependent response theory,49, 83 γ IJ

pq is given by

γ IJpq =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

−∑a tJa

p tIaq for p, q ∈ occupied orbitals,∑

i tIp

i tJq

i for p, q ∈ virtual orbitals,

γ(1),IJpq for p ∈ virtual orbitals, q ∈ occupied orbitals,

γ(2),IJpq for q ∈ virtual orbitals, p ∈ occupied orbitals.

(C3)

The occupied-virtual components of γ IJ are obtained bysolving

[(A BB A

)+ �E

(I 00 −I

)](γ (1),IJ

γ (2),IJ

)= −

(Y(1)

Y(2)

),

(C4)

where �E = EJ − EI,

Biajb = ⟨�ab

ij

∣∣OKS

∣∣�DFT

⟩(C5)

= abij , (C6)

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024114-12 Ou et al. J. Chem. Phys. 141, 024114 (2014)

Y(1)bi =

∑pq

pbqiDIJpq +

∑jcd

t Icj tJd

i bcdj +∑j lc

t Icj tJb

l lcji

+∑j lcd

t Icj tJd

l �cljdbi , (C7)

Y(2)bi =

∑pq

pbqiDIJqp +

∑jcd

t Idi tJ c

j bcdj +∑j lc

t Ibl tJ c

j lcji

+∑j lcd

t Idl tJ c

j �cljdbi . (C8)

We now want to compare our derivative couplings withEq. (C1). We will assume a complete basis and ignore Pulayterms (S[x]) and the antisymmetrized AO overlap derivatives(SA[x]). Hence, our derivative coupling expression in a com-plete basis limit (dCB

IJ ) becomes

dCBIJ = 1

EJ − EI

∑pq

v[x]pq DIJ

pq − 1

EJ − EI

∑bi

Ybi�[x]bi ,

(C9)

where the difference density matrix DIJpq and the Lagrangian

Ybi in the MO basis can be represented as

DIJpq =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−∑a tJa

p tIaq for p, q ∈ occupied orbitals,

∑i t

Ip

i tJq

i for p, q ∈ virtual orbitals,

0 otherwise,

(C10)

Ybi =∑pq

pbqi

(DIJ

pq + DIJqp

)

+∑jcd

(t Icj tJd

i + t Idi tJ c

j

) bcdj

+∑j lc

(t Icj tJb

l + t Ibl tJ c

j

) lcji

+∑j lcd

(t Icj tJd

l + t Idl tJ c

j

)�cljdbi . (C11)

As illustrated in Sec. II E,∑

bi Ybi�[x]bi is obtained ac-

cording to the “z-vector” method∑bi

Ybi�[x]bi =

∑ja

zajM[x]aj (C12)

= −∑jaib

(∂2E

∂�aj∂�bi

)−1

YbiM[x]aj . (C13)

With an infinite basis, the only term in M[x] that con-tributes to the final result is the v[x] term. Hence,

∑bi

Ybi�[x]bi = −

∑jaib

(∂2E

∂�aj∂�bi

)−1

Ybi(v[x]aj + v

[x]ja )

(C14)

=∑jaib

(A + B)−1jaibYbiv

[x]aj . (C15)

Thus, our derivative coupling can be rewritten as

dCBIJ = 1

EJ − EI

∑pq

v[x]pq �IJ

pq , (C16)

where �IJpq is given by

�IJpq =

⎧⎪⎪⎨⎪⎪⎩

DIJpq for p, q ∈ occupied orbitals or p, q ∈ virtual orbitals,

− 12

∑bi(A + B)−1

qpibYbi for p ∈ virtual orbitals, q ∈ occupied orbitals,

− 12

∑bi(A + B)−1

pqibYbi for q ∈ virtual orbitals, p ∈ occupied orbitals.

(C17)

Finally, if we compare our derivative coupling expression(Eq. (C16)) to the Chernyak-Mukamel formula (Eq. (C1)), itis clear that the two expressions will agree if �IJ

pq = γ IJpq . To

that end, note that Ybi = Y(1)bi + Y

(2)bi if we compare Eqs. (C7)–

(C8) and Eq. (C11). Furthermore, in the limit that �E → 0,one finds that from Eq. (C4),

γ(1),IJaj + γ

(2),IJja = −

∑bi

(A + B)−1jaibYbi = �IJ

aj + �IJja .

(C18)

Therefore, we may now conclude that near a crossing ora conical intersection (�E → 0), our derivative couplings in acomplete basis (dCB

IJ ) agree with the Chernyak-Mukamel for-mula (dCM

IJ ), provided that the transition density matrix (γ IJ)is computed with time-dependent response theory.

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