A multiconfigurational time-dependent hartree-fock approach

Volume 65. number 1

A XIULTICONFIGURATIONAL

CHEMICAL

PHYSICS LETTERS

TIME-DEPENDENT

HARTREE-FOCK

1

August i979

APPROACH

Danny L. YEAGER DepartJJJeJrt of Chemistr). Texas A & AI Chiwrsity. College StatioJJ.Tesns 77,523. USA

and Pot.11JORGENSEN DepartJJJeJJt of CfJeJnisiry.AadJus 8JJiversit>.SO00 AarlJJrsC. DeJJJJlark

Received 30 Apri: 1979; in titu1 form I3 June 1979

An cltension of the time-dependent

Hartree-Focii

approximation

as reference state has been devefoped- Preliminary multicontigxation

to employ a multimnti~ur~ttin Ihrlree-Fock state time-dependent Hartree-Fock calculations wirh the

round state of the Be atom using the configurations Is’ and 7~’ she\\a ms_simumdeviation from ihe experiments1 excication enegies of 0.31 eV for the lo\iest 16 ewitations. v.ith an aterag? deviation of 0.16 eV_

1. Introduction During the past decade direct calculations of excitation energies and second order response properties have proven very useful and various formuhtions have been given to describe direct calculational procedures, e-g_ the equation of motion approach [I ] and the Green’s function or polarization propagator approach [r?] _The simplest direct approach is the random phase (RPA) or the time-dependent HartreeFock (TDHF) approximation where a single configuration Hartree-Fock state is used as reference state_ The TDHF approximation has been improved successfully using a perturbative scheme where terms through second and through third order in the electronic repulsion have been included in the calculation [3] _ An alternative way of improving the TDHF approsimation may be to use a multiconfiguration HartreeFock state as reference state- This has been suggested by several authors [a] and in a recent publication Simons et al_ [5] presented a formulation of a multiconfigurational time-dependent Hartree-Fock scheme. In this publication we describe an approach for estending the TDHF approximation to employ a multiconfiguration Hartree-Fock state as reference state and we discuss the advantages of the present scheme

relative to the one suggested by Simons et al. Preliminary calculations have been performed wirh the ground state of the Be atom using the 1~~2s’ and the ls22p2 configurations, and satisfactory agreement with experiment has been obtained. In the next section we present t!le derivation of the multiconfiguration time-dependent Hartree-Fock (WCTDHF) approuimation, while the last section contains the numerical result and some concluding remarks.

2_ The multiconfigurational Fock approximation

time-dependent

Hartree-

The equation of motion f-or the spectral polarization propagator may_ within the superoperator formalism [2], be written as G(E)=(rl(~~-$)-ljr>.

(1)

where r is the position operator and land fi denote the superoperator identity and hamtltonian respectively_ The superoperator hamiltonian and identity are defined within an operator space X as fix=

iX=X

yf,X],

(3 (3

procedure and the projection manifold thus becomes

and the superoperator binary product is given as c-~~t-u,)=.

(3

where [OO,refea :a the wetectron reference state_ The supecopemtor resolvent may be approximated via inner projection techniques_ Thus eq. (1) becomes G(E) = ~~rffr)(lifm=-

f&k)-’

(kir)

*

(9

where h is a projwtion manifold_ The equation of motion in eq_ (5) may be approximated in two ways: (a) by choosing an approximate reference state IO), and (b) by using a !imited set of excitation operators to span the projection ma&old_ In this communication we approsimatc the reference state 10, with a multicontiguratiunti Harrree-Fock st3te io> = c

13s’c~* _

(61

where C,n denotes P!:e coefficient i+t,, = &

for the states /i&l

+ac>

(7)

and iI rfy =F refer to an tirdered product of creation operators. The muftifnn~gur~t~onttl reference state [61 is determined through optimtring the set of coerSicients fC,& and the orbiktls, i.e. the creation operators (*f 1. %he coefficient optitniution determines d set of strttes On>] IlZ) = c

iq1 C&.,

ir={Q*.Rf,Q.R)_

optin~ir3tion

invokes

r

the escita-

(9)

_

rhr wbitaf optimization invoks tion Q’ and deex&t.ktion Q operz:ors

rvhere3s

0’

= {~zp).

ttte excita-

r>s.

(10)

Elimination cX redundant operators fo: specific reference stzues has been discussed in detail in mf_ [6]_ We chose tfte projection m3nifold k to be the set of-eseit3tion

occupation

=+O.

03

To obtain a hermitean propagator matrix for E= 0 and to reproduce the frequency independent polarizability given in ret [6] the operation of the superoperator hamiltonian has to be defined such that the superoperator hamiltonian always fit operates on the set of operators {Qt ,Q) when coupling eIements between the contiguratbn and the orbital space are evaluated. The propagator ntatrix in eq. (5) thus becomes Wl=

f(N?f),(~fRf),

@IQ), (NW

where the matrices A, B, and S are defined s

R=

tion R$ 3nd dcexcit.ttion R operators

iirt)(Oll

from any orbit31 with nonzero

A stnightforwwd appliccltion of this msnifold leads to a propagator mat& which is nonhermitean for tr’ = 0 since

number.

(8)

nntrix

R* =

011

Included among our Q* operators are all possibte single excitations

and the coeiiicients in eqs. (6: and (S) form 3 unitary The coefficient

I August 1979

CiIEMICAL PHYSICS LE-I-ERS

Voiutuc65. number i

3nd deexcit3tion

rhe muhicontiguntion

operators

H3rtree-Fock

&voived

optimizatkm

in

(OIK?. [H.QlllO)

torrrQ,HJ.Rlfo)

(OIIRR, EfLQllfO>

(OflR fftRf1fO~

I

s = (hlrl)

_

I’

(t5) (16)

Eq_ (13) defines the muIticonfiSurational time-dependent Hartrce-Fock (MCXDHF) procedure_ The excitation energies and transition moments can thus be obtained by solving a nonhermitean eigenvslue probIem Gmilar to the one given in the standard time-dependeut thrtree-Fock (TDHF) appro_xim~tion_

Volume 65. number 1 3. Results and discussion

Table 2 Beryllium excitation energies 0) (in eV)

&citation energies have been calculated for the Be atom within the MCTDHF approximation_ The muiticonfiguration Hartree-Fock reference state _included the l&s2 and the ls22p2 configurations and the MCTDHF calculation considered all states tifvarious symmetry that resulted from the limited configuration interaction (CI) caIc&tion between the set of states {IGg>j and all excitation operators aia that are non redundant according to the procedure described in ref_ [6] _ In table 2 the excitation energies calculated in the MCTDHF. in the TDHF, and in the second order polarization propagator [7] approximation (SOPPA) are gRen_ All calculations were performed using the basis given in table I _The TDHF calculation shows a triplet instability that is removed in both the SOPPA and the MCTDHF approximation as the reference state is correlated in both these calculations_ The triplet spectrum obtained in SOPPA and in hfCTDHF seems to be of equal quality and has an average deviation from experiment of0.32 eV and O-17 eV respectively for the excitation energies given in table 2_ The singiet spectrum in SOPPA and in MCTDHF is of the same quality and shows a substantial improvement relative to the spectrum in the TDHF npproximation. The average deviation from the experimenta singlet spectrum is 0.78 eV in TDHF. 0.19 eV in hlCTDHF_and 0.39 eV in SOPPA. The MCTDHF results are all slightly high compared with esperiment wvhiIethe SOPPA results are low. The highly correlated excitations 3_p2 t D; Ip’

‘P

Table 1 The basis set for Be

#

Tn.=

E\poncnt

Type

L‘xponent _~

Is

k.36861 3.47116 271830 1.48725

zp

I.5 1.0 0.8

2s

3P

0.94067 3s 4s

0.77810 0.6 0.4 0.4 O-3

1 All!2ust1979

CHEBIIC~L PIIYSICS LEi-l-ERS

-tP

3d 4d .____

0.6

c.\p.b)

?p 3po Ip ‘PO 3s 3s 3s ‘S 2pz ‘D 3p 3Po Ip’l3P 3p IPo 3d 3D 3d ‘D

1.73 5.28 6.46 6.78 6.99 7.28

-Is 3s

-k’s 4p3po Ap ‘PO 5s 3s 5S’S

TDIiF e) JICTDIIF ~ 4.80 6.11 -

7.71 x-t3 6.50 6.95 7-17 7.35

7.10

-

7.66

7-46 7.69 7.99 8.00 8.09 8 17 8.33 8~55 8.59

6.72 6.94 7.26 7.58 7.85

7.60 7.90

SOPPA d) 2.32 5.19 6.12 6.53 5.98 7-18

7.40

8.17

7.44

8.15 8.26 8-48 8.50 S-77 8.85

7.6 1 7.73 7.96 8.06 823 8.34

4 The TDHF, MCRPA .md SOPPA c&ulntions \\rerecarried out using the basisgiren in table 1.

W Ref. [S]. The time-dependent Hsrtree-rock (TDHF) .tppro%imation is triplet unstable_ d) Tbe serwnd order polarization propaytor calculations (SOPPA) were carried out ds described in ret: [ 7 I_ 4

and 2pz I.5 cannot be described wthin the SOPPA approximation_ In the MCTDHF approximation these e_xcitations are introduced through the set of escitation operators R?.R and through the additional escitation operarors 07, Q rhat appear 3s 3 result of using a muIticonfiguration reference state. The 2~’ ‘P escitation has an amplitude of 0.95 for the 12~~ 3PX1s7_ ‘Sl excit.nion operator and arnplirudes of about 0-I ior the excitation operators ax anda? G, _ a,pxpa$v_

The MCTDHF 2p2 3Po ex?ttatlon IS oft tram c$ea ment by 0.26 eV. The 2p2 t D transition has an amplirude of O-73 for the /2p2 1D)(2s2

ISI e.xcitation operaror

and arnpli-

-

tudes of 0.55 and 0.2s for the azdnls and aidnzs ewzitarion operators respectively. The 3d ID transition 113s amplitudes of 0.40 for rhe 12~’ 1D)Qs’ 1St excirarion operator 2nd 0.81 and O-41 for the Q&z~~ and the

-

~~~~~~

I

0.5

O-35 l-3 0.6 1.0 0.5 _ - _ _-___

Symmetry

excitation

operators

respectively-

The 2~’

‘D

and rhe Sd ‘D escirsrions are thus both highly correlated and differ from the experimental result by 0.1 S and 0.33 eV respectively in the MCTDHF caiculation. 79

CHEMICAL

VoIume 65, number 1

PHYSICS LETI-ERS

The inclusion of all the states of various symmetry that @rutted from the fimited Cl calcuhtion is thus obviously very important in order to get a reasonab!e description of the highIy correkted states_ Ahhough the agreement betweeo the e.uperimental and the MCXDHF result is exceknt, a huger and optitied basis set may improve the agreement with experiment .md is needed to describe the higher Iying excited stirtes that have not shown up in the present calculntion _ Simonset al_ have suggested the use as projection manifold of the set {1r#r4___) obtained from the contiguration with maximum coefficient. In Lxse of several runfigurations with brge coeftkients, a partitioning of the Projection manifold rebtive to a sinpie configtiration becomes unntisfactory,c_g_ Parts of /r6 may be ~xpraliy important as the/r, set. Our scheme appears to give good excitation en&es by including onfy oprntors from our single excitation manifofd (which difks from :he h-, nnnifi~fd ot~Simons)_ Further, the dhision of the pro$ction mrmifold suggested by Simons

et d.

which

negkcrs

the Iimircd

Ci stares

makes

it vw\; difficult

to describe highiy correbted excita(ions such as the 3pz ;I’. Sd * D, and 3p3 t D excitation in Re_ Our preliminary cxkuhrions seem to indicate that the resuhs obtained irr the MCTDHF procedure are cf J little better quality than the ones obtained in the SOPPA approximation_ The RlCTDHF approximation is. holxever. in contrast to the SOPPA, Lxrpable ofdescribing highly carreLned elrcitations. More calcuiations are planned and have, of course, to be carried out be-

SO

1 August 1979

fore definite conchrsions can be made about the quality of the resuIts in MCTDHF caIcuiations_

Achnowkdgement We are grateful to Dr_ E. Dalgaard for v&able discussions and helpful comments concerning this paper_

References

[ t 1 C-W. McCurdy, T-N_ Rescigmo. D-L_ Yeager and V_ YcKoy. in: Modern theoretical chemistry. Vol. 3. ed. II-F- Schaefer II! (Pienum Press, New York, 1977). and references therein. 121 P. Jdrgensen, Ann_ Rev. Phys_ Chem_ 26 (1975) 359; I, Oddershede. Advan- Quantum Chem_ I I (1978) 275, and references therein131 J. Oddershede and P. JBrgensen, J. Chem. Phys. 66 (1977) 1541; D-L Yeager and K_ Freed, Chem. Phys_ 32 (1977) 415. I41 D_f_ Rowe. Nude collective motionr models and theory (b!ethuen, London, 1970); D-L- Yqer, Ph.D. Thesis. C;llifornQ Institute of Technology ( 1975). 151 A_ Banerjee. J. Kenny III and J_ Simons. Intern. J_ Quantum Chem., to be published. [61 E_ Dzdgxd and P_ Jdgensen. J_ Cbe-m Phyr 69 t1978) 3833_ [7j J. Oddershede. P_ Jdrgensen and N.ILF_ Beebe. Intern. J. Quantum Chem. I3 (1977) 65% [I31 C-E_ Moore. Atomic Enet-gy LereIs, NatI. Bur_ Std. Circuhr 4 (US Gort_ Printing Office, Washington, 1949) P_ 967_