Density-functional theory of linear and nonlinear time-dependent molecular properties

JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 21 1 DECEMBER 2002

Density-functional theory of linear and nonlinear time-dependentmolecular properties

Paweł SałekLaboratory of Theoretical Chemistry, The Royal Institute of Technology, SE-10044 Stockholm, Sweden andDepartment of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway

Olav VahtrasLaboratory of Theoretical Chemistry, The Royal Institute of Technology, SE-10044 Stockholm, Sweden

Trygve HelgakerDepartment of Chemistry, University of Oslo, P.O. Box 1033 Blindern, N-0315 Oslo, Norway

Hans ÅgrenLaboratory of Theoretical Chemistry, The Royal Institute of Technology, SE-10044 Stockholm, Sweden

~Received 10 June 2002; accepted 4 September 2002!

We present density-functional theory for linear and nonlinear response functions using an explicitexponential parametrization of the density operator. The response functions are derived using twoalternative variation principles, namely, the Ehrenfest principle and the quasienergy principle,giving different but numerically equivalent formulas. We present, for the first time, calculations ofdynamical hyperpolarizabilities for hybrid functionals including exchange-correlation functionalsat the general gradient-approximation level and fractional exact Hartree–Fock exchange. Samplecalculations are presented of the first hyperpolarizability of the para-nitroaniline molecule and ofa porphyrin derived push–pull molecule, showing good agreement with available experimentaldata. © 2002 American Institute of Physics.@DOI: 10.1063/1.1516805#

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I. INTRODUCTION

The interest in density-functional theory~DFT! has in-creased steadily since Hohenberg and Kohn1 provided itstheoretical justification and Kohn and Sham2 its practical for-mulation for calculations. The independent-particle KohSham model, which describes correlation in terms ounique, albeit unknown, functional of the electron denshas had an enormous impact on computational chemiand the available approximate exchange-correlation futionals have provided a level of accuracy that is adequatmany situations.

The generalization of the Kohn–Sham equationsRunge and Gross3 represents an important milestone in tdevelopment of time-dependent DFT~TDDFT!. Thus, al-though most applications of DFT have focused on structand static properties, the interest in time-dependent respproperties has grown considerably in recent years.4–43At thesame time one has witnessed a great success ofab initiowave function quantum chemistry in computing molecuproperties for nonvariational and variational wave functiousing time-independent analytic gradient theory and timdependent response theory. Recently, time-dependent prties have been obtained also from the quasienergy anwhere properties of any type~linear, nonlinear, static, dynamic, magnetic, electric, internal, external! are obtainedfrom the solution of a common set of equations, withofurther parametrizations or manipulations than those appto the reference wave function. Considering the broad raof systems that nowadays can be treated at an acceplevel of accuracy by DFT, the implementation of such

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‘‘toolbox approach’’ for DFT calculations of molecular properties would have broad ramifications. An implementatishould then involve all the common forms of the exchancorrelation functional, also those that include some proption of the Hartree–Fock exchange. In this paper, sucprogram is carried out by invoking two time-dependevariation principles: one based on the Ehrenfest theore44

and the other based on the quasienergy ansatz,45,46 where aparametrized time-evolution operator acts on a referestate, here the Kohn–Sham determinant. When the elecdensity is calculated as an expectation value of the denoperator, all the changes in the density to a given order inperturbation are obtained as the terms of a BakeCampbell–Hausdorff expansion.

To the best of our knowledge, the implementation psented here is the first complete implementation of Dfrequency-dependent nonlinear properties beyond the lodensity approximation~LDA !. Thus, in the previous implementation of frequency-dependent hyperpolarizabilitiesGisbergenet al.,18 the calculations are carried out at the adbatic LDA ~ALDA ! level, in which the response of the density is always obtained from LDA functional derivativeeven when the density itself has been calculated ingeneralized-gradient approximation~GGA!. By contrast, inour implementation, the response of the density is alwobtained using the stated DFT functional. We do invokeadiabatic approximation, however, meaning that the timdependent functional is assumed to depend on the tidependent density in the same manner that the tiindependent functional depends on the time-independdensity.

0 © 2002 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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9631J. Chem. Phys., Vol. 117, No. 21, 1 December 2002 DFT of time-dependent properties

This paper is organized as follows: In Sec. II we presDFT linear and quadratic response functions—in the Ehrfest form in Sec. II E, and in the quasienergy form in SII F. Our computer implementation is outlined in Sec. III.Sec. IV, we present two applications to illustrate the useness of the theory. The first application concerns the stand dynamic hyperpolarizabilities of para-nitroaniline withdetailed analysis of the basis-set and functional dependof the results; the second application addresses the seharmonic generation coefficient for a porphyrin derivpush–pull molecule, which has the potential to be an efftive material for this particular property. The paper is sumarized in Sec. V. Finally, in the Appendices, we provideexplicit formulas for the functional derivatives, together wia variable transformation that has been useful in derivthem.

II. THEORY

A. Linear and quadratic response functions

Response theory is perturbation theory with emphasisproperties rather than on states and energies, aiming toscribe how the properties respond to internal or externalturbations. Consider the expectation value of a timindependent operatorA, which for the unperturbed systemgiven by ^0uAu0&. In the presence of a time-dependent pturbation V(t), the expectation value ofA becomes time-dependent and may furthermore be expanded in powerthe perturbation,

^tuAut&5^tuAut& (0)1^tuAut& (1)1^tuAut& (2)1¯ . ~1!

Each term in this expansion has a Fourier representatiothe form

^tuAut& (1)5E ^ A;Vv1&&v1exp~2 iv1t ! dv1 , ~2!

^tuAut& (2)51

2 E E ^ A;Vv1,Vv2&&v1 ,v2

3exp@2 i~v11v2!t# dv1 dv2 , ~3!

and so on, whereVv is the Fourier transform ofV(t),

V~ t !5E2`

`

Vv exp~2 ivt ! dv. ~4!

In practice, we work with monochromatic perturbations; tintegrals in Eqs.~2!–~4! are then replaced by discrete sum

In this paper, we shall derive and implement expressifor the linear and quadratic response functions^ A;Vv1&&v1

and ^ A;Vv1,Vv2&&v1 ,v2within the framework of Kohn–

Sham DFT. In this particular formulation of DFT, the eletron density r(r ,t) is parametrized in terms of a timedependent reference determinantut&. There are thus noproblems associated with the calculation of expectationues of one-electron operators~even differential ones!, as as-sumed in Eq.~1!. We note that, in this paper, we shall onconsider properties of spin-restricted closed-shell states a


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ciated with real singlet operators; the generalizationimaginary and triplet operators is fairly straightforward awill be treated in a subsequent paper.

B. Kohn–Sham theory

In Kohn–Sham theory, we assume that the timdependent densityr(r ,t) is represented in terms of a timedependent reference Slater determinantut&. The Kohn–Shamenergy is then written as a functional of this density in tfollowing manner:

E@r,t#5Ts@r#1Vext@r,t#1J@r#1Exc@r#1VNN . ~5!

The first term is the kinetic energy evaluated as an expetion value

Ts@r#521

2 (i

^tu¹ i2ut&, ~6!

whereas the second and third terms represent, respectithe classical Coulomb interactions of the electron denwith the external potential and with itself,

Vext@r,t#5VN@r#1V@r,t#

5E r~r ,t !S (I

ZI

ur2RI u1v~r ,t ! D dt,

J@r#51

2 E E r~r1 ,t !r~r2 ,t !

r 12dt1 dt2 . ~7!

In Eq. ~7!, we have split the external potential into a stanuclear potential and an explicitly time-dependent perturtion. The exchange-correlation functionalExc@r# in Eq. ~5!contains all two-electron interactions except the Hartree teJ@r#, that is, it includes the effects of exchange and corretion. In addition, it corrects for the error made in the evaation of the kinetic energy according to Eq.~6!. The last termin Eq. ~5! represents the classical nuclear–nuclear repulsenergy.

In the widely used adiabatic approximation, which whave adopted here, the time-dependence of the exchacorrelation energy is contained in the density, that is,exchange-correlation functional is approximated usingsame functional form in the time-dependent and timindependent cases. It is not obvious that this approximaholds for other than slowly varying external fields, but it hbeen verified that the adiabatic approximation is adequatecalculating excitation energies.12

In Kohn–Sham theory, the time evolution of the sporbitals is governed by the differential equation

@ f ~r1 ,t !1v~r1 ,t !#f j~r1 ,t !5 idf j~r1 ,t !

dt, ~8!

where we have introduced the Kohn–Sham operator,

f ~r1 ,t !5h~r1!1 j ~r1 ,t !1vxc~r1 ,t !, ~9!

which we choose to define without the explicit perturbatiterm ~it still depends implicitly on the perturbation througthe density!. The first term in Eq.~9! contains the one-electron combined kinetic and nuclear-attraction operato


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9632 J. Chem. Phys., Vol. 117, No. 21, 1 December 2002 Sałek et al.

h~r1!521

2¹1

21(I

ZI

ur12RI u, ~10!

whereas the last two terms are the Coulomb and exchacorrelation potentials, respectively,

j ~r1 ,t !5E r~r2 ,t !

ur12r2udt2 , ~11!

vxc~r1 ,t !5dExc

dr~r1!U

r(r1)5r(r1 ,t)

. ~12!

Note that these terms are themselves functionals of thesity.

C. Time evolution of the reference determinantand the density

The Kohn–Sham reference determinant satisfiesequation

~H1V!ut&5 id

dtut&; H5(

if ~r i ,t !, ~13!

where the total Kohn–Sham Hamiltonian has been pationed into a termH implicitly dependent on the perturbatioand the perturbation itselfV @cf. Eq. ~8!#. In our second-quantization formulation of time-dependent Kohn–Shtheory, we adopt an exponential parametrization of the timevolution operator, representing the time-dependent reence Kohn–Sham determinant in the following manner:

ut&5exp@2k~ t !#u0&. ~14!

Here u0& is the unperturbed Kohn–Sham determinant,

u0&5)k,s

aks† uvac& ~15!

and k(t) the anti-Hermitian operator,

k~ t !5(rs

k rs~ t !Ers5(rs

k rs~ t !(s

ars† ass ;

k rs* ~ t !52ksr~ t !, ~16!

wherears† and ass are the creation and annihilation oper

tors, respectively, of orbitalsr ands of spin s. This particu-lar representation of the Kohn–Sham determinant has prously been used in connection with relativistic foucomponent theory.47

The fundamental variational parameters of our theare the elements of the rotation matrixk rs(t). As in Hartree–Fock theory, the nonredundant rotations are those betwoccupied and unoccupied orbitals. Equation~14! implies thatthe individual Kohn–Sham spin–orbitals obey the transfmation law,

aks† ~ t !5exp@2k~ t !#aks

† exp@ k~ t !# ~17!

and that orthonormality is preserved.To study the time-dependence of the electron density,

introduce the second-quantized density operator,


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r~r !5(pq

fp* ~r !fq~r !Epq . ~18!

Recognizing that the density is an expectation value ofoperator and given the parametrization in Eq.~14!, we obtainthe following expression for the time-dependent density:

r~r ,t !5^tur~r !ut&5^0uexp@ k~ t !#r~r !exp@2k~ t !#u0&.~19!

A Baker–Campbell–Hausdorff expansion of the exponentime-evolution operator gives for the density~and similarlyfor other operators!,

r~r ,t !5r~r ,0!1^0u@ k~ t !,r~r !#u0&

1 12 ^0u@ k~ t !,@ k~ t !,r~r ##u0&1O~k3!, ~20!

which will be used frequently in the following sections.

D. Perturbation expansion of the densityand Kohn–Sham operators

In the absence of the perturbation, the time-evolutoperator produces only a dynamical phase factor, which ccels out for expectation values. In the presence of the pebation, it is assumed that an expansion in powers ofperturbation exists such that

k~ t !5k (1)~ t !1k (2)~ t !1¯ ~21!

with Fourier representations

k (1)~ t !5E kv1 exp~2 iv1t ! dv1 , ~22!

k (2)~ t !51

2 E E kv1 ,v2 exp@2 i~v11v2!t# dv1 dv2 .

~23!

For monochromatic perturbations, the integrations areplaced by summations. Using the expansions Eqs.~20! and~21!, we introduce the perturbed density matrices up to sond order as

Dpq(0)5^0uEpqu0&, ~24!

Dpq(1)5^0u@ k (1),Epq#u0&, ~25!

Dpq(2)5^0u@ k (2),Epq#1 1

2 @ k (1),@ k (1),Epq##u0&, ~26!

which allow us to write thenth order correction to the density as

r (n)~r ,t !5(pq

fp* ~r !fq~r !Dpq(n)~ t !. ~27!

We now expand the second-quantized Kohn–ShHamiltonian in orders of the perturbation,

H5(n

H (n), ~28!

H (n)5(pq

f pq(n)Epq , ~29!

where


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f pq(n)5dn0 hpq1 j pq

(n)1vxc,pq(n) . ~30!

The first contribution tof pq(0) is the one-electron integral ove

the kinetic-energy and nuclear-attraction operators of~10!,

hpq5K fpU2 1

2¹21(

I

ZI

ur2RIU fqL . ~31!

The electron-repulsionnth order Coulomb interaction integrals

j pq(n)5(

rsgpqrsDrs

(n) ~32!

constitute the second contribution to the Kohn–Sham maelement f pq

(n) and are obtained from the two-electron intgrals,

gpqrs5 K fp~1!f r~2!U 1

r 12Ufq~2!fs~2!L ~33!

and from the time-dependent density-matrix elements in E~24!–~26!. The last contribution to Eq.~30! is the integralover the exchange-correlation potentialvxc@r#,

vxc,pq(n) ~ t !5^fpuvxc

(n)~r ,t !ufq&. ~34!

Since the potential depends on the densityr, this contribu-tion to the Kohn–Sham matrix element depends on theturbation,

vxc(1)~r ,t !5E dvxc~r !

dr~r 8!r (1)~r 8,t ! dt8, ~35!

vxc(2)~r ,t !5E dvxc~r !

dr~r 8!r (2)~r 8,t ! dt8

11

2 E E d2vxc~r !

dr~r 8!dr~r 9!r (1)~r 8,t !r (1)

3~r 9,t ! dt8 dt9. ~36!

E. The Ehrenfest method

Let us consider the time-development of the expectavalue of an operator of the form,

Q~ t !5exp@2k~ t !#Q exp@ k~ t !#. ~37!

Differentiating the expectation valuetuQ(t)ut&5^0uQu0&@see Eq.~14!# with respect to time and invoking the Schr¨-dinger equation~atomic units!

S H~ t !1V~ t !2 id

dt D ut&50 ~38!

we obtain the Ehrenfest theorem

^tu iQ~ t !1@Q~ t !,H~ t !1V~ t !#ut&50, ~39!

which forms the basis for the time-dependent variatprinciple.44 Collecting the exponential operators in Eqs.~14!and ~37! and using the identity


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n

exp@ k~ t !#S d

dtexp@2k~ t !#Q exp@ k~ t !# Dexp@2k~ t !#

5Fexp@ k~ t !#S d

dtexp@2k~ t !# D ,QG , ~40!

we find that the Ehrenfest theorem may be written inmore convenient form

K 0UF Q,exp@ k~ t !#S H~ t !1V~ t !2 id

dt Dexp@2k~ t !#GU0L 50.

~41!

This equation holds for any time-independent one-electoperatorQ. In particular, it holds for the spin-averaged ecitation operatorsEpq in the expansion ofk(t) in Eq. ~16!.Collecting these operators in the column vectorq, we arriveat a set of nonlinear equations from which the timdependence ofk(t) may be determined. In the following, wshall use these equations to determine the first- and secorder terms in Eq.~21! and thereby the linear and quadratresponse functions.

1. Linear response

The linear response equations are obtained by expanEq. ~41! to first order, yielding a differential equation in timfor k (1),

^0u@ q,@ k (1),H (0)#1H (1)#u0&1 i^0u@ q,k (1)#u0&

52^0u@ q,V~ t !#u0&. ~42!

In the frequency domain, this equation becomes an algebequation forkv,

^0u@ q,@ kv,H (0)#1Hv#u0&1v^0u@ q,kv#u0&

52^0u@ q,Vv#u0&. ~43!

We have here introduced the Fourier transform of the pturbed Kohn–Sham matrix,

Hv5(pq

f pqv Epq , ~44!

where

f pqv 5 j pq

v 1vxc,pqv

5(rs

gpqrsDrsv

1 K fpU E dvxc

dr~r 8!rv~r 8! dt8UfqL , ~45!

Drsv 5^0u@ kv,Ers#u0&, ~46!

rv~r !5^0u@ kv,r~r !#u0&. ~47!

To bring out the matrix structure of Eq.~43!, we may expressthe first-order parameters in matrix form,

kv5¯Epq¯S Akpq

v

AD [q†kv ~48!


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such that we formally solve the linear equations

~E2vS!kv5Vv. ~49!

In Eq. ~49!, the matrixE has the structure of an electronHessian,

E5^0u@ q,@H (0),q†#u0&1 (pqrs

^0u@ q,Epq#u0&F 1

ur2r 8u

1dvxc~r !

dr~r 8! Gpqrs

^0u@Ers ,q†#u0&, ~50!

where, in the last term, we have introduced two-electrontegrals defined by analogy with Eq.~33!. In Eq. ~49!, wehave introduced the generalized overlap matrix,

S5^0u@ q,q†#u0&, ~51!

and the perturbation vector on the right-hand side of~43!,

Vv5^0u@ q,Vv#u0&. ~52!

From the solutionkv to Eq. ~49!, the linear responsefunction is easily obtained for any one-electron operatorA,

^ A;V&&v5^0u@ kv,A#u0&52A†~E2vS!21Vv, ~53!

where

A5^0u@ q,A#u0&. ~54!

It should be noted that the working equations of lineresponse theory resemble Eq.~43! more closely than Eq.~49!sinceE andS are never constructed explicitly and since,the iterative algorithms we use,k corresponds to trial vectors. Furthermore, the exchange-correlation potentialswe consider are local and never give rise to two-electintegrals—even the GGA functionals are local in this sen

2. Quadratic response

The quadratic response to a perturbation requires thelution of k (2) from the second-order expansion of Eq.~41!,

^0u@ q,@ k (2),H (0)#1H (2)#u0&1 i^0u@ q,k (2)

1 12@ k (1),k (1)##u0&

52^0u@ q,@ k (1),V~ t !1H (1)#1 12@ k (1),@ k (1),H (0)###u0&.

~55!

With the Fourier transform Eq.~23!, this equation can berearranged to give

^0u@ q,@ kv1 ,v2,H (0)#1Hv1 ,v2#u0&

1~v11v2!^0u@ q,kv1 ,v2#u0&

52 P12 0u@ q,2@ kv1,Vv21Hv2#1@ kv1,@ kv2,H (0)##

1v2@ kv1,kv2##u0&. ~56!

Here the operatorP12 symmetrizes with respect to the frequenciesv1 andv2 ,

P12A~v1 ,v2!5 12 @A~v1 ,v2!1A~v2 ,v1!# ~57!


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atn.

o-

and the elements of the second-order perturbed Kohn–Smatrix,

Hv1 ,v25(pq

f v1 ,v2Epq ~58!

are given by

f pqv1 ,v25 j pq

v1 ,v21vxc,pqv1 ,v2 ~59!

with Coulomb and exchange-correlation parts,

j pqv1 ,v25(

rsgpqrsDrs

v1 ,v2, ~60!

vxc,pqv1 ,v25 K fpU E dvxc

dr~r 8!rv1 ,v2~r 8! dt8UfqL

1 P12K fpU E E d2vxc

dr~r 8!dr~r 9!rv1~r 8!rv2

3~r 9! dt8 dt9UfqL , ~61!

The solution of Eq.~56! requires the separation of thsecond-order density matrix elements into first- and secoorder parts,

rv1 ,v2~r !5(pq

fp* ~r !fq~r !Dpqv1 ,v2, ~62!

Drsv1 ,v25Drs

v1 ,v21D% rsv1 ,v2, ~63!

Drsv1 ,v25 P12 0u@ kv1,@ kv2,Ers##u0&, ~64!

D% rsv1 ,v25^0u@ kv1 ,v2,Ers#u0&, ~65!

such that the second-order Hamiltonian may be written assum of two operators that depend on the first- and secoorder parameters, respectively,

Hv1 ,v25HC v1 ,v21H% v1 ,v2. ~66!

As in the linear case, the equation for the second-orderrameters Eq.~56! may then written in matrix form,

@E2~v11v2!S#kv1 ,v25Vv1 ,v2, ~67!

where all first-order parameters have been collected onright-hand side,

Vv1 ,v252P12$^0u@ q,@ kv1,@ kv2,H (0)###u0&

1^0u@ q,@ kv1,v2kv2##u0&12^0u@ q,@kv1,Hv2

1Vv2##u0&1^0u@ q,HC v1 ,v2#u0&%. ~68!

The quadratic response function may now be calculated

^ A;V,V&&v1 ,v25^0u@ kv1 ,v2,A#u0&

1 P12 0u@ kv1,@ kv2,A##u0&, ~69!

where the first-order responseskv1 and kv2 are obtainedfrom Eq. ~43! and the second-order responsekv1 ,v2 fromEq. ~56!. We note that the first term in Eq.~69! resembles theevaluation of the linear response function,


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^0u@kv1 ,v2A#u0&52A†kv1 ,v2

52A†@E2~v11v2!S#21Vv1 ,v2. ~70!

An alternative approach is to solve the adjoint linearsponse equation forA at frequencyv11v2 , as done in ourimplementation.

In short, for a given propertyA and two periodic pertur-bationsB and C with associated frequenciesvb andvc re-spectively, the evaluation of the quadratic response equa

^ A;B;C&&vb ,vcis carried out by first solving the three linea

response equations,

kA†@E2~vb1vc!S#5A†, ~71!

~E2vbS!kvb5B, ~72!

~E2vcS!kvc5C, ~73!

and then evaluating

^ A;B;C&&vb ,vc5kA†Vvb ,vc1 Pbc^0u@ kvb,@ kvc,A##u0&.

~74!

The main complication associated with extending a HartreFock quadratic response code to the Kohn–Sham DFT isevaluation of the exchange-correlation contributionVvb ,vc. Section III is devoted to this implementation in dtail.

F. The quasienergy method

The quasienergy formulation provides an alternattime-dependent variation principle for deriving dynamicresponse functions. It is arguably more attractive thanEhrenfest method in that it provides a unified frameworktreating variational and nonvariational wave functions,analogy with time-independent theory, to which it naturareduces in the limit of static perturbations.45,46 As an addi-tional advantage over the Ehrenfest method, the permtional symmetries~with respect to the exchange of operato!becomes manifest in the quasienergy method.

The concept of quasienergy arises naturally in TDDFTindeed, it turns out to be nothing but the action integralRunge and Gross,3 where the integration limits are chosenspan a period of the perturbation, scaled by the inverse operiod length. In the following, we discuss the applicationthe quasienergy method to the calculation of response futions in Kohn–Sham DFT. Since the Kohn–Sham quasiergy method follows closely the corresponding HartreFock method, we here consider only those aspects of KoSham theory that differ from Hartree–Fock theory.

In Hartree–Fock theory, the quasienergy and its timeerage are defined as follows:

Q~ t !5 K tUH2 id

dt Ut L , ~75!

$Q%T51

T E2T/2

T/2

Q~ t ! dt, ~76!


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-

whereT constitutes one period of the time-dependent perbation. In Kohn–Sham theory, there is an additional conbution to the quasienergy from the exchange-correlatfunctional,

Qxc~ t !5Exc@r~ t !#. ~77!

The perturbation is expanded in its~discrete! Fourier com-ponents, each of which is further expanded in field strengex

vk coupled to a quantum-mechanical operator,

V~ t !5(kx

exp~2 ivkt !exvkVx

vk. ~78!

Included in the summation is the operatorA, the responsefunctions of which we are calculating. We associate a nzero frequencyv but zero perturbation strength with thoperator.

The periodic perturbation Eq.~78! induces a change inthe quasienergy, which is expanded in orders of the perbation,

Q~ t !5Q(0)~ t !1Q(1)~ t !1Q(2)~ t !1¯ . ~79!

The response function of ordern is then recovered by differ-entiating the time-averaged perturbed quasienergyQ(n11)(t)with respect to the frequency-dependent field strengths,

^ A;B&&v5d2$Q(2)%T

deA2vdeB

v , ~80!

^ A;B,C&&v1 ,v25

d3$Q(3)%T

deA2v12v2deB

v1deCv2

. ~81!

The frequency associated withA is set equal to minus thesum of the perturbing frequencies so that the time-averaquasienergy does not vanish.

To calculate the perturbed quasienergy Eq.~79!, we mustfirst determine the perturbed orbital-rotation parametk(t), which are likewise expanded in orders of the perturbtion,

k (1)~ t !5(kx

exp~2 ivkt !exvkkx

vk, ~82!

k (2)~ t !51

2 (klxy

exp@2 i~vk1v l !t#exvkey

v lkxvkky

v l. ~83!

From the 2n11 rule, it follows that, to calculate the quasieergy to order 2n11, we need only determinek(t) to ordern, the contributions from higher orders being zero. Thus,calculate the linear and quadratic response functions,need only determinek (1). Note that, even though the contrbution of for examplek (2) to Q(2) vanishes, its contributionto the exchange-correlation part of the quasienergyQxc

(2) isnonzero but cancelled by a similar contribution toQ(2)

2Qxc(2) . In our discussion ofQxc

(2) andQxc(3) we shall therefore

ignore all contributions that do not depend onk (1), in accor-dance with the 2n11 rule.

1. Linear response

In linear response theory, the exchange-correlationergy is expanded to second order ink (1),


th

ie

ea

hi

de

ared

e-ero.ua-

ofon-sitylh-ala

intan-

onsum

oftionionsis

ad-


Exc(2)5E dExc

dr~r !r (2)~r ,t ! dt1

1

2 E E d2Exc

dr~r !dr~r 8!

3r (1)~r ,t !r (1)~r 8,t ! dt dt8. ~84!

The perturbed densities are given by Eq.~27! except that thecontribution fromk (2) to r (2) is ignored in accordance withthe 2n11 rule, as indicated by the overbar inExc

(2) and r (2).Next, we expandk (1)(t) according to Eq.~82! and obtain

Exc(2)5

1

2 (kl

exp@2 i~vk1v l !t#(xy

exvkey

v l

3F E dExc

dr(r )rv1 ,v2(r ) dt1E E d2Exc

dr(r )dr~r 8!

3rv1(r )rv2(r 8) dt dtG , ~85!

where we have introduced

rxv5^0u@ kx

v ,r #u0&, ~86!

rxyv1 ,v25 P12 0u@ kx

v1 ,@ kyv2 ,r ##u0&. ~87!

The first-order parameterskxv1 andky

v2 are determined by avariational condition on the second-order quasienergy,result of which is an equation identical to Eq.~43!. Differ-entiating the time-averaged exchange-correlation quasergy with respect to the field strengthsea

va ,ebvb , according to

Eq. ~80!, we obtain

^ A;B&&vb

xc 5E dExc

dr~r !rva ,vb~r ! dt1E E d2Exc

dr~r !dr~r 8!

3rva~r !rvb~r 8! dt dt8 ~88!

for the exchange-correlation energy contribution to the linresponse function.

2. Quadratic response

The quadratic response function is obtained as the tderivative of the time-averaged quasienergy. The programthen to expand the energy to third order in the first-orparameters,

Exc(3)5E dExc

dr~r !r (3)~r ,t ! dt

1E E d2Exc

dr~r !dr~r 8!r (1)~r ,t !r (2)~r 8,t ! dt dt8

11

6 E E E d3Exc

dr~r !dr~r 8!dr~r 9!

3r (1)~r ,t !r (1)~r 8,t !r (1)~r 9,t ! dt dt8 dt9, ~89!

where

r (3)~r ,t !5 16^0u@ k (1)~ t !,@ k (1)~ t !,@ k (1)~ t !,r~r !###u0&.

~90!

Expanding the parameters as in the linear case Eq.~85!, weobtain


e

n-

r

rdisr

Exc(3)5

1

6 (klm

exp@2 i~vk1v l1vm!t#(xyz

exvkey

v lezvm

3F E dExc

dr~r !rvk ,v l ,vm~r ! dt

12PxyzE E d2Exc

dr~r !dr~r 8!rvk~r !rv l ,vm

3~r 8!dt dt81E E E d3Exc

dr~r !dr~r 8!dr~r 9!rvk

3~r !rv l~r 8!rv l~r 9! dt dt8 dt9G , ~91!

where the first- and second-order perturbed densitiesgiven by Eqs.~86! and ~87! and the third-order perturbedensity by

rxyzv1 ,v2 ,v3~r !5 P123 0u@ kx

v1 ,@ kyv2 ,@ kz

v3 ,r~r !###u0&.~92!

As in the linear case, time averaging cancels the timdependent phase factor if the sum of the frequencies is zThe exchange-correlation energy contribution to the qdratic response function then becomes

^ A;B,C&&vb ,vb

xc 51

6F E dExc

dr~r !rva ,vb ,vc~r ! dt

13PABCE E d2Exc

dr~r !dr~r 8!

3rva~r !rvbvc~r 8! dt dt8

1E E E d3Exc

dr~r !dr~r 8!dr~r 9!rva

3~r !rvb~r !rvc~r ! dt dt8 dt9G . ~93!

III. IMPLEMENTATION

The implementation of the response module consiststwo major parts: The computation of the DFT response ctribution expressed in terms of commutators of the denmatrix and response vectorski , and the computation of al19 functional derivatives appearing in the formalism. Tecnically, these two parts have very different computationcharacters. The functional-derivative evaluation involveslimited amount of data and a fixed number of floating pooperations, including evaluation of power operators and trscendental functions~arcsin, etc.!. The evaluation time for asingle grid point may be substantial but does not dependthe system size. The DFT contribution is expressed as aof matrices. The evaluation of the prefactors and the sumthe matrices is a memory-intensive operation. The evaluatime of this part of the code per single numerical integratgrid point scales formally as the squared number of bafunctions but can be reduced to linear scaling by takingvantage of the matrix sparsity.


to

fo

o

de

fol-

–

st

x-

ues


The DFT contribution

EpqDFT5DpqFpq ~94!

is in general a sum of 16 matrices with appropriate prefac~Einstein summation overx),

F5r VBC1qxVxBC1r BVC1r CVB1qx

BVxC1qx

CVxB

1r BCV1qxBCVx . ~95!

HereV is the matrix containing the elementsV i j 5f if j ~inthe spin–orbital basis! and we use the notation

Vx5]V

]x, ~96!

VB5@kB,V#, ~97!

VBC5 12 ~@kB,@kC,V##1@kC,@kB,V## !. ~98!

The double-index transformation of the response vectorslows Eq. ~89!. The coefficientsp and q are expressed interms of functional derivatives and expectation valuescommutatorsrB5^VB&, rx

B5^VxB&.

We express the functional in terms of densityr andsquare of the density gradientz5¹r•¹r5rx

2 ,

r 51

2

]F

]r, ~99!

qx51

2

]F

]rx, ~100!

r B5rB]2F

]r2 1rxB ]2F

]r]rx, ~101!

qxB5rB

]2F

]r]rx1ry

B ]2F

]rx]ry, ~102!

r BC5rBrC]3F

]r3 1~rBrxC1rCrx

B!]3F

]r2]rx

1rxBry

C ]3F

]r]rx]ry1rBC

]2F

]r2 1rxBC ]2F

]r]rx, ~103!

qxBC5rBrC

]3F

]r2]rx1~rBry

C1rCryB!

]3F

]r]rx]ry

1ryBrz

C ]3F

]rx]ry]rz1rBC

]2F

]r]rx1ry

BC ]2F

]rx]ry.

~104!

Note thatq, qB , andqBC vanish for functionals that depenonly on the density. The functionals do not depend indepdently on the density gradient componentsrx . The first-,second-, and third-order derivatives ofF with respect torx

are given by

]F

]rx52

]F

]zrx , ~105!

]2F

]rx]ry54

]2F

]z2 rxry12]F

]zdxy , ~106!


rs

l-

f

n-

]3F

]rx]ry]rz58

]3F

]z3 rxryrz14]2F

]z2 ~dxyrz1dxzry

1dyzrx!. ~107!

The complete coefficients are expressed in terms of thelowing auxiliary variables:

SB5(x

rxBrx , ~108!

SBC5(x

rxBCrx , ~109!

PBC5(x

rxBrx

B . ~110!

In this way, we arrive at the final coefficients in the spinorbital basis,

r 51

2

]F

]r, ~111!

qx5]F

]zrx , ~112!

r B5rB]2F

]r2 12SB]2F

]r]z, ~113!

qxB5S 2rB

]2F

]r]z14SB

]2F

]z2 D rx12rxB ]F

]z, ~114!

r BC5rBrC]3F

]r3 12~rBSC1rCSB!]3F

]r2]z

14SBSC]3F

]r]z2 1rBC]2F

]r2 12~SBC1PBC!]2F

]r]z,

~115!

qxBC52rBrC

]3F

]r2]zrx14~rBSC1rCSB!

]3F

]r]z2 rx

12~rBrxC1ra

CrxB!

]2F

]r]z18SBSC

]3F

]z3 rx

14~rxBSC1SBrx

C1PBCrx!]2F

]z2 12rBC]2F

]r]zrx

14SBCrx

]2F

]z2 12rxBC ]F

]z. ~116!

The original expression Eq.~95! for the DFT contribu-tion is formulated in terms of matrix commutators at a coper integration point that scales asN3. However, we canutilize a special structure of theV i j 5f if j matrix and trans-form the working formula to a form that uses only matrivector operations scaling asN2. Even further optimizationmaking use of the sparse structure off i can be accom-plished. The required matrices and their expectation valare expressed in terms of transformed vectors,

~f iN!p5~ki !pqfq , ~117!

~f iT!p5~ki !qpfq , ~118!


tre

nrdan

alanamuu

i-abealusm

th

o-n

nsbamormto

-re

anabae

thethecor--

-ll

re-er-

aticnddlere-ld

ons

en-rP,

ne

annfor

torthee

th

er-er-

ility


~fNN!p5~kB!pq~fCN!q1~kC!pq~fB

N!q , ~119!

~fTT!p5~kB!qp~fCT !q1~kC!qp~fB

T!q , ~120!

whereN andT stand for untransposed and transposed matransformations, respectively. The expectation valuespressed in terms of these vectors are

r i5^ri&5fpDpq~f iN!q2~f i

T!pDpqfq , ~121!

rBC5^rBC&5 12 fpDpq~fNN!q1 1

2 ~fTT!pDpqfq

2~fBN!pDpq~fC

T !q2~fCN!pDpq~fB

T!q . ~122!

The matrix elements contributing to the DFT matrix are

~rBC!pq5fpNNfq1fpfq

TT22~fCN!p~fB

T!q

22~fBN!p~fC

T !q , ~123!

~ri !pq5~f iN!pfq2fp~f i

T!q . ~124!

The derivation and implementation of first-, second-, athird-order functional derivatives is a fairly straightforwatask, complicated only by the length of the expressionsthe number of required derivatives~see Appendix A!.

The functional evaluator module provides functional vues and functional derivatives with respect to densitiesdensity gradients up to third order. This part of the progrhas been coded in a fashion enabling later extension torestricted open-shell calculations. The integrator modevaluates the property of interest~the energy, the Kohn–Sham matrix, etc.! for each grid point, summing up contrbutions with an integration weight associated with the pticular grid point. This requires the atomic orbitals toevaluated at each grid point. Derivatives of atomic orbitwith respect to spatial coordinates are needed when thefunctional contains a gradient correction, or when one coputes the molecular gradient of the energy.

The formalism derived has been implemented withinDFT part of theDALTON quantum chemistry package.48 Thispart was recently initiated to allow for computation of mlecular energies, optimization of molecular structures acomputation of molecular properties at the linear respolevel.32–34 The DFT extension has been accomplishedadding the DFT contribution to the energy, the Kohn–Shmatrix, and the one-index transformed response vectThese property-evaluating modules make use of a comsubset of routines. The numerical-integration grid generain DALTON has been generously provided by the authorsthe CADPAC program.49

IV. SAMPLE APPLICATIONS

A. The hyperpolarizability of para-nitroaniline

As a first illustration of the TDDFT methodology derived above for nonlinear optical properties, we presentsults for the hyperpolarizability of para-nitroaniline~PNA!.With an NH2 donor and an NO2 acceptor substituted onphenyl ring, this molecule shows an exceptionally strocharge-transfer character accompanied by a large polarizity and hyperpolarizability. It has served in the past asimportant test system for experimental and theoretical inv


ixx-

d

d

-d

n-le

r-

sed-

e

de

y

s.onorf

-

gil-ns-

tigations of hyperpolarizabilities.51–56 There is therefore asignificant accumulated knowledge for this system abouteffects of frequency dispersion and solvation, as well asrole of computational parameters such as basis-set andrelation effects.50,51 The applications of DFT have been restricted to the LDA and LB94 levels.

Compared with experimental observation,52 the Hartree–Fock random-phase approximation~RPA! and several standard correlatedab initio techniques give much too sma~static and dynamic! hyperpolarizabilities for PNA. This dis-crepancy was tentatively attributed to solvation.53 However,with the recent availability of gas-phase values,54 it is morecorrectly attributed to an incomplete treatment of thesponse of the isolated PNA molecule to strong external pturbations. In the present work, we reinvestigate the stand dynamic hyperpolarizabilities of PNA at the GGA aB3LYP levels of theory, to see how these methods hanthis difficult system. Among the many optical processeslated to the hyperpolarizability, we focus on the electric-fieinduced second harmonic generation~ESHG!, denotedb~22v;v,v!, using the convention of Ref. 51:bz5(k(bzkk

1bkzk1bkkz)/3, wherek5x,y,z.

1. Computational details

We have used six different basis sets in our calculation PNA: the 4-31G* basis with@8s5p1du3s3p1d# func-tions on the nonhydrogens and@4su2s# on the hydrogens;the 6-31G* basis57,58contracted as@10s4p1du3s2p1d# and@4su2s#; the Ahlrichs VDZ basis contracted as@7s4pu3s2p#and @4su2s#; the cc-pVDZ basis59 contracted as@9s4p1du3s2p1d# and @4s1pu2s1p#; the aug-cc-pVDZbasis60 ~cc-pVDZ augmented with diffuse functions! con-tracted as@10s5p2du4s3p2d# and @5s2pu3s2p#; and theSadlej basis61 ~optimized for polarizability calculations! con-tracted as@10s6p4du5s3p2d# and @6s4pu3s2p#. We haveused three exchange–correlation functionals: the local dsity approximation~LDA !,57,62 the Becke–Lee–Yang–Par~BLYP! GGA functional,63,64 and the three-parameter BLY~B3LYP! hybrid functional.65,66 As in previous calculationswe adopt a planarC2v geometry.50,51 The coordinates ofPNA are obtained from crystallographic data,55 supple-mented with two additional coordinates to determiuniquely the geometry56 ~see Table I of Ref. 51!. For allfrequencies considered here, the assumption of Kleinmsymmetry holds well, the deviation being negligible eventhe largest frequency of 1.494 eV.67

2. Excitation energies

The calculated excitation energies and oscillastrengths are listed in Table I. The accurate description ofhyperpolarizability is strongly linked to the description of thintensive amino-to-nitro charge-transfer~CT! transition at4.35 eV,55 which collects nearly all of the oscillator strengfor the manifold of low-lying states~see Table I!. This facthas in earlier studies motivated a truncation of the sum-ovstate expression for the hyperpolarizability and a considation of the characteristics~transition energy, dipole, andtransition moments! only of the ground and CT states.53 Infact, the strong solvent dependency of the hyperpolarizab


S

0.26220.01910.11830.00220.03940.01980.00000.00000.0002


TABLE I. The lowest excitation energies~eV! and oscillator strengths of PNA.a

Sym root

4-31G 6-31G* Sadlej cc-pVDZ

B3LYP LDA B3LYP LDA B3LYP LDA B3LYP LDAEE OS EE OS EE OS EE OS EE OS EE OS EE OS EE O

1 1 3.80 0.3511 3.43 0.2901 4.13 0.3418 3.71 0.2712 3.84 0.3436 3.47 0.2825 4.13 0.3324 3.722 5.90 0.0478 5.56 0.0319 6.38 0.0077 5.52 0.0133 5.78 0.0185 5.47 0.0353 6.32 0.0020 5.383 6.42 0.0336 5.54 0.0788 6.46 0.0424 5.87 0.1047 6.45 0.0188 5.63 0.1194 6.62 0.0281 5.89

2 1 4.57 0.0001 4.14 0.0017 4.74 0.0019 4.35 0.0000 4.51 0.0004 4.12 0.0005 4.71 0.0066 4.362 5.03 0.0459 4.60 0.0318 5.27 0.0574 4.72 0.0400 4.94 0.0474 4.47 0.0322 5.23 0.0547 4.663 5.97 0.0852 5.49 0.0026 6.18 0.0653 5.58 0.0208 5.94 0.0780 5.48 0.0261 6.12 0.0637 5.50

3 1 4.24 0.0003 4.08 0.0002 4.44 0.0000 4.17 0.0000 4.20 0.0003 3.93 0.0002 4.42 0.0000 4.142 5.12 0.0029 4.62 0.0000 6.05 0.0000 4.50 0.0000 4.78 0.0017 4.32 0.0000 5.88 0.0000 4.363 6.11 0.0057 5.03 0.0027 6.50 0.0004 6.21 0.0003 5.71 0.0060 4.72 0.0020 5.95 0.0003 5.74

aExperimental excitation energy is 4.35 eV~Ref. 55!.

mnth.e

n

eFthti

en

isca

ree

ofl-

ip-is

-

ic

toT/ri-tlyltsessonef-for

can be traced to a lowering of the CT excitation energy cobined with a change in the dipole moments. Polar solvestabilize the donor and acceptor levels by reducingHOMO–LUMO gap, thereby redshifting the CT transition

It should be noted that the CT excitation energy obtainat the multiconfigurational self-consistent field~MCSCF!level using thep correlation space is actually higher thathat obtained with RPA, that is, 5.18 versus 5.04 eV,50 indi-cating thatp electron correlation is more important for thground state than for the excited state. By contrast, Drepresents a significant improvement on RPA, reducingexcitation energy, as expected from the increased correlacontribution tob from thes electrons. Still, DFT results arclearly dependent on the functional, the local exchange futionals giving too low excitation energies~and too largeb!.The hybrid B3LYP functional gives the best result, that4.13 eV in the cc-pVDZ basis, consistent with the nonlonature of the CT excitation.

3. ESHG values

Theb values are summarized in Tables II and III, whethe frequencies have been chosen to coincide with thos


-tse

d

Te

on

c-

,l

in

the experiments of Teng and Garito.52 The results of differentbasis sets and functionals show a fairly consistent picturethe size of theb values~average values and component vaues!. The 4-31G* basis is too small for an accurate descrtion of hyperpolarizabilities, while the aug-cc-pVDZ basgive the best results~comparable with Sadlej basis set!. Anoticeable increase inb is obtained by adding diffuse functions to the basis set.

In Table IV, our DFT results for the static and dynamhyperpolarizabilities of PNA are compared with RPA,53 mul-ticonfigurational quadratic response~MCQR!,53 MP2,68 pre-vious DFT results,69 and experimental results.52,54 Whereasthe hyperpolarizability increases by about 40% from RPAMCQR, it increases by several factors from the RPA to DFB3LYP, dramatically improving the agreement with expement. The description of dispersion, in particular, is greaimproved at the DFT level. One notices that the LDA resuare almost on top of the experiment, while B3LYP go20%–25% below for the best basis sets. In this comparione must, however, qualify a hidden factor, namely, thefect of zero-point vibrational averaging that may make upa considerable fraction of the total ESHG value.70

TABLE II. B3LYP results for theb components of the PNA in units of 10230 esu (1 a.u.58.639 418310233 esu).

v ~eV! bzxx bzyy bxxz byyz bzzz bzxx bzyy bxxz byyz bzzz

4-31G* 6-31G*

0.650 21.28 20.71 21.20 20.70 20.76 21.38 20.02 21.30 20.02 14.341.170 21.70 20.88 21.29 20.80 32.24 21.79 20.03 21.40 20.03 20.571.361 22.09 21.01 21.35 20.86 43.06 22.13 20.04 21.46 20.03 25.721.494 22.59 21.16 21.40 20.91 57.09 22.51 20.05 21.52 20.03 31.56

AhlrichsVDZ cc-pVDZ

0.650 21.58 20.11 21.49 20.10 16.47 21.48 20.08 21.41 20.08 13.471.170 22.07 20.15 21.61 20.11 24.08 21.90 20.10 21.52 20.09 19.461.361 22.47 20.18 21.68 20.12 30.58 22.24 20.11 21.59 20.09 24.411.494 22.95 20.22 21.75 20.12 38.20 22.63 20.13 21.65 20.10 30.03

aug-cc-pVDZ Sadlej

0.650 21.36 20.63 21.27 20.62 17.80 21.21 20.64 21.14 20.63 18.111.170 21.82 20.76 21.41 20.71 26.93 21.60 20.78 21.24 20.73 28.001.361 22.23 20.87 21.49 20.77 35.10 21.95 20.90 21.30 20.80 37.161.494 22.73 20.99 21.56 20.83 45.16 22.39 21.04 21.35 20.86 48.81



TABLE III. b-average values of PNA for different basis sets/functionals/frequencies, given in 10230 esu (1 a.u.58.639 418310233 esu), using theB-convention.

v ~eV!

4-31G* 6-31G*

B3LYP BLYP LDA B3LYP BLYP LDA

0.650 9.41 9.47 9.28 6.50 6.69 6.611.170 14.99 17.66 16.93 9.50 10.90 10.761.361 20.28 27.68 26.10 12.00 15.03 14.831.494 27.14 45.87 42.19 14.84 20.67 20.36

AhlrichsVDZ cc-pVDZ

0.650 7.42 7.47 7.37 5.98 6.13 6.071.170 11.10 12.55 12.45 8.86 10.19 10.061.361 14.25 17.83 17.71 11.25 14.17 13.971.494 17.94 25.51 25.37 13.97 19.60 19.28

aug-cc-pVDZ Sadlej

0.650 7.94 8.71 8.21 8.16 8.70 8.231.170 12.33 15.30 14.36 12.94 15.86 14.941.361 16.28 22.69 21.14 17.40 24.33 22.771.494 21.16 34.50 31.73 23.10 38.95 35.97

egep

nt

in

ge

,

eee

nenne

esetiontheehe

ure-

nts

e

ntocaltheng

i-,

Correlation does not affect all components of the hyppolarizability tensor equally. While correlation may chanthe sign of some of the smaller components, other comnents (bzyy) differ only slightly from the RPA values~seeTable II!. Overall, the contribution tobz is dominated bybzzz, corresponding to the strong in-plane CT excitatioindicating that the simple two-state model can be appliedthis component only.50

Comparing with the other correlated calculationsTable IV, we note that Simet al.68 report MP2 values~as-suming Hartree–Fock dispersion! of 12.0310230 esu at 1.17eV, greater than MCQR but smaller than DFT. van Gisberet al.69 obtained LDA and LB94~optically corrected func-tional! values of 16.99 and 21.16310230 esu, respectivelyfor bvec(22v;v,v) at v51.17 eV. Their LDA/ALDA re-sult is consistent with our LDA values.

Concerning the experimental comparison it has bmuch debated in the last 10 years how to compare theperimental extrapolated microscopic property with the oobtained from quantum chemistry calculations in solutioMany issues, such as conventions, experimental referestandards, and the form of local field factors, have been


r-

o-

,o

n

nx-s.cex-

tensively discussed~see Refs. 71–73, and referenctherein!. It was claimed71 that a special convention might bused in the electric-field induced second harmonic genera~ESHG! measurements for PNA molecules in solution, asresult of the experimental values forb was suggested to bmultiplied by a factor of 3 in order to be compared to tresults defined by the Taylor convention.71 However, this hasnot be confirmed by the subsequent experimental measments. Recently, the ESHG results forbz of the PNA mol-ecule in gas phase at 1064 nm (v51.17 eV) was found to be15.4460.5 (10230 esu),54 which is quite close to the valueof 16.9 extrapolated from the ESHG solution measuremeby Teng and Garito.52

For v51.17 eV we thus note only a small differencbetween the gas-phase value54 and the value obtained fromlocal field corrections of the solvent value,52 amounting toabout 1.5310230 esu. This indicates that the strong solveeffect can be accounted for to a large degree by such a lfield factor correction. This agreement is good news frompoint of view that solvent measurements indicate a strodependence ofb values with respect to solutions with varous~larger! dielectric constants for CT molecules like PNA

TABLE IV. The hyperpolarizabilityb average of PNA in units of 10230 esu (1 a.u.58.639 418310233 esu), using the B-convention.

v ~eV! RPAa MCQRa MP2b LDA/ALDA c LB94/ALDAc B3LYPd B3LYPe Exptf Exptg

0 4.09 5.93 6.72 6.850.65 4.50 6.52 7.94 8.15 9.660.51.17 5.68 8.20 12.0 16.99 21.16 12.33 12.94 16.960.4 15.4460.631.364 6.50 9.38 16.28 17.41 25611.494 7.24 10.43 21.16 23.09 4063

aFrom Ref. 53.bFrom Ref. 68 using RPA dispersion.cFrom Ref. 69.dPresent results using the aug-cc-pVDZ basis.ePresent results using the Sadlej basis.fExperiment from Ref. 52 extrapolated from solvent measurements.gExperiment from Ref. 54 in the gas phase.


d-a

ol-

oruty

ioWeinreG

ro

e V

ce ato

artial,tor.

twondunc-antriddy-

thel-torlts-p-leg-heetalinger-

y-rtyen-dy,rp-eful

en


and extrapolation toe51 becomes therefore uncertain. Moeling with the reaction field theory support the notion ofstrong influence on the hyperpolarizability of PNA by a svent, even for an unpolar solvent like dioxane.74

B. ESHG of a push–pull porphyrin

Increasing the size and conjugation of the chromophand increasing the dimensionality of the donor–acceptor sstitutions are two measures that are known to increasehyperpolarizability and effects that are derived from the hperpolarizability tensor, like second harmonic generat~ESHG! and the two-photon absorption cross section.have therefore extended the test on ESHG to includdouble substituted ‘‘push–pull’’ type porphyrin studiedRef. 75~molecule IIa in Table I of Ref. 75, reproduced hein Fig. 1!. We see from the table that indeed the ESHvalues are increased by at least an order of magnitude f

FIG. 1. Structure of the push–pull porphyrin molecule. The substitugroups are NH2 and NO2 .


eb-he-nea

m

the corresponding values for PNA. We see also from Tablthat a very similar pattern to PNA is obtained forb compo-nents and ESHG average values. For instance, we notidecrease of the average ESHG values going from LDAhybrid type functionals.

V. DISCUSSION

We have developed a formalism for linear and nonlineresponse functions at the DFT level based on an exponensecond-quantization, parametrization of the density operaThe DFT response functions have been derived fromalternative variation principles—the Ehrenfest method athe quasienergy method. Linear and quadratic response ftions were derived and implemented for the most importclasses of functionals, that is, for the LDA, GGA, and hybfunctionals. We have presented the first calculations ofnamical hyperpolarizabilities for hybrid functionals.

The theory has been illustrated by calculations ofstatic and dynamic hyperpolarizabilities for the PNA moecule and for a push–pull two-dimensional donor–accepsubstituted porphyrin molecule. The LDA and B3LYP resuare found to differ significantly for ESHG; the B3LYP functional gives good agreement with experimental values. Aplications on the push–pull porphyrin derived molecushow equally good applicability, predicting an order of manitude increase over PNA for the ESHG values. With tpresent method, the role of substituents, the role of matoms in the center, and the role of bridging and conjugatgroups can easily be investigated for the nonlinear propties, something we intend to do for hyperpolarizabilitrelated properties, thereby building up structure-to-properelations that can be useful for the synthesis and experimtal characterization of photonic materials. In a coming stuwe will apply the presented method to two-photon absotion, which serves as an exceptionally interesting and usproperty for a large class of photonic materials.

t

TABLE V. ESHG b-average values for the push–pull porphyrin of Fig. 1, in units of 10230 esu (1 a.u.58.639 418310233 esu).

v~eV! bx;x,x by;y,x bz;z,x bx;y,y bx;z,z bav v~eV! bx;x,x by;y,x bz;z,x bx;y,y bx;z,z bav

4-31G:B3LYP AhlrichsVDZ:B3LYP0.30 281.99 267.88 0.20 264.43 0.20 2148.52 0.30 282.51 267.96 0.22 264.54 0.23 2149.110.40 287.28 275.55 0.20 268.56 0.21 2160.30 0.40 287.84 275.60 0.23 268.70 0.24 2160.91

4-31G:BLYP AhlrichsVDZ:BLYP0.30 285.71 280.03 0.18 272.60 0.18 2163.08 0.30 286.24 279.90 0.23 272.62 0.24 2163.480.40 294.06 297.81 0.18 279.54 0.19 2185.59 0.40 294.63 295.48 0.24 279.62 0.25 2184.58

4-31G:LDA AhlrichsVDZ:LDA0.30 285.55 281.42 0.16 273.59 0.17 2164.19 0.30 286.46 275.64 0.22 268.65 0.23 2159.540.40 293.89 297.86 0.17 280.66 0.18 2185.84 0.40 294.91 290.34 0.22 275.05 0.23 2179.93

6-31G* :B3LYP cc-pVDZ:B3LYP0.30 260.17 251.11 0.10 248.46 0.11 2110.29 0.30 257.36 248.81 0.19 246.24 0.20 2105.120.40 263.45 256.19 0.10 250.99 0.11 2117.80 0.40 260.54 253.71 0.19 248.69 0.20 2112.38

6-31G* :BLYP cc-pVDZ:BLYP0.30 268.60 258.97 0.09 253.63 0.10 2125.69 0.30 261.51 256.19 0.20 251.06 0.21 2115.7870.40 274.40 268.79 0.10 257.56 0.11 2139.34 0.40 266.51 265.59 0.20 254.89 0.22 2128.327

6-31G* :LDA cc-pVDZ:LDA0.30 268.68 264.53 0.10 258.58 0.10 2131.12 0.30 265.25 261.20 0.21 255.58 0.22 2124.360.40 274.50 275.88 0.10 263.15 0.11 2146.03 0.40 270.83 271.97 0.21 259.96 0.22 2138.58


nchr

c

u

e-

in

e-

nc-


ACKNOWLEDGMENTS

This work has been supported by the Swedish ScieResearch Council~O.V.! and by the EU training and researcnetwork MOLPROP~‘‘Molecular Properties and MoleculaMaterials,’’ Contract No. HPRN-CT-2000-00013! ~P.S.!. Weacknowledge valuable discussions with Dr. Yi Luo.

APPENDIX A: THIRD-ORDER FUNCTIONALDERIVATIVES

1. Dirac–Slater exchange functional

The Dirac–Slater functional is one of the simplest funtionals,

F@r#5Cr4/3; C523

4 S 6

p D 1/3

. ~A1!

The nonzero derivatives needed to compute the qdratic response DFT contribution for this functional are

]F

]r5Cra

1/3, ~A2!

]2F

]r2 5C1

3ra

22/3, ~A3!

]3F

]r3 52C2

9ra

25/3. ~A4!

2. LYP correlation functional

The LYP functional is an example of a nontrivial corrlation functional couplinga andb densities.

The energy contribution of the LYP functional is67

F@r,¹r#52aE f 0 dt2abE v@rarb f 1~r,¹r!

1 f 2~r,¹r!#dt, ~A5!

where

f 054

11dr21/3

rarb

r, ~A6!

f 15211/3CF~ra8/31rb

8/3!14727d

18¹r22

452d

18~ u¹rau2

1u¹rbu2!2d211

9 S ra

ru¹rau21

rb

ru¹rau2D , ~A7!

f 252 23 r2¹r21~ 2

3 r22ra2 !u¹rbu21~ 2

3 r22rb2 !u¹rau2,

~A8!

v5exp~2cr21/3!

11dr21/3 r211/3, ~A9!

d5cr21/31dr21/3

11dr21/3. ~A10!

One of the constants is derived analytically, the remainones come from fitting to analytical results,

CF5 310 ~3p2!2/3, a50.04918, b50.132,


e

-

a-

g

c50.2533, d50.349. ~A11!

3. First order derivatives of the BLYP

The derivatives ofF can be expressed in terms of drivatives of auxiliary functions,

]F

]ra52a

] f 0

]ra2ab

]v

]ra@rarb f 1~r,¹r!1 f 2~r,¹r!#

2abvFrb f 11rarb

] f 1

]ra1

] f 2

]raG ~A12!

or

]F

]u¹rau52abvFrarb

] f 1

]u¹rau2

] f 2

]u¹rauG . ~A13!

The relevant auxiliary function derivatives are

] f 0

]ra5

4rb~dra13rb~d1r1/3!!

3r5/3~d1r1/3!2 , ~A14!

]v

]ra5

211r4/31r~c210d!1r2/3cd

3r16/3~r1/31d!2 e2cr21/3, ~A15!

d852dr2/31c~d1r1/3!2

3r4/3~d1r1/3!, ~A16!

] f 1

]ra5211/3CF

8

3ra

5/311

18S u¹rau21u¹rbu227¹r2

22S ra

ru¹rau21

rb

ru¹rbu2D D ]d

]ra

1112d

9

rb

r2 ~ u¹rau22u¹rbu2!, ~A17!

] f 2

]ra52

2

3u¹rbu~4ru¹rau13rau¹rbu!, ~A18!

] f 1

]u¹rau5

4727d

9¹r1u¹rau

d2451~2222d!ra

r

9,

~A19!

] f 2

]u¹rau52

4

3r2¹r12S 2

3r22rb

2 D u¹rau2. ~A20!

4. Second-order derivatives

The interesting second order derivatives are

]2F

]ra2 52a

]2f 0

]ra2 2abS v9r 12v8

]r

]ra1v

]2r

]ra2 D , ~A21!

]2F

]ra]u¹rau52abS v8

]r

]u¹rau1v

]2r

]ra]u¹rau D , ~A22!

]2F

]u¹rau252abvS rarb

]2f 1

]u¹rau21

]2f 2

]u¹rau2D . ~A23!

These derivatives are expressed through the auxiliary fution derivatives,


fo


d954r~c1d!12d~6r1/3cd12cd21r2/3~6c1d!!

9r7/3~d1r1/3!3 , ~A24!

r 5rarb f 11 f 2 , ~A25!

s5ra

ru¹rau21

rb

ru¹rbu2, ~A26!

]s

]ra5~ u¹rau22u¹rbu2!

rb

r2 , ~A27!

]2f 0

]ra2 528rb

rad~d12r1/3!13rb~d1r1/3!~2d13r1/3!

9r8/3~d1r1/3!3 ,

~A28!

]2f 1

]ra2 5211/3CF

40

9ra

2/32]s

]ra

d8

91

1

9~112d!

]2s

]ra2

11

18~ u¹rau21u¹rbu227¹r222s!d9, ~A29!

]2f 1

]ra]u¹rau5F u¹rauS 2

ra

r21D27¹rG d8

9

1F ~112d!rb

r2G 9

2u¹rau, ~A30!

]2f 1

]u¹rau2 5226d

91~112d!

2ra

9r, ~A31!

]2f 2

]ra2 52

u¹rbu3

~6u¹rbu18u¹rau!, ~A32!

]2f 2

]ra]u¹rau52

8

3ru¹rbu, ~A33!

]2f 2

]u¹rau2 522rb2, ~A34!

]r

]ra5rb f 11rarb

] f 1

]ra1

] f 2

]ra. ~A35!

Other second-order derivatives that do not vanishLYP are

]2F

]ra]rb52a

]2f 0

]ra]rb2abFv9r 1v8S ]r

]ra1

]r

]rbD

1v]2r

]ra]rbG , ~A36!

]2F

]ra]u¹rbu52abS v8

]r

]rb1v

]2r

]ra]u¹rbu D , ~A37!

]2F

]u¹rauu¹rbu52abvS rarb

4727d

92

4

3r2D , ~A38!

which refer to

]2f 0

]ra]rb5

4ra~drb13ra~r1/31d!!

3r5/3~r1/31d!2 , ~A39!


r

]2r

]ra]rb5 f 11rb

] f 1

]rb1ra

] f 1

]ra1rarb

]2f 1

]ra]rb

1]2f 2

]ra]rb, ~A40!

]2f 1

]ra]rb5sd91

d1d8211

9r2 ~ra2rb!~ u¹rau22u¹rbu2!,

~A41!

]2f 2

]ra]rb52

8

3u¹rauu¹rbu, ~A42!

]2f 1

]ra]u¹rbu5

d8

18F2u¹rbu214~ u¹rbu2u¹rau!

24u¹rbura

r G12d211

9

rb

r2 u¹rbu2,

~A43!

]2f 2

]ra]u¹rbu52

8

3ru¹rau24rau¹rbu, ~A44!

]2r

]ra]rb5 f 11rb

] f 1

]rb1ra

] f 1

]ra1rarb

]2f 1

]ra]rb

1]2f 2

]ra]rb, ~A45!

]2r

]ra]u¹rbu5rb

] f 1

]u¹rbu1rarb

]2f 1

]ra]u¹rbu

1]2f 2

]ra]u¹rbu. ~A46!

5. Third-order derivatives

The interesting third order derivatives are

]3F

]ra3 52A

]3f 0

]ra3 2abS v-r 13v9

]r

]ra23v8

]2r

]ra2

1v]3r

]ra3 D , ~A47!

]3F

]ra2]u¹rau

52abS v9]r

]u¹rau12v8

]2r

]ra]u¹rau

1v]3r

]ra2]u¹rau D , ~A48!

]3F

]ra]u¹rau252abFv9S rarb

]2f 1

]u¹rau21

]2f 2

]u¹rau2D1vrbS ]2f 1

]u¹rau2 1ra

]3f 1

]ra]u¹rau2D G ,~A49!

]3F

]u¹rau350. ~A50!

Again, we use a set of auxiliary functions,



d-522r2/3d2~84c15d!14d~r~28c18d!17cd3!128r1/3~4cd31r~c1d!!

27r10/3~d1r1/3!4 , ~A51!

es

]2f 1

]rb]ra52

7

9¹rd81

u¹rau9 S d82~2222d!

ra

r2

22ra

r D , ~A52!

]3f 1

]ra3 5211/3CF

80

27ra

21/321

9 S ]2s

]ra2 d81

]s

]rad9D

1d-18

~ u¹rau21u¹rbu227¹r222s!

2S s1]s

]raD d9

922

]2s

]ra2

d8

91

112d

9

]3s

]ra3 , ~A53!

]3f 1

]ra2]u¹rau

5~ u¹rau27¹r!d9

91S d9

ra

r2d8

rb

r2

22~112d!rb

r3 2d8rb

r2 D 2u¹rau9

, ~A54!

]3f 1

]ra]u¹rau252

2

3d82d8

2ra

9r1~112d!

2rb

9r2 , ~A55!

]3f 2

]ra2]u¹rau

528

3rb

2. ~A56!

For the LYP, other nonvanishing third-order derivativare

]3F

]ra2]rb

52a]3f 0

]ra2]rb

2abFv-r 1v9S 2]r

]ra1

]r

]rbD

1v8S ]2r

]ra2 12

]2r

]ra]rbD 1v

]3r

]ra2]rb

G , ~A57!

]3F

]rb]u¹rau252abFv8S rarb

]2f 1

]u¹rau21

]2f 2

]u¹rau2D1vS ra

]2f 1

]u¹rau2 1rarb

]3f 1

]rb]u¹rau2

1]3f 2

]rb]u¹rau2D G , ~A58!

]3F

]ra]rb]u¹rau52abFv8

]r

]u¹rau1v8S ]2r

]ra]u¹rau

1]2r

]rb]raD1v

]3r

]ra]rb]u¹rauG ,~A59!


]3F

]ra2]u¹rbu

52abS v9]r

]u¹rbu12v8

]2r

]ra]u¹rbu

1v]3r

]ra2]u¹rbu D , ~A60!

]3F

]ra]u¹rau]u¹rbu52abv8S rarb

]2f 1

]u¹rauu¹rbu

1]2f 2

]u¹rauu¹rbu D2abvS rb

]2f 1

]u¹rauu¹rbu

2rarb

7

9d82

8

3r D . ~A61!

These use

]3r

]ra2]u¹rbu

5rarb

]3f 1

]ra2]u¹rbu

12rb

]2f 1

]ra]u¹rbu

1]3f 2

]ra2]u¹rbu

, ~A62!

]3f 1

]ra2]u¹rbu

5~d81d211!2u¹rburb

9r2

1d9

9 S u¹rbu27u¹rau24rb

r U¹rbU D ,

~A63!

]3f 2

]ra2]u¹rbu

524u¹rbu28

3u¹rau, ~A64!

]3r

]ra]rb]u¹rau5

]2f 1

]ra]u¹rau1ra

]2f 1

]ra]u¹rau

1rb

]2f 1

]rb]ra1rarb

]3f 1

]ra]rb]u¹rau

1]3f 2

]ra]rb]u¹rau, ~A65!

]3f 1

]ra]rb]u¹rau52u¹rauFra

rd91

d1d8211

9r2

3~ra2rb!G , ~A66!

]3f 2

]ra]rb]u¹rau52

8

3u¹rbu. ~A67!


or

ivl-

om

on

h-

en

. L

Ghy

Ph

Ph

ys.

.

ds,

em.

hys.

Phys.

ys.


APPENDIX B: VARIABLE TRANSFORMATION

We want to transform derivatives expressed in termsu¹ru to derivatives ofz5u¹ru2 because it is usually simpleto compute derivatives in terms ofu¹ru while the responseformulas are simplest when expressed in terms of derivatwith respect toz. The transformation is carried out as folows:

]F

]z5

]F

]u¹ru]u¹ru

]z5

]F

]u¹ru1

2u¹ru, ~B1!

]2F

]z2 5]2F

]u¹ru21

4u¹ru2 2]F

]u¹ru1

4u¹ru3, ~B2!

]3F

]z3 5]3F

]u¹ru31

8u¹ru3 2]2F

]u¹ru23

8u¹ru4

1]F

]u¹ru3

8u¹ru5. ~B3!

This transformation is a convenient place to transform frspin–orbital coefficients to closed shell ones.

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Density-functional theory of linear and nonlinear time-dependent molecular properties

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