non-linear electric and magnetic properties obtained from cubic response functions in the random...

Chemical Physics

E L S E V I E R Chemical Physics 203 (1996) 23-42

Non-linear electric and magnetic properties obtained from cubic response functions in the random phase approximation

Patrick Norman, Dan Jonsson, Olav Vahtras, Hans/~gren Institute of Physics and Measurement Technology. LinkiJping University, S-58183, Linki~ping. Sweden

Received 18 September 1995

Abstract

We derive cubic response functions in the Random Phase Approximation for calculations of non-linear frequency- dependent properties of molecules, and demonstrate an implementation of these functions using efficient computational algorithms. Illustrations are given by calculations of the frequency-dependent second hyperpolarizabilities, the frequency- dependent excited state polarizabilities and the three-photon transition amplitudes of the para-nitroaniline molecule, and by calculations of the frequency dependent magnetic second hyperpolarizabilities and their anisotropies for the first row hydrides isoelectronic with neon.

1. Introduction

In non-linear optics one can describe the polarization of a system as a power series in the strength of the perturbing field. The expansion coefficients define the molecular properties corresponding to dipole moment, polarizability, first hyperpolarizability, and second hyperpolafizability, etc. With an improved laser technology, experimental progress in determining non-linear molecular spectra and properties has been achieved. Effects in connection with the first hyperpolarizability are well studied both in theory and experiments and, for instance, second harmonic generation (SHG) is routinely used in order to convert laser frequencies. The counterpart for the second hyperpolarizability is called third harmonic generation (THG) and represents one out of a great number of effects that are collectively named four-wave mixing. As the name implies, in general, up to three different incident waves interact in the medium, leading to a fourth resulting wave. THG, unlike SHG that vanishes in fluids unless the molecules are optically active (chiral), occur in all media and the search for suitable compounds is not limited by the condition of noncentrosymmetry. For example, atomic gases have proven useful to triple laser frequencies from the visible into the vacuum UV region with an conversion efficiency of about 10 -4 . Another third order effect of technical interest is the intensity-dependent refractive index. Certain materials then acts as a positive lens, which causes light beams with nonuniform transverse intensity to self-focus.

The list of third order properties can be made long and clearly it is worthwhile to develop a microscopic un- derstanding for these phenomena by a theoretical treatment. One could hope not only to reproduce experimental

0301-0104/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved SSDI 0301-0104(95)00419-X

24 P. Norman et al./Chemical Physics 203 (1996) 23-42

results but also make structure-to-property predictions for groups of compounds. Direct non-approximate calculations of the second hyperpolarizability and other non-linear properties have both formal and computational advantages [ 32,19,27,51,47,58,52,53]. The present work is carried out within the general response formalism where we derive analytical cubic response functions in the random phase approximation including the working expressions for the second hyperpolarizability, for three-photon transition matrix elements, for two-photon matrix elements between excited states, and for excited state polarizabilities at the SCF level of approximation. In the response terminology our method would be coined as the Cubic Random Phase Approximation (CRPA). The merits of the response formalism for various linear and non-linear properties is well-known, see reviews given in Refs. [40,44,13,39,35], and motivates a further step in terms of order. The generality of this formalism is reflected by that each new property arising from the non-linear response to the applied field, be it internal or external, magnetic or electric fields, can be accounted for by a mere change of operators in the response function.

2. Method

The present RPA implementation is double-direct in the sense that the RPA eigenvalue equations are solved by iterative techniques without explicitly forming matrices and that the handling of integrals are carried out in the atomic orbital (AO) basis. The method is formulated in terms of general Fock matrices linearly transformed by vectors. This approach does not assume canonical Hartree-Fock orbitals and allows the use of AO driven routines. The contraction of the Hessian-type matrices is then formulated as multiply one-index transformed Fock matrices. The one-index transformation is the transformation the integrals of an operator undergo when a commutator is formed with the operator and a generator of orbital rotations. This has been the key operation in development of direct MCSCF [41,22]. In this section we derive the necessary expressions for the third order response function of a single determinant wave function. The derivation follows the notation used by Olsen and JOrgensen [42] for response function theory and is also in analogous with their derivation for the first [25] and second [ 18] order response functions. A detailed description of our starting point is given there and we refer to this work and references therein for the necessary background.

2.1. Time development o f the SCF state

The time dependence of the SCF state is introduced as [33,24]

10(t)) = eiK(t)10),

K(t) = ~ ( K . ( t ) q tn + K; ( t )q . ) , (1) n

where {qj, q.~ } are singlet one-electron excitation operators and n runs over non-redundant orbital operators. For a closed-shell system the non-redundant operators transfer electrons from occupied to unoccupied orbitals. The time development is given by the time-dependent SchrSdinger equation

7-/l()(t) ) = i d l o ( t ) ) ,

7-I = 7-[0 + V t. (2)

The Hamiltonian 7-/ is the sum of the Hamiltonian for the unperturbed system 7-[0 and a time-dependent perturbation operator V t. The perturbation will in general be assumed as decomposed in the continuous frequency

P. Norman et al./Chemical Physics 203 (1996) 23-42 25

domain, but in many cases only a few discrete frequencies are involved. The operator x(t) can be expressed in a more general operator basis

K(t) = O a ( t ) ,

O = ( q t , q ) X,

a(t) = X -I ( K ( t ) "~ K*(t) J ' (3)

It is shown in Ref. [42] that an appropriate time development of the SCF state can be determined from a set of Ehrenfest equations for the operator basis O which can be seen as the solution to the time-dependent SchriSdinger equation projected onto allowed wave function variations. The Ehrenfest equations in matrix notation which determines ~ ( t ) then read

..,,,,I,+,1 . ( , ) , . , 1 - - r ( , ) , . , Z i n+' ( E l"+'l V t{"+ll a}~)(t) , (4) I O.jltl2...l,,Ollt k l ) 11Of/,, k[) = -- ~, jlll2...l, "~- jlll2...ln

n---I #=2 n=0 u.=l

where S I''1, E I"l, and V t['l are defined in Ref. [42]. We will have use of the following matrices:

St2~ = (oi [o~, Ok] 10) jk . ,

S~3~ -½(o1[o~, [Oz, Oklll0), jkl =

s t4~ = ~(o1[o~, [o , , [Om,Okl l ]10) , jkhn

E I'l-- (Ol[O~, Ho] I0)--0, J

Et21 -(0l[O~,[Ok, Ho]]10), ik =

El31 1 t jkl ---- 9~(01 [ O j , [Ok, [Ol , H 0 ] ] ] [0),

E141 1 O4 [ O k , [ O l , [ O m , Ho]]] ] [O) ~k~,,, = -- ~(01[ , ,

v/~' ~ =/oL[o~, v'[ Io/,

v/2 ~ = -<ot[o.~, [ok, v'] ] IO), V/~131 i 1 • = ~ ( O l [ O s , [ O k , [ O t , V ' ] ] ] l O ) . ( 5 )

We have here assumed that the reference state, ]0), is optimized with respect to the unperturbed Hamiltonian so that that the Brillouin condition is fulfilled, i.e. E I11 vanishes.

2.2. E x p a n s i o n in order o f the pe r tu rba t ion

The parameters c~(t) that determine the response of the SCF state to the perturbation V t are expanded in orders of the perturbation

Otk(t) = OL~l)(t) + a~2)(t) + a~3)(t) + ... (6)

The terms of (4) that will contribute to third order in the perturbation are

iSl~16k(t) S l 3 t l a k ( t ) o t t ( t ) • [4] • iVjt[ll ., -- __ 1SJklmOtk(t)Otl(t)Otm(t) + • -- EJ2klak(t)

iEJ3tlC~k ( t )ol l ( t ) --iStk[?] ffk (t)Oll(t) + E i 4 1 m a k ( t ) a l ( t ) a m ( t ) = 0. (7) t)


Inserting (6) in (7) and collecting terms of third order gives

iSI2,.~(3)(t ) _ cl31 (&~l)(t)a~2)(t) + & ~ 2 ) ( t ) a ~ l ) ( t ) ) • [4] . ( I , (1 , (1) • - IS)klmOt k ( t ) a t ( t ) a m ( t ) jk t~k ~]kl

r'[2l ( 3 ) : . \ wt[2]~(2){/) :1C'[3]( 1)(t)C r ( t ) + a ~ 2 ) ( t ) a ~ ) --1~.i k Ce k i t ) "jk --k t ,r . jk l a~ ~2) l)(t ) ;,,t[31 (1) . . . . ( l ) t , , I --[41 (1). +, a~ 1) (t)a(1)

- , v j k t a k tf)~x t ~., + t~)ktmCe k tr) (t) =0 . (8)

This set of coupled linear inhomogeneous differential equations determine the third order response of the SCF state. However, the differential equations separate in the representation where both E I21 and S t21 are diagonal. In diagonal basis we have E}j 21 = wj6ij and S~j 21 = sgn( j )6q , where also wj = w_). Thus (8) turns into

~ ( 3 ) ( t ) -~ S~k 3] . [4] . (1) (1) (1) L&kl) ( t )cq2)( t ) +&~2) ( t ) a} l ) ( t ) ) +]S)klmOtk ( t )ot I ( t )ot m ( t ) i sgn(j)&~3)(t) - ,oj~j"

;t7131( I 2)(t)O t )__ i~,[4] . ~ ( l ) ( t ) o t } l l ( t ) O t ( ] ) ( t ) + ' ~ i k t a~ ) ( t ) a ~ 2 ) ( t ) + a ~ ~l)(t) • ~jklm ~'k

lztl21Ol<2) ( t ) -Jr ;l:t[3]~(I) ) + "j* * "'jkl ~ k ( t ) a ~ 1 ( t ) . (9)

To solve (9) we introduce the Fourier transform of the time-dependent parameters

i i S (3) e-i(aJl +a~.+a.,3) aJ3)(t) = crj (Wl,W2,w3) do91 dw2dt.03, --00 --00 --00

77 ol}2) (t) = a}2) (w i, o)2) e-i('°'+'°2) do)l d0)2,

--0(3 - - ~

ffJl)(t) = . f ff( l)(wl) e -i~°l d~! j ,

OG

gj~[21 = i ~'j~l[2] e-i~°l d ° ) l '

- oo oo

VJ~<~31 = S Vjk~'[31 e-i°J' d~l. (10) - - 0 , 0

Inserting the Fourier transform expressions (10) into (9) gives

i sgn( j ) ( - i ( w l + o92 + w3)) 0~3) ((.01,~2, o)3) - ~jOt'~3) (¢-O1, ¢O2, O93)

= +.,k'+131 ((__ioJi)ot~l) (OJl)Ce~2) (~2,093) __ i(o) l "4- ~2)~(k2) (~l , ~2)0~I) (0~3))

• 41 • ( 1 ) ( 1 ) ( 1 ) +lSj,t.,C-~oJ )a k (oJl)a t (o~2)a., (oJ3)

:.131 (0~I)(0)1)0~2)(~02, aJ3)_i_ 0~(k2>(0)i,0)2)0~I,(093)) + it.Ski __ / 7 1 4 1 ,~,(1>(¢.01 )0~1>( ( ,02 )~ ( ] ) (~03) ~./klm ~ k

vwi121 ,a,(2) + .jk ~k (O)2'O')3) @'lVj~/l[3]~(ki)(o)2)~}l)(('O3)' (11)

P. Norman et aL /Chemical Physics 203 (1996) 23-42 27

which after collecting terms can be rewritten as

i sgn( j ) ( - i ( w l + w2 + w3) - 0)j) _(3) c~j (0)1,0)2,0)3)

= i (Ei3I 17131 el31 ) ,., + - - ( 0 ) ,

_ (El4] . d41 "~ ot~l)(0)l)a~l)(0)2)ot~))(0)3 ) \ .iktm - Wl Ojklm j

• il/a~l[31~(l)r . x (1), x +iVj~l[21ce~ 2 ) (w2 '0 )3 ) - "jkl "k ~'°2)ett tW3) . (12)

In order to simplify the notation we introduce

(17131 -131 _ woSJl3]) T~i~ j (0) , , , 0 )b ) = ~,~j(~t) - 0)~a;-kt _

141 . ¢,[4] _ , £,[41 ) Z~klm(0)a, rOb, rOc) = //1714] r,[41 . _ ~ ~j(klm) -- rOaO)k(tm) mb°fl(kra) wC°jm(kl) ' (13)

where indices enclosed in parenthesis are to be permuted. To obtain unique definitions the response functions are defined to be symmetric under the interchange of integration variables, wl, w2 .... For this purpose we symmetrize (12). Using that a~2)(Wl,W2) is symmetric in 0)1 and 0)2 we can write a symmetric expression

for O'.13)(O)1,602,O)3) as

6i sgn( j ) ( - i ( w l + w2 + o93) (Oj) (3)- - a j (0)], w2, 0)3)

= Zlljkl,',','.'13] (0) 1,0)2 -+- 0)3) ~1) (0)1) ~'~2) (~2, 0)3) -+" 2iT)3] (0)1,0)2, 0)3) (0)2, 0)1 + 023) 0¢~1) (022) ot~2) (0)1 , o93)

+ 2iT)31(ro3,0)l + 0)2)o~1)(0)3)o,~2)(0)1 0)2) __ ,1".[4] ̂ (l)r . )~1)(0)2)Ot(1)(¢..O3 ) , ljklmtXk ~tOl

i~312]_ (2), 0)2)) + 2 (~'~'121oz~2)(0)2, 0)3) + Vj'~zI21ae~ 2) (0)1,0)3) Jr" "jk 61Ok 1"0)1'

+ i/V"I31 + ) a~') (0)2)o~I') ((03) + i V ml'3, (Vj~/213, --}- Vj~ 213]) ~1) (0)1)01'~1) (0)3) \ .ikl .irk

+ i (W[31 + ~[31) ~1)(0)1)a}1)(0)2). (14)

The first and second order response are taken from Ref. [42] to be

- i sgn(j ) Vj °'dll a!l)(0)|) = ( 1 5 )

.i (0)1 - s g n ( j ) 0 ) j ) '

i 1(0)| ,0)2) o'~l)(0)l)ot}l)(0)2)-+'Fjk Ol k [0)2)" l -Vjk Ol k ~tOl) _(2) (16) u j (0)1,0)2) = sgn( j ) 2(0)! + 0)2 - sgn(j) toj)

2.3. Time dependent expectation value

We are now ready to identify the third order response function by comparing the time dependent expectation value for an operator A in the single determinant approximation with the general expression defining the response functions. With the use of the Baker-Campbell-Hausdorf expansion we have

oo tl (6(t) IAI0(t)} = (01 e -i'¢(t) A eiX(t) [0) = E i"A},ntl...t,, I I ct& ( t ) , (17)

n----O g,=l

28

where

P Norman et al./Chemical Physics 203 (1996) 23-42

t) - (-1)"(olHOt,,AIO).-- (18) Alnl

lll2...I, II] /~=1

The terms of (17) that will contribute to third order in the perturbation are

(~)( t)la]O( t) ) = (01al0) + ia}llce6 ( t) _ a}~ce6 ( t )a6 ( t ) : A t 3 ] o~ 'nllld3 l, ( t ) a 6 ( t) 0% ( t ) . (19)

Inserting (6) in (19) and keeping terms of at the most third order in the perturbation gives

• .[1] (2)... Ai2]ot(1)rDa(i ) t t ~ (O(t) lAlO(t) ) =(OlAlO)+iA} , l ia~l l ) ( t )+l , ' t t l at, t t ) - lit2 Ii ~'! 12 ~ , ' 1

• --Ill ( 3 ) . . . A[21 (ol}12)(t)ot~zl l( t)+ce}l)( /)~}2)(t)) _ia[3] ot ( l ) t t ,o t ( l ) t t , o t ( l ) t t , "Ji-I't~I/i OQI [ [ ) 1112 t d z h tl ~" ! 12 ~" ] 13 ~" I " (20)

Inserting the Fourier transform expressions (I0) of the time dependent a parameters gives

f All]if(I)- ~ e-i~ol ( O ( t ) l A l O ( t ) ) = ( O I A l O ) + i t, i to91) do91 --o,o

f a[21~(1)" )a}211 (0./2) e -i(wl+°*2) do91 d0./2 ÷ ia}llcel2)(wl,m2)--,.lll2~ll I. Wl --OO --OC)

f AI21 ~I?)(O)l , 0./2)O¢}21)(0./3)+ 0¢I11)(O91)~12)(0./2, 0)3) + iAI, 1 ]or}| 3) (0./1,0)2, 0-/3) - Ill2

-oo -oo -oo

• .131 (l)(0./1)a~2')(w2)ot}3')(0./3)] e -i(°~'+°J2+~°3) do91 do92d0./3. (21) - - l /~l l l2130l l )

The definition of the three first response functions is [42]

7 77 1 (O(t)lAlO(t)) = (0[A[0) + ((A; V°")) e -i~°' d~l ÷ ~ ((A;V~',V°~Z))e-i( '°'+~2)dogldo92

--OO --0(3 --OO

' / + ~ ((A; V '°' , V ~°z, V~°3)) e -i(~°l+'°2+'°~) do91 d~o2 doJ3. (22)

By comparing (21) and (22), and demanding a symmetric expression we identify the third order response function as

((A; V '°' , V ''z , V'°3)) = ((A; B, C, D) ),,,,,,,,2,,,, 3

= P(1 ,2 ,3 ) \/{iAlllcr(3)'t, t, I.°)1,0./2,°)3)- AI2]lll2 (ff}12)(('/')1' ¢"02)O/}2 l) (('03) "~- O/}ll) (('Ol)O¢} 2)(0)2' 0)3))

- ) 31 , (o.,2),,},> -- l / - i l l l2l~OQl

r-~[ll (3). --01/~.j otj ~o91,o92, w3)

~(1) (2) ^(1) ) __9A121 (~(1)(O91)~2)(O92, O93) ÷ j (t_O2)Ot k ((.O1,¢..O3)_t_~j (O93)O¢~2)(O91 0.)2 ) "''(./k) ~ .i

-iA'3'(.i,~l) (crJl) (o9') oral) (w2) °:~ 1) (~3 ) ) . (23)

P Norman et a l . /Chemica l Physics 203 (1996) 23 -42 29

2.4. Elementary basis

In order to avoid the complete diagonalization of E I21 and S 121 that would be necessary for carrying out the sum over state expressions for the response function value we return to the elementary operator basis. The general operators Oj then turns into the normal pair of creation and annihilation operators in the orbital basis used for a particular calculation. Moving between the two representations involves the transformation matrix X. As long as there is no danger of confusion we will denote matrices the same in the different representations although their elements change, e.g. in the elementary basis S I21 looks like

--A* -Y~* '

E = (Ot[qi, q~]lO ), A = (0[[qi, qjll0). (24)

For a closed-shell system in orthogonal orbital basis A = 0 so that S |21 becomes diagonal. Other results that we will need to express the third order response function in elementary basis are

iAI I la, t 1) e I IINB(tol), .i j (wl) = - A j

2Alll~x(j2)(wl,w2)=e I11N~C(wl,W2),Aj (25)

Combining (15), (16), (14), (23), and (25) yields an expression for the third order response function in the elementary basis.

[4] ((A; B,C,D)),o,,o2~3 = NA(w, + w2 + w3)T)ktm(o~,,to2, w3)N~(wi)N~t (w2)ND(o~3)

NA(o ,

[o121 ~,rCD. 121 BD 121 BC ] 131 D C)(kt)Nk Dj(kt)N~[31 B - + (o~1) ND (to3) + (OJl) N~Tt (w2)]

a l2 l CD D BC + " ik)[U~(tOl)U k (w2, w3) + NC(to2)N~D(wl, to3) + N~ (to3)N k (tOl,t02)]

13] B C - a(./kt)Ni (Wl) N k (w2)N/D (w3), (26)

NX(wa) = (E l21 - waSl21)~l / ~ 1 ] X E { A , B , C , D } , (27)

N XY t ~o , (E { 2 ] j , a wb) = -- (Wa+t°b)S[21)jklxy;ll(°oa, tOb),

XyJ ,I (tOo ' COb) -- [31 X r y[21 ~,X, - V[21 ~rY Tjkl (Wa'Wb)Nk (r'°a)Nl (Wb) - jk lvk tWa) --"jk "k (wb)

XY E {BC, BD, CD}. (28)

The evaluation of the response function can thus be separated into two steps; first solving a set of seven linear equations (27) and (28) to obtain the corresponding seven response vectors to be followed by the matrix multiplication (26) for the response function value. As described before the third order response function can be used to express a number of molecular properties. The theory is quite general and no specific assumptions have so far been made about the perturbing operators. However, the case with dipole operators shows particular interest since it connects to the field of non-linear laser spectroscopy. The response function itself becomes

30 P Norman et al./Chemical Physics 203 (1996) 23--42

minus the value for the second hyperpolarizbility, but also several other properties are within reach through a residue analysis.

2.5. Residue analysis

We will focus on a single and a double residue of the cubic response function with dipole operators

lim (cos caf)((It~;Itb, ¢ d - It ,I t ) ) ~ , , ~ 2 , ~ s , ( 2 9 ) ¢03 "-'*tO/

lim (ca2+0)e) lim (w3 _caf)((ita;itb,it ,It ))~,,~2,~3 , (30) ~0 2 ~ -- r~ e ~¢03 ----+ ¢O f

from which the three-photon matrix element and the two-photon transition amplitude between excited states can be identified [42]. The eigenvectors that simultaneously diagonalizes E [21 and S t21 are paired in nature; if the row-vector e T = (1X, 2X) is an eigenvector of this diagonalization then so is also e T_j. = (2X*, iX*) that we label with a negative index. Using the eigenvectors for resolving the matrix inverse we get

eke~ (31) ( E I 2 I - wSI21)-' = - ~ " w s g n ( k ) - w t '

k

where k now runs over both the positive and negative indices and cat are the eigenvalues of the matrix (E 12! - 0)812]). With the use of this resolution we obtain the residues

lim (caz + c a e ) N C ( c a 2 ) = --e-eeT-e Cll], (32) o~2~--oJ e

lim (ca3 - 0)f)N°(ca3) = efeff DIll (33) ¢O3 ---* O/./

Here we identify - e ~ C t~l with (01Cle) and effl)t~J with (flDlO) in agreement with the exact limit, so that with the notation e_~ = Ne( -ca : ) and ef = NF(caf) we get for the three-photon matrix element in the SCF approximation

r.CBA . N : ( OAf [4] t0~.f ,cal , ca2) = -- 0)1 -ca2)Tjklm(cal,ca2, ca3)N~(-cal)N~l (--ca2)NF(caf)

- Na(0)f-wl-~2) [T)Sl(-cal,caf-ca2)Nl](-0)l)NCtF(-w2,0)f)

131 _ ] + T'/~, ( - c a : , 0): ca, ) NC ( -ca: ) u : F ( -ca~ , ca : ) + TJ3) ( ,,, : , -ca , - ca2)N~fca:)N:cf-ca,,-0),,2) 131 I31 B N((0) f ) ] -- N : ( c a f - - O31 - - 092) [nj(kl)N~k ( - c a 2 ) g : ( c a f ) + C3(kl)g~ ( - - c a l )

[B[21 ,~F , , :[21 ~rBF: . ] "FN: (Ca f - -Ca l - - ca2) I jk lVk I,--ca2,caf)"q-~jk "'k t - - W l , c a / )

+ .a121. (./k) [N}B ( _ca, ) NkCY ( --0)2, 0)f) + NC(-caz)N~F(-0), , roy) + N:(caf)N~C( -ca', -0)2)]

_ a13) ~IBr_Cal)NC(_Ca2)NF(Ca/), • =(jkl)"./ ,,

uX(ca.) = (E - 0)oSt2 ).7 ' x c (A ,8 ,C} ,

(34)

(35)

(El2) _ toyS 121 )J* Nff (0)f) = 0, (36)

P Norman et al . /Chemical Physics 203 (1996) 23-42 31

(E[21 _ (COl + CO2)SI2])j k NBC(COl,CO2) = ~[31, tilmttol,to2)N~(wl)N~m(co2) r ~I2} ~'Br - - k , , , , , c o , ) - ( C O 2 ) ,

(37)

-~131. NX(-COa)NFm(CO/) - X~}INF(COf), (38) ,~121~ N X F ( _ C O a , C O f ) = l~ lm~_COa,COf) (E TM - (COl- CO,,)o ).jk 1

x {B, C}.

Note that the sign of the eigenvector and therefore also the sign of the three-photon matrix element is arbitrary. The expression for the two-photon amplitude between excited states looks like

r ee f (COlBA) ---- Na(COf _ toe _ c o l ) T j k l m ( _ c o l , [ 4 1 _ C O e , C O f ) N ~ ( _ c o l ) g E ( _ c o e ) N F ( c o f )

-- N.a(COf--COe- COl)[T)3](--COI,COf--COe)NB(--COl)NEF(--COe,COf)

Nn~( -COe)] + T ) k~I I ( -- CO e , COy" -- COl ) N E ( -- W e ) N B F ( -- 091, CO f ) + T) 31 ((,Of, --COl -- co e ) N ff ( co f ) 1 - C O l ,

~BI21 ~rEF z CO , o 31 N~(_COe)NF(COf) + N~(CO/-- COe - - COl, jk irk t - -COe,COf) - N (coS-CO - lJ ' - ' . j (kt>

F BE a121 [ NB ( _CO, ) NffF ( _COe, CO f ) + N~ ( _COe ) N~F ( _COl , CO y ) + N; ( CO f ) N~ (-CO,,--O.,)e) ] + ,,(.jk)

,131 ~,Br CO.)N~(_CO~)Nr(CO/), (39) - - /~(jkl)JVj k - - I

NX(COa) = (E TM - COaSI2l)~ ' X~" , X E { A , B } , (40)

(E 121 - COysIZl)j k N[(COy) = 0, (41)

(EI21 (CO,, COl ) S[21"~ NBY ,-,-[31. -- -- l kl m ~--COl • , j~ : . , (-CO,,COy) = ,COy)Nln(-COi)Nrm(COy) - B~21]N)/(COy), (42)

Y e {E, F},

( El21 - (COl - COe) S{2]]/.ik NE g(-coe'cof) = T[3l~-coe'COf ) N E ( - c o l k l m ~ )N~,(COf).F (43)

When the two excited states are equal the two-photon absorption amplitude (39) will become the difference in polarizability between the reference state and the excited state.

3. Implementa t ion

The expressions derived have been implemented in the DALTON program package for molecular properties [ 17]. Essentially two new features appear for the coding of the cubic reponse functions; the higher order matrices (third order property and fourth order Hessian and overlap matrices) and the coupled linear set of equations. The latter is readily taken care of by first solving the equations for the vectors NX(w) to obtain the XY Ill and thereby also the NXr(tol,co2) vectors. This therefore does not in principle encounter new programming. The fourth order matrices, however, demand further development.

3.1. One-index transformations

We need a way of multiplying E [41 and S [41 on three vectors, and A TM on two vectors without explicitly forming these large matrices. In analogy with previous implementation the matrix multiplications can be written in terms of doubly and triply one-index transformed operators

AI311 1,, 2~, 1 ((Ol[q~,a(IK, 2 K)]10)'~ (44) .ikl ,vk , v t = - - ~ \ ( O I [ q j , A ( I K 2 K ) ] I O ) I ,

32 P Norman et al./Chemical Physics 203 (1996) 23-42

El41 IA, 2~r3xr 1 ((OI[qj , H(Ix ,2K,3K)]IO)~ jkln, lVk lVl 1¥m ----" - - g (01 [qJ, H ( I K , 2 K, 3 ~ ) ] 1 0 ) / ' (45)

SI41 ~ , 2~,3 ~, 1 ( (01[q~, [(3Kkq~ -t-3K~qk), [ (2,,~q] + 2,,lq,) ' [(iKmqL + lx~q,, ,)] l l l lO) '~ (46) iktm ,*k ,,t '*m = g (0l[q~, [(3Kkq~ -k-3K~qk), [(2K/q/ -t-2KIql), [(IKmqtm n t- 'K'mqm)]]]]10)J '

where the transformed operators are given by

I t A(~K, 2 K) = [(]Kkq~ + KkqD, [(2Ktq~ + 2K~ql),AI], (47)

I t H ( IK,2 K, 3 K) = [( lKkq ~ -t- Kkqk) , [ (2K/q] q- 2K~ql), [(3Kmqtm -t" 3K~mqm),/4111 (48)

and

i N = ( iK ) iK , , i = 1 , 2 , 3 .

3.2. Double direct construction o f Fock matrices

(49)

Since Eq. (45) is of the form of a gradient vector containing a triply one-index transformed Hamiltonian it can be expressed in terms a Fock matrix with triply one-index transformed integrals. Forming this Hamiltonian in (45) in a non-direct fashion involves tedious two-electron integral transformations. Also the need to store integrals would restrict our applications to small systems in comparison to what one deals with in direct SCF methods. We therefore wish to construct the necessary Fock matrices over atomic integrals as well as using a direct linear transformation for the triply one-index transformed Fock matrix. Denoting inactive MOs with j, general MOs with p, q, r, s, and AOs with Greek letters we have

Frs = hrs + £rsjj, (50)

" Er~,:/./ = 2(rs l j j ) -- (r j [ j s ) . (51)

The one index transformed one- and two-electron integrals with I K take the form

Fr," = hr., + Zrsj./, (52)

-hrs = I K r p h p s - I Kpshrp, (53)

--~rsl, j = I Krq£qspj -- l Kqs£rqpj "~- I Kpqff~rsqj - I Kqj£rspq. (54)

Using (52), (53), and (54) repeatedly we obtain after expanding £ in atomic orbitals

~(IK,2K,3t¢) = F 123 -t- [3to, F12] + [2K, FI3 + [3K, FI]] n t- [1K, F23 + [3K, F2] -t- [2K, F3 a t- [3K, F ] ] ] ,

(55)

where the intermediate Fock matrices and the density matrices needed for their construction are given by

F123 r'~ 123p rs = uctfl "L'rsalff'

DI23 = 1 2 3 1 2 3 aft K./p Kpq KqtCatCf l j - Kjp Kpq Kt jCaqCf l t - l Kjp2Kqj3KptCatCflq

I 2 3 "~- K.jp Kqj KtqCapCflt -- 1Kpj2Kjq3KqtCatCt~p -t- 1Kpj2Kjq3KtpCaqC~t

I 2 3 1 2 3 -t- Kp. i Kqp Kj tCatCf lq- Kpj Kqp KtqCajCt3t,

Frm, m, = DctB£.rsal~,


nln 111 11 D a , = K.ip KpqCetqC~Sj -- m Kjpn KqjCetpC~q -- m Kpjn KjqCotqC[Jp "~ m Kpjn KqpCetjCl~q,

O a,BErsaO,

D~% ="'Kjt, c,~pcflj - mKnjcajCflp.

If we permute (55) in 1K, 2~¢, and 3K we obtain an expression for ~,[4] times three response vectors ~j(ktm)

(,i) /(k/m) Nk2NI3Nm = - 2 - F i s ' (56)

where

F = P(I K, 2K,3K)F(IK, 2K,3t¢) = [IK, [2a¢, [3K, F] + 3F 3 ] -q-[3K, [2tc, F] q-3F 2] + 3F <z3)]

+ [2K, [IK, [3K,F] + 3F 3 ] + [3t¢, [IK, F] + 3F 1 ] + 3F ~13)]

-+- [3t¢, [2K, [ltc, F] + 3 F 1 ] + [lK, [2tc, F] + 3 F 2] + 3 F O~)] + F (123). (57)

We note that the expressions for the intermediate Fock matrices in (57) have the same structure as the ones in direct SCE so that well-known techniques can be used for their construction.

4. Applications

We have chosen para-nitroaniline (PNA) to serve as a model system for the calculations of third order electronic response properties. The non-linear optical properties of this molecule have been widely studied already both by experiment and theory, see Refs. [30,12,28,36,10,2,54] and references therein. This attention is connected with the fact that PNA shows particular large non-linear effects as a result of its strong donor- acceptor character. The 4bl HOMO orbital (Tr) and the 5bl LUMO orbital (~'*) are localized on the NH2 and the NO2 groups, respectively, giving rise to a low-lying charge-transfer (CT) state of Al symmetry. The low energy gap and high oscillator strength for the ~- ~ 7r* transition together with the fact that the long-in-plane component of the hyperpolarizability is heavily dominating have been used as justifications for using a simple two-state model (TSM) including only the ground and the CT state [36] in the summation (Eq. (59)). Since the long-in-plane component completely dominates for the second hyperpolarizability as well, see Fig. 2, it is our intention to investigate whether the TSM can be applied also here.

The third order response function also enables an analytical way of calculating the second hypermagnetizability. We demonstrate this feature of the cubic response function for the 10 electron series; Ne, HF, H20, NH3, and CH4. In the literature we have found hypermagnetizability calculations for all molecules but NH3, see below, but this will be the first attempt to employ a purely analytical calculation; we also use a common type of basis functions for the full series. Along with the angular momentum operator comes the problem of gauge origin dependence. Various approaches have been used to deal with this and for lower order magnetic properties gauge independency has been reached through so-called field dependent London orbitals [49], molecular orbitals with individual gauges (IGLOS) [31,50,37,55], and also through localized orbitals with local origin (LORG) [ 14]. For small molecules, however, it is possible to assure gauge independence through the use of large basis sets since in the basis set limit results are by definition gauge invariant.

4.1. Basis sets and symmetry axes

Second hyperpolarizability calculations are very sensitive to the choice of basis set, especially considering polarizing and diffuse functions. We have previously explored this dependence for the thiophene molecule [ 38]


Table 1 RPA values for excitation energies (eV) for the first two excited states in each symmetry for para-nitroaniline

1 2

A j 5.02 6.95 B2 5.64 6.19 B I 4.88 6.23 A2 4.50 6.83

finding a well suited basis set of moderate size. The basis set is a 4-31G [ 16,11] with added polarizing and diffuse function on the heavy atoms with exponent 0.05. Comparing excitation energies with the extensive basis set investigation on the first hyperpolarizability performed for PNA in Ref. [30] we find our basis set to give results in agreement with their largest basis sets, especially for the Al symmetry. We obtain excitation energies 5.02 and 6.95 eV for this symmetry, see Table 1, to be compared with their best result of 5.01 and 6.89 eV. Other values differ from 0.19 up to 0.50 eV. A discussion of RPA results for the excitation spectrum of PNA in comparison with other calculations and with experiment has been given in Ref. [30]. The geometry for PNA has been assumed to be planar in the C2v molecular point group in agreement with crystallographic data [3] (slightly non-planar) and geometry optimizations [ 10]. The molecule is placed in the xz plane with the long axis (C2 axis) along the z direction and the short axis in the x direction.

In order to achieve the large basis sets required for gauge invariant results of the hypermagnetizability and still use one overall common type of basis functions we employed uncontracted basis sets with exponents from the A N t basis sets by Widmark et ai. [56,57] and Pierloot et al. [45]. The sizes of these atomic basis sets are; [ 14s9p4d3f] for C, N, O, F, Ne and [8s4p3d] for H. Using Cartesian functions the total number of orbitals for the molecules Ne, HF, H20, NH3, and CH4 are 95, 133, 171,209, and 247, respectively. All molecules have been placed with the center of mass at gauge origin and geometries are taken from Harmony et al. [ 15]. HF is placed along the z-axis; hydrogen with negative z-coordinate. H20 is placed with the C2v symmetry axes along the z-axis; oxygen with negative z-coordinate. NH3 has the z-axis as principal rotational synunetry axis and the yz-plane as symmetry plane; nitrogen with positive z-coordinate. Finally, CH4 is placed with the hydrogens in the corners of the cube.

4.2. Results

4.2.1. Electronic properties Expanding the dipole moment of a molecule perturbed by an electric field in a power series

[*i = I-ti + o~iFj + ½,BijkFjFk + ~YijklFjFkFl + . . . . (58)

the polarizability, first hyperpolarizability, and second hyperpolarizability, are defined by the expansion coefficients ot0.,/3,-j~, and yijkt, respectively. A different convention is sometimes used, including the factorial factors in the definition, making/3,jk and Yijkt differ by a factor of 2 and 6 respectively. In order to compare with other results our values are corrected accordingly. From perturbation theory we have [43]

r l r2

ar,,r2 (¢ol; w2) = P(1 ,2) Z I'£OflObfO ' f sa0 ( .Of - - O9 2

r l - - - r2 r3 tZo f tZ f gtZ so

~rl,r2,r,(OJl;O92,(.03) = P(1 ,2 ,3 ) Z (wf-~w,---3)(w'--f~--- w3)' f, g4,0


Yrl ,r2,r3,r4 ( 0)1; O92, (-03,0")4) ---- P ( 1,2, 3,4)

].ZO f/.I. fg]A'gh]..thO I.'.O f/£, fOP~Og~,tg 0 × ~ - , (59)

j,g,h4:O (O)f-~-0)l)(0)g - 0 ) 3 - ( ' 0 4 ) ( 0 ) h - 0 ) 4 ) , (O')f'~-(l)l)(OAf--OA2)(O)g "Jr-t03)

where 0)1 equals the negative sum of the other frequencies and

#rig = ( f l t x r lg ) , (60)

- - r r ~ f ,e = Iz f g - 8 f g tZ~o. (61)

Comparing with the spectral representation of the exact state response function gives the following connection to the electronic properties:

O'r, ,r2 (0)1 ; 0)2) = -- ((/tgrl ;/tLr2))w2, (62)

A~rl ,r2,r3 (0)1; (-02, O93 ) ---- __((/../rl ;/t/,r2, ].~,r3})to2,ta3, ( 6 3 )

Yrhr2,,'3,r4 ( 0)1; 0)2, 0 )3 ,0 )4 ) ---- - - ((/-/fl ; t ' tf2, / ttr3,/tff4))to2,(o3,to3 • ( 6 4 )

An orientationally averaged scalar y-value

1 (y) = - ~ (2yiijj + Yi j j i ) , i , j e { x , y , z } , (65)

t,.l

is suitable for comparison with experiment in gas phase or solution. We illustrate computations of electronic properties from cubic response functions by the frequency-dependent

second hyperpolarizabilities, the frequency-dependent excited state polarizabilities and of the three-photon transition amplitudes of the para-nitroaniline (PNA) molecule. For the second hyperpolarizability of PNA we consider five different optical processes; third-harmonic generation (THG), electric-field-induced second-harmonic generation (EFISH), degenerate four-wave mixing (DFWM), electric-field-induced Kerr effect (EFIKE) and electric-field induced optical rectification (EFIOR). The dispersion of the averaged values of 7 for these effects are shown in Fig. 1. The ordering in magnitude observed in the graph also holds for the individual components: Yijkl(--30);0) ,0) ,0)) > Yi jk l ( - -20);0,0) ,0)) > Yijkl(--0);to, to , - - to) > Yijkl(O;O, to, - t o ) ~' Yijkl(--to;O,O, to) > Yijkt (0; 0, 0, ). The individual components of y( -30); to, to, to) (THG) are shown in Fig. 2. The behavior of the components of the other processes are similar. The long in-plane component z z z z is much larger then the others, dominating the contribution to the averaged value. The in-plane components have the strongest dispersion and the x x z z and z z x x components have opposite signs. The large transition moment of the first excited state of z symmetry and the position of the first pole of the response function at one third of the transition energy explains the dominance and dispersion of Yzzzz ( - 3 0 ) ; to, to, 0)). For low frequencies we observe Kleinman symmetry, Y~xzz = Yzzx~, Yx~yy = Yyyxx and Yyyzz = Yzzyy, to hold. A comparison with a previous calculation of Karna et al. [26] using the time-dependent coupled perturbed Hartree-Fock approach, can be found in Table 2. We obtain a larger static value 8.33 x 10 -36 esu compared to 6.63 x 10 -36 esu. The dispersion does not differ much, our being larger. They use almost the same geometry as we do but a different basis set of comparable size. This explains the difference in results between the two methods which should be equivalent, and only be different in formulation.

An experimental y-value of 15 + 3 x 10 -36 esu is presented in Ref. [8]. The EFISH/THG experiment was performed in acetone solution at a wavelength of 1.91 /zm. This is to be compared with our calculated values of (y) ( -30); 0), to, 0)) = 2.7 x 10 -36 esu and (y} ( -20); 0, to, to) = 2.5 x 10 -36 esu at to = 0.65 eV. The experimental value are thus 6 times larger than the calculated ones. When comparing these values we have to consider that the measurement was performed in solution, and that RPA is applied to an uncorrelated SCF


15.0

0

A

v

10.0

5.0

o THG EFISH DFWM

A EFIKE=EFIOR

i i i

0"00.0 0.5 1.0 1.5

frequency (eV)

Fig. 1. Dispel~ion curves for averaged "g values of the para- nitroaniline molecule for the following processes: Third-harmonic generation (THG), electric-field-induced second-harmonic generation (EFISH), degenerate four-wave mixing (DFWM), electric-field-induced Kerr effect (EFIKE) and electric-field induced optical rectification (EHOR).

Table 2 RPA values for Y:zzz ( 10 - ~ esu) for para-nitroaniline

to (eV)

0.000 0.650 0.905 I. 170 1.364 1.494

y:== ( - 3to; to, to, to) (a) 8.33 10.47 13.40 20.35 33.81 59.75 (b) 6.63 8.45 10.96 17.03 29.17 54.00

),:::: ( -2to; 0, to, to) (a) 8.33 9.31 10.40 12.30 14.53 16.68 (b) 6.63 7.46 8.39 10.02 11.95 13.83

7:::: ( -to; to, to, -to) (a) 8.33 8.97 9.64 10.74 11.92 12.98 (b) 6.63 7.17 7.74 8.68 9.70 10.62

y:-= ( -to; 0, 0, to) (a) 8.33 8.64 8.94 9.39 9.83 10.18 (b) 6.63 6.89 7.15 7.53 7.90 8.20

yz:=: (0; 0, to, -to) (a) 8.33 8.64 8.94 9.39 9.83 10.18 ( b ) 6.63 6.89 7.15 7.53 7.90 8.19

(a) Our calculation. (b) From Ref. I261.

wave function. A correlated calculation of /3 for PNA by Luo et al. [36] showed that the correlation effect is significant and increases the static values by 40%. In the same article it is also found that the solvent increases the static value as well as the dispersion. In a two-state model the shift in transition energies and transition and dipole moments more than doubled the static/3-value. The corresponding changes for the polarizability a was in the order of a few % for the correlation effect and of about 20% for the solvent effect (MCSCF values) [36]. Thus, if we perform a simple extrapolation of the solvent and correlation effects on ot a n d / 3 to predict 7, the deviation of a factor of 6 for ~, as obtained from vacuum RPA calculations and measurements in solution can be understood.

P Norman et al./Chemical Physics 203 (1996) 23-42 37

l0 -~

avera 1

- - o

1.5 >.

~0.5

,.o xxxX

V~/'/

.\ \

~zzxx i i k

-3 0.5 1 L.5 frequency (eV)

300

(a)

i

o o c o o o- <~ (b )

1000 0.02 O. frequency (a.u.)

Fig. 2. Dispersion curves for the different components of 3,( -3to; to, to, to) for the para- nitroaniline molecule.

Fig. 3. Dispersion curves for the zz component of the polarizability for (a) the ground state and (b) the first excited state of AI symmetry (the charge transfer state) for the para-nitroaniline molecule.

Table 3 Accumulated sum-over-state values (a.u.) for the ten first excited states of Al symmetry compared to response values for the long-in-plane component of ct, /9 and 3' for the para-nitroaniline molecule

Excited states a z: flzzz 3,zzzz

1 39.6946 1159.0 6151 2 45.9854 1882.9 -33701 3 60.8611 1776.5 --41450 4 66,7032 1684.6 --09356 5 67,0722 1629.7 20148 6 67.1953 1518.6 22408 7 67.3605 1483.4 20639 8 67.3852 1467.9 17494 9 68.1890 1335.6 20131

10 68.2318 1338.6 22338

response 131.4854 1379.3 99252


Table 4

RPA values for the three photon transition matrix element (a.u.) for the first two excited states in each symmetry for para-nitroaniline

AI B2 Bl A2

I 2 1 2 1 2 1 2

zzz zxx

zyy XXX XZZ xyy

3'!'!' yZZ yx

xyz

7601 27018 419.3 931.4 113.2 -496 .9

148.3 -68 .4 335.8 -71 .7

10.4 -6 .32 9.54 - 112.9

25.4 -1151 1.74 - 145.4

5.39 335.8

Table 5 RPA values (a.u.) for hypermagnetizabilities of Ne

ijkl { (ri; r j , Lk, Li) ) ( (ri; r j , Qla) ) 71ij~l 71ijkt a

x x x x 0 1.91 - 1.91 - 2 . 0 4

xxyy -3 .71 4.06 -3 .13 -4 .25

~r/ i .22 2.20

a From Ref. [ 23 l.

Table 3 shows the sum-over-state and response results for the dominating component of the static a, fl and y values. The quantities entering in Eq. (59) were determined using response theory; the transition energies and moments from a linear response function and the moments between excited states from a residue of a quadratic response function. The fl value converges most rapidly. Already the first excited state gives the main part, confirming the validity of the two-state model for the first hyperpolarizability. The convergence of a is very slow whereas the y value does not to seem to converge at all. Thus few state models seem to have much less significance for the second than for the first hyperpolarizability.

In the C2v symmetry the long-in-plane dipole operator is of Al symmetry, the short-in-plane is of B2 symmetry, and the out-of-plane dipole operator is of B1 symmetry. So for the three-photon absorption there are no symmetry forbidden dipole transitions. However, we observe that the transition matrix elements are order of magnitudes lower for non-symmetric states, see Table 4. Nevertheless, the completely dominating CT transition is well-known from single-photon absorption and our results with dominating Al transitions and especially the z z z component, are in that perspective not surprising.

As discussed above the double residue of the cubic response function gives the polarizability difference between reference and excited states. In Fig. 3 we display the z z component of the polarizability for the ground state and the CT state. The static value for the CT state is about 60% larger than for the ground state and the dispersion is stronger. The latter can be understood by that the energy gap between the first (CT) and second excited state of Ai symmetry is much smaller than between the ground and the CT state, see Table 1 (See also Ref. [30] for discussion of experimental and computed excitation energies of PNA).


4.2.2. Magnet ic propert ies In analogy to the polarizability of a molecule also the magnetizability is expressed as a power series in the

perturbing electric field

1 )(i.i = X i j + ~i.i, kFk + ~rlkl,i.iFkFl + ... (66)

Here the expansion coefficients ~i.k and rlkt,i j define the molecular first and second hypermagnetizability. The latter is related to the quadratic (diamagnetic part) and cubic (paramagnetic part) response function in the following way,

rlii kt( --c.o; c.o, 0, 0 ) = - -1 ((r i ; r j , Lk , Lt) ),o,O,O - ( (ri; r j , Qkl) )oJ,o, (67)

where r is the dipole moment operator, L is the angular momentum operator and Q is the quadrupole operator (diamagnetic magnetizability operator) defined by

1 (r2•kl _ r k r l ) . (68) Qkl = -

The experimental quantity in connection with the hypermagnetizability is the Cotton-Mouton constant, related to the birefringence of light in a constant magnetic field known as the Cotton-Mouton effect. The Cotton-Mouton constant is proportional to the anisotropy of the hypermagnetizability, At/,

1 A'rl = - ~ E ( 37"]i.i'ij -- 77ii ' j j) ' i , j E { x , y , z } . (69)

i j

There are still comparatively few experimental determinations of Cotton-Mouton constants available, see recent compilation of Bishop [4]. Comparison between different experimental data and comparison with theory have been aggravated by the several different conventions in use for ~r/, see Refs. [29,34,4]. Except for the computations by Jamieson on hydrogen and helium with the fourth-order coupled Hartree-Fock method [20], calculations of these constants have been accomplished by means of finite field calculations. However, such calculations are quite elaborate for the hypermagnetizability and the precision of values might depend on the particular differentiation scheme. In the present work it is shown that cubic response theory offers an analytical approach to the calculations of hypermagnetizabilities that is generally applicable.

Neon is the most investigated species concerning hypermagnetizabilities in the ten-electron series [ 46,5,21,23 ]. Using the basis set which in size and type is common with the rest of the basis sets used for the isoelectronic series, we obtain an anisotropy value (At/) of 1.22 a.u. This can be compared with the experimental value of 1.36 +0.007 a.u. given by Cameron et al. [7]. However, very large basis sets push up the RPA value in the region between 2 and 3 a.u. (also given in Ref. [23] ), see Table 5. Furthermore, correlated calculations using numerical differentiation [46,5,21] end up at about Ar/ = 2.7. Thus the agreement with experiment of our original value must be fortuitous in light of the basis set and correlation dependence.

For the HF molecule both vibrational and correlation effects have been investigated by Cybulski et al. [9]. Vibrational corrections are minor but electron correlation changes the results by about 70%. The ordering of absolute values is in agreement with ours, see Table 6, and our value for the anisotropy of 3.40 a.u. (RHF = 1.750 a.u.) is in fair agreement with Cybulski's 6.64 a.u. (RHF = 1.733 a.u.) considering the sensitiveness of these calculation to basis sets and internuclear distances.

Rizzo et al. [48] have estimated the hypermagnetizability for H20 using gauge invariant London atomic orbitals and a numerical differentiation technique. The absolute values for all but the zzzz component as well as the anisotropy are about 5-15% lower than those of Rizzo et al., see Table 7. Our analytically calculated hypermagnetizabilities are gauge dependent which can only be made up for by a large basis set expansion. The fair agreement with the results of Rizzo et al., anyway indicates that this is a viable approach for small molecules for which the number of basis functions could easily be saturated.


Table 6 RPA values (a.u.) for hypermagnetizabilities of HF

(jkl ((ri; r j , Lk, Li) ) ( (ri; r j , Qkl) ) rlijkl rliykl a

x x x x 19.35 5.92 - 10.76 -6.96 xxyy 6.37 14.05 - 15.64 - 17.75 xxzz - 14.76 15.31 - 11.62 -14.45 zzxx - 16.68 11.55 -7.38 -6.51 zzzz 0 6.01 -6.01 -5.83 xyxy b 6.49 -4.06 2.44 5.40 xzxz 11.78 -4.79 1.85 2.44

At/ 3.40 6.64

a From Ref. 19 I. b xyxy = ½(xxxx -- x x y y ) due to the axial symmetry of the HF molecule.

Table 7 RPA values (a.u.) for hypermagnetizabilities of H20

(jkl ((ri; r j , Lk, LI ) ) ( (ri; r j , Qkl) ) rlijkl rl(ild a

x x x x -21.35 16.17 --10.83 -11.28 xxyy - 110.72 39.69 -- 12.01 - 12.70 xxzz -75.16 38.12 - 19.33 -21.60 yyxx -23.81 53.17 -47.22 -51.42 yyyy 18.69 18.40 -23.07 --23.66 yyzz -- 10.37 50.78 --48.19 --51.70 zzxx -70.99 43.97 --26.22 -29.50 zzyy - 17.77 43.05 --38.61 -41.32 zzzz -4.97 17.68 - 16.44 - 14.61 yxxy 28.14 -14.18 7.14 8.02 xzzx 46.11 - 15.24 3.72 4.25 yzzy 10.52 - 14.18 11.55 13.82

At/ 15.02 17.71

a From Ref. 148 I.

We have found nei ther theoret ical nor exper imental results for NH3 in the literature. However , the results

presented in Table 8 seem to fo l low a trend o f increasing hypermagnet izabi l i ty wi th increas ing number o f

hydrogens and decreas ing nuclear charge for the heavy atom. A qual i ta t ive explanat ion for this t rend could be

that the molecu le becomes more polar izable as one goes f rom the intact noble gas a tom to the molecu les wi th

the lower nuclear charge o f the heavy atom and a more diffuse electron cloud.

For CH4 we have found two values for A t / i n the literature. In an art icle by Bishop et al. [6] an exper imenta l

value o f 48 + 3 a.u. received f rom private communica t ion with W. Htit tner is presented. A publ i shed result for

A t / o f 140-t-80 a.u. at a = 632.8 nm has been given by Buck ingham et al. [ 1 ] . This is to be compared wi th our

calculated static value o f 32.79 a.u., see Table 9. It is hard to qual i fy the compar i son with exper imenta l data,

but we can anyway conclude that the value for CH4 falls neatly in the trend for the neon isoelect ronic series

with increasing hypermagnet izabi l i t ies for decreasing nuclear charges.

Table 8 RPA values (a.u.)

P Norman et a l . /Chemical Physics 203 (1996) 23-42

for hypermagnetizabilities of NHa

41

i jkl ((ri; r j , Lk, LI ) ) ( (ri; r j , Qld ) ) rlijld

xxxx -28.79 20.05 - 12.85 x x y y -153.98 51.10 -12.61 xxzz - 103.05 43.85 - 18.09 y y x x - 153.98 79.17 -40.68 3'3'3'3' -28.79 31.73 -24.53 yyzz - 103.05 68.36 -42.60 zzxx -97.98 140.37 - 115.87 zzyy -97.98 140.37 - 115.87 ,'zzz 72.59 46.36 -64.51 x y x y 63.16 -15.90 0.11 xzx z 48.32 -23.37 11.29 yzyz 46.83 -36.59 24.88

A T 23.97

Table 9 RPA values (a.u.) for hypermagnetizabilities of CH4

i jkl ((ri; r j , Lk, Ll) ) ( (ri; r j , Qkl) ) "qijkl

x x x x -22.64 36.46 -29.80 xxyy - 111.94 100.36 -72.37 xvxv 94.59 -36.78 13.13

At/ 32.79

5. Discussion

We have derived and implemented analytical cubic response functions in the Random Phase Approximation.

Our motivation to do so was based, on one hand, on the number of interesting non-linear third order properties that then become within reach, and, on the other hand, on the success of response theory for linear and quadratic

response properties. A third reason is associated with the RPA method itself, namely that it can be implemented

in such a way that the computational effort scales with direct SCE This means that the same extended species can be applied for cubic RPA as for direct SCF, and, having the same bottlenecks, that future efforts improving the direct SCF techniques (more efficient integral handling, density screening etc.) are transferable to cubic

RPA. The formal merits of RPA for linear properties are well-known, and so also the limitations from being a

low-order correlation method. It remains to evaluate these merits/l imitations for the cubic response case, and to find out to what extent a multi-configuration implementation of cubic response functions is called for.

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non-linear electric and magnetic properties obtained from cubic response functions in the random...

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