ELSEVIER

22 December 1997

Physics Letters A 236 (1997) 525-532

PHYSICS LETTERS A

Variation-perturbation method in time-dependent density-functional theory Arup Banerjee, Manoj K. Harbola

Laser Programme, Centre for Advanced Technology, Indore 452 013, India

Received 7 August 1997; revised manuscript received 14 October 1997; accepted for publication 17 October 1997 Communicated by B. Fricke

Abstract

We develop the variation-perturbation approach for periodic time-dependent perturbations within the Kohn-Sham method of density-functional theory. Our formulation parallels a similar formulation developed by Gonze for static perturbations. The theory is demonstrated by calculating the dynamic polarizability and hyperpolarizability for the helium atom. @ 1997 Elsevier Science B.V.

Since its inception, density-functional theory (DFT) [ 1 ] has proved to be highly versatile in handling a range of many-electron systems. Further, our understanding of its fundamental as well as application aspects has increased much. The theory is exact in principle, and can be applied with relative ease within certain approximations, among these the local-density approximation (LDA) [ 1,2] being the most widely used one. Because of these attractive features, there have also been situations where the theory has been applied without any rigorous justification for it at the time. Thus, Zangwill and Soven [ 31, and Stott and Zaremba [ 41 applied it to calculate the frequency-dependent polarizabilities of atoms without there being any formal density-functional theory for time-dependent potentials, and found the results to be satisfactory. The formal proofs followed later in the pioneering works of Deb and Ghosh [ 51, Bartolotti [ 61, and Runge and Gross [ 71. The situation, as it stands today, is that time-dependent density-functional theory (TDDFT) [ 81 is a well-established theory, although it is not as well explored as its time-independent counterpart.

One of our interests is in applying density-functional theory to obtain the response [9] - linear as well as nonlinear - of a system to an applied electromagnetic field. These responses are important in describing the interaction of light with matter, and are generally calculated using perturbation theory. Density-functional perturbation theory (DFPT) [lo] provides a method of calculating these quantities for many-electron systems and has been applied extensively to obtaining both the static and the dynamic response. The connection between the variational principle and the perturbation expansion for static perturbations within DFT has been explored by Gonze [ 111 and by us [ 121. This leads to a variation-perturbation approach within DFPT for time-independent situations. On the other hand, such a general formulation does not exist for time-dependent potentials, although variation-perturbation calculations up to the second order by employing constructed functionals have been performed [ 131 in the past. A complete development of this approach to higher-order potentials is thus highly

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526 A. Banerjee, M.K. Harbola/Physics Letters A 236 (1997) 52.5-532

desirable and forms the basis of our present work. In the following we develop, within the rubric of TDDR, the variation-perturbation formalism for perturbations periodic in time. Assuming the applied perturbation to be periodic is not a severe restriction since for most practical situations such a treatment is sufficient. Furthermore, the extension of the method to a more general time dependence is straightforward.

Our motivation in developing variation-perturbation theory is to make use of the advantage offered by variational methods over the direct solution of the Schriidinger equation. The standard perturbation approach in TDDFT is to calculate [ 8,14,15] the screening potential and the perturbed orbitals self-consistently by solving the appropriate inhomogeneous differential equation. This is done either by the Green’s function method [ 141 or by summing over [ 151 the complete set of KS eigenstates. In contrast, a variational approach circumvents the complexity of the above-mentioned methods by directly making an appropriate energy or action functional stationary with respect to a parametrized wave function. This also makes it computationally easy to implement.

To develop a variation-perturbation method for periodic time-dependent potentials we work within the formulation given by Sambe [ 161 for Hamiltonians which are periodic in time. (For a general discussion of time- dependent perturbation theory and its connection with Sambe’s formalism, we refer the reader to Ref. [ 171.) It is on the basis of this formulation that Deb and Ghosh also proved [ 51 the Hohenberg-Kohn theorem [ 181 for potentials periodic in time. We first derive the time-dependent Kohn-Sham (KS) equations employing Sambe’s formulation and then expand the orbitals, the eigenenergies and the total energy perturbatively. Following Gonze’s treatment [ 111 of static perturbations we then derive the corresponding expressions for energies up to the fourth order in the energy and demonstrate the general rule of the (2n + 1) theorem and its even- order variational corollary. This completes our goal of the formal development of the variation-perturbation method within TDDFT. We then demonstrate the formalism with calculations of the dynamic polarizability and hyperpolarizability of the helium atom from its Hartree-Fock and Kohn-Sham orbitals. We note here that the (2n + 1) theorem has been used [ 191 in TDDFT earlier to calculate the second-order nonlinear response of semiconductors. However, a complete description of the variation-perturbation expressions for the energies, and their relation to the perturbation equations via the even-order corollary of the (2n + 1) theorem is lacking. The present paper fills this gap.

The steady state solution ?P (rl , . . . , TN, t) of a time-dependent Hamiltonian periodic in time satisfies the equation (atomic units, lel = me = fi = 1 are used throughout this paper)

(H(r) -i$)?P(r,t) =O,

where H( t + 7) = H(t), and is of the form

P(r, t) = @(f, t)e@‘, (2)

where @(rl,..., r~, t) is periodic in time with time period 7, i.e. @(t + 7) = Q(t) and E is called the quasi-energy of the system. Substituting Eq. (2) in Eq. ( 1) gives

(H(r) - i$)@(r,r) = E@(r,r), (3)

for @(r, t). Thus, in Sambe’s formulation, H(t) - ia/& is treated as the Hamiltonian ‘H( r, t) for the steady state in a composite Hilbert space (R + T). This Hilbert space consists of all possible functions c$( r, t) which

are periodic with period 7 and for which sIf,* /4( r, t) I* drdt is finite. The inner product of two functions 41 and ~$2 in (R + T> is defined as

((41 (r, f)l42(f, ~)))a”erage~ (4)

where the curly brackets denote ( l/7) cz2 dt, the average over a period. Thus, the quasi-energy becomes

A. Banerjee, M.K. Harbola/Physics Letters A 236 (1997) 525-532 527

As pointed out in Ref. [ 161, there is a one-to-one analogy between the “Hamiltonian” FL, the “wave function” @(r, t) and the “energy” E in (R + T) for the steady states to the same quantities in R for static cases. Thus, Eq. (3) for Q, can be derived [ 161 from the variational principle which makes E[@] stationary with respect to @. Also, because of the Hohenberg-Kohn theorem for time-dependent potentials, the stationary principle leads [‘7] to the Euler equation for the time-dependent density. Notice that the time-averaged energy for periodic potentials is equivalent to the action integral employed in Ref. [ 71. Now, by assuming the density to be non-interacting v-representable, the variational principle can be further employed [ 71 to derive the time- dependent Kohn-Sham equations as follows. Consider a system of non-interacting fermions in a potential us(t) periodic in time with period T which gives the same density as the interacting system, so that

dr, f) = c l4(r, t>1*, i

where the ui (r, t) satisfy

( - iV2+Us(r,r) -ii Ui(r,f) CO. )

Then, analogous to the static case [ 11, Ui(r, t) are the time-dependent KS orbitals and

u,(r, t) = Dext(T, f> + EH[P(~,~)I

&(r, t) + Exc[p(r, t) 1

@(r,O ’

(6)

(8)

where EH is the Hartree energy and E,, is the exchange-correlation energy. Since u,(t) is periodic, ui( r, I) = #i (r, f)e-iq’, where the 4i (r, t) are also periodic (time period T) and satisfy

( -iV*+U,(r,t) -ii 4i(r,t) =Ei&(r,r). >

(9)

We refer to Ei, which are time independent, as quasi-Kohn-Sham eigenvalues. Note that the expression for the density in terms of 4i(r, t) or Ui(r, t) is the same. The total energy in terms of the Kohn-Sham orbitals is given by

where ~~~~ represents the sum of the Hartree, exchange, and correlation energies. Let us now consider u,,r(r, r) to be made up of two parts,

uext(r, t) = uo(r) + u(‘)(r 2) 7 1

where uc(r) is a time-independent potential and

(11)

J*)cr , f) = Lv(l)(r)(eiW’+e-i~f) 2

is the applied perturbation. The Hamiltonian is accordingly

‘H = 7-h + H’(r, t),

where

(12)

(13)

(14) IFlo = --iv* + UO(T, t) -i&,

528 A. Banerjee. M.K. Harbola/Physics Letters A 236 (1997) 52.5-532

and H’( r, t) is the sum of ~(‘)(r, t) and corrections to the Hartree and the exchange-correlation potentials arising from the self-consistent change in the density due to the applied perturbation. Given the simple form of the equation for #~(r, t), it is now straightforward to develop [ 161 the perturbation theory by expanding H’, +i(T, t) and pi. However, for the time-dependent potentials, the +i( r, t) also contain a complicated phase factor. It is therefore better to separate out [ 17,201 the phase part and work in terms of the transformed wave functions [ 161 xi( r, t), which are the same as the normalized regular function of Ref. [ 171. Separating out the phase part makes the structure of the resulting perturbation equations similar to that of the static case, and also ensures the correct zero-frequency behavior of the perturbed orbitals [ 16,17,20]. The transformed Kohn-Sham equation is

Ho+~H(“)-~i(f) Xi(r,f)=O, II=1 >

where [ 16,171

Ei( t) = EO + Re (xl C,,=, ~8”) (t) Ix’)

(x1x0) ’

where the subscript “0” refers to the unperturbed state. The quasi-energy Ei is given as

712 1 Ei = - dtEi(t). 7 J

-712

(15)

(16)

(17)

Each of the transformed orbitals satisfies [ 16,171 the normalization condition

(XIX) = 19 (18)

and

(x1x0) = (x01x>- (19)

Notice that the expressions for the density and the total energy in terms of transformed orbitals remain the same as given above by Eqs. (6) and ( 10) respectively, although the Ei in Eq. ( 10) are now replaced by their time-dependent counterpart as given by Eq. (16). The perturbation expansion for the orbitals, the density, the perturbed Hamiltonian H’, and the C: is

x(r,t) = gyk’(r,t), p(r,t> = -&qf,f), k=o &=o

H’(r,t) =FH”‘(r,r), Ei(t) =fQ’(f), k=l !f=o

and the equation for xCn) is

d, H(O) - e. - iz ,ycn)

>

+ -&” - ,(i) jX(n-i) = 0,

i=l

(20)

(21)

where by Eqs. (18) and (19)

A. Banerjee, M.K. Harbola/Physics Letters A 236 (1997) 525-532 529

(22) i=l

Now following the steps given in Ref. [ 111, we get

E(l) = #)(r, t)p(‘)(r) dr , average

Ec2) = (,y”‘~7-fo - l oJ,yc1))

+ (X(l)b%‘(o)) + (X(“)l~(l)l~(l)) + f /- 8;;r$;;;2j #)(r,, t)pc1)(r2, t) dr, dr2} , average

E(3) = (,ppp _ E(‘qp)

E(4) = (p’~‘Ho _ ~oJxw) + (*Wp(‘) - ,(‘)I*(‘)) + (X(‘)p(‘) - ,wp)

@(rl )‘$(r2)&‘(r3)+(r4) P(1’(rl,t)p’1’(r2,t)P’1)(r3,t)p(1)(r4,t) dri dr2dr3dr4 .

average (23)

In the expressions above, the sum over occupied orbitals is implicit. As is clear from these expressions for energies to various orders, only x(” is required to obtain the energy up to order 3. Also due to the variational nature of the energy functional, the even-order energies Ec2) and Ec4) are stationary [21] with respect to x(l) and xc2), respectively. This is evident from the quadratic nature of Ec2) with respect to x(l) and that of EC41 with respect to xc2). Stationarity of Ec2) and Ec4) leads to the equations for the first- and second-order correction to the orbitals. Alternatively, it can also be used to perform variational calculations of the induced orbitals and energies. The expression for H(l) is

zP(r, t) = u(‘)(r t) + s a2E~xc 9 Sp(r)Gp(rl > d’)h 7 t) dn.

Similar expressions for the higher-order corrections of the Hamiltonian can be derived and involve higher-order derivatives of the Hat-tree and exchange-correlation energy functionals. The procedure adopted above is quite general and can be employed to calculate energy changes to any desired order. Furthermore, if the orbitals are known correctly to order (n - l), even-order energies E(2n) are stationary with respect to ,$“), thus leading to the perturbative equation for the nth-order correction of the orbitals. This completes the formal development of variation-perturbation theory for periodic time-dependent potentials. We note, however, that if we wish to derive a minimum even-order energy principle, this would be possible only in the normal dispersion region. This is because the minimum of the energy exists [6,17] only below the first transition frequency.

530 A. Banerjee, M.K. HarbolalPhysics Letters A 236 (1997) 525-532

We now demonstrate the theory by calculating variationally the dynamic polarizability and hyperpolarizability for the helium atom. For this purpose we obtain the second-order (EC*)) and the fourth-order ( Ec4)) energies by minimizing these quantities. Therefore, we restrict ourselves to frequencies below the first excitation energy for the second-order and half of that for the fourth-order energies. The dynamic polarizability a(o) and hyperpolarizability y( --w; w, w, -w) are related to these energy changes by

a(w) = 2[2E’2)(o)] 9 y(-w;w,w,-w) = $[24E(4)(w)], (25)

where the extra factors (compared to the static case) of 2 for (Y and S/3 for y arise from the time averaging of the energy. Further, since the hyperpolarizability is calculated from the average energy shift, the only component of this quantity which is obtained directly is y( --CO; w, w, -0).

For the calculation of the polarizabilities, V(‘)(r) in Eq. (12) is taken to be of the form

V(‘)(r) = Ercost?, (26)

where E is the field strength in the z direction. Since the phase part has been removed from the orbitals, perturbative solutions of x have the following simple time dependence [ 16,171,

~(‘)(r, t) = Xy/(r)eiu’ + X?l)(r)emio’,

*(*) ( r, t) = *$Y.i ( r)e2io’ + *$ ( r)e-2iwf + ~a) (r) , (27)

where the x&, depend only on the spatial co-ordinates. These are the functions which we determine variation- ally. Normalization conditions satisfied by the various components of the induced orbitals are

tx:‘; I/p’) = 0 , LY:21W0)) = -‘(Xl”qxJ”q 2

LdwO’) = -&&“y&g + (,&“y&“g)~ (28)

The variational forms we choose are

,yy’ = A&)X(~)(~) cos8,

x:2,b= [A:,,(r) +A~,o(r)~os2~],y(o)(r) +Ak,o,y(o)(r), (29)

where

A’(r) =air+bir2+Cir3+... (30)

is a polynomial with ai, bi, ci . . . being the variational parameters, and Ah,0 is chosen such that Eq. (28) is satisfied. Such forms of variational wave functions have been applied [22] earlier for the static response functions. In the present calculation we have chosen five parameters for each of the A’. This amounts to a ten- parameter calculation for the polarizability and a thirty-parameter one for the hyperpolarizability. We perform our calculations for both the Hartree-Fock and the Kohn-Sham orbitals. Within Hartree-Fock, the exchange energy for helium is given exactly as

Ex = -+E”[p]. (31)

For the Kohn-Sham calculation we employ the LDA parametrized by Gunnarsson and Lundquist [ 21. We do note, however, that an LDA for the frequency-dependent linear response also exists [ 8,231, although no such kernel has been constructed, to the best of our knowledge, for the nonlinear response. A detailed discussion of the frequency-dependent kernel and its comparison with the static LDA is given in Ref. [ 81.

A. Banerjee, M.K. Harbola/Physics Letters A 236 (1997) S25-532 531

Table 1 Table 2 Polarizability (Y and hyperpolaxizability y of helium as a function Polarizability (2 and hyperpolarizability y of helium as a func- of angular frequency calculated from the Hartree-Fock density. tion of angular frequency calculated from the Kohn-Sham LDA Atomic units are used density. Atomic units are used

w do) y(-~;~,w -w) w a(o) Y( --w; w, w, -w)

0.0 1.32 36.04 0.0 1.63 83.30 0.1 I .34 39.26 0.1 1.65 94.65 0.2 1.38 52.19 0.2 1.73 149.9 0.3 1.46 98.07 0.3 1.88 1018 0.35 I .52 195.1 0.31 1.90 19462 0.38 1.57 613.5 0.4 2.15 0.385 1.575 1068 0.5 2.72 - 0.39 1.58 5832 0.6 5.12 0.4 1.60 0.62 7.66 0.5 1.83 0.64 42.1 0.6 2.28 0.7 3.43 0.75 5.37 0.78 10.1 0.79 15.9

Shown in Table 1 are the results obtained from the Hartree-Fock wave function of helium. Our results for both linear and nonlinear polarizabilities match well with those [24,25] in the literature. Notice that LY increases sharply from w = 0.7 a.u. onwards and becomes singular at about 0.79 a.u. This is very close to the eigenvalue difference of the 1s and the 2s orbitals; we attribute the slight difference to numerical inaccuracy. The hyperpolarizability as a function of w increases faster than the linear polarizability, becoming singular at w = 0.39 a.u., again very close to half the eigenvalue difference between the lowest orbitals. The reason for it being little less than half could be due to the fact that within the dipole approximation there is a two-photon resonance state (1~2s) which is slightly below the single-photon excited state (1~2~).

In Table 2 we give the polarizabilities obtained using the Kohn-Sham orbitals. It is well known that the LDA overestimates both the polarizabilities (linear as well as nonlinear), as is also seen from the table for all the frequencies. The trend is the same as for the Hartree-Fock numbers except that a(w) is singular at about w = 0.65 a.u. and y( --o; w, w, -w) at 0.32 a.u. This is consistent with the LDA 1s orbital eigenvalue being at -0.59 a.u. and the 2s orbital being unbound.

For low frequencies, the dispersion of LY( w) is given [ 261 as

a(w) = a(O)( 1 + C2w2). (32)

Results obtained by us are C, HF = 1 10 and CkDA = 1.56 (Gunnarsson-Lundquist parametrization) ; the experi- . mental value [26] is 1.16. Thus, the Hartree-Fock number for C2 compares well with experiment. On the other hand, the LDA numbers obtained here and in other calculations [ 14,151 overestimate the experimental value. They match well with each other - the slight difference arises due to the different parametrizations used for the exchange-correlation functionals.

To conclude, we have derived variation-perturbation theory within TDDFT, and demonstrated it with a calculation on the helium atom. We note that there is a choice [ 171 of working with orbitals which satisfy the intermediate normalization condition, or with those satisfying the orthonormality condition. In DFT, however, it is natural that normalized orbitals are used as has been done by us. The use of normalized orbitals also keeps [ 161 the functions within the extended Hilbert space, which is necessary while working within the Sambe formulation. Finally, although the theory has been applied to obtain only y( -w; w, w, -w), it is possible to obtain the third-harmonic generation coefficient also by wave functions up to the second order. This

532 A. Banerjee, M.K. Harbola/Physics Letters A 236 (1997) 525-532

follows [ 25,271 from the (2n + 1) theorem in time-dependent perturbation theory. Such a calculation is in progress and the results will be reported in the future.

Discussions with Dr. K.C. Rustagi are gratefully acknowledged.

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