Simulated transition state for the hartree-fock and hartree-fock-slater methods

Volume

CHEMICAL

109, number 4

SlMULATED

TRANSITION

FOR THE HARTREE-FOCK

PHYSICS

24 August 19S4

LETTERS

STATE AND HARTREE-FOCK-SLATER

METHODS

*, E. ORTIZ * I;bcultad de QuiJJJica. UJJiversidad NaciorJal AutbJroma de hfkico, Cd. UJJisersitaria. hl&ico, D_F.04-i 10,:%xirO DeparralJlCJltO de QuiJJlica. DiPixi6JJde CieJrcias Bdsicas e IJJgeJlieria, Uriversidod AuldJJonJa Zlerropolirana-I=tapalapa. C.P. 55.534, M&ico. D.F. 09340. Blerico

JosC L. GAZQUEZ

and J. ROBLES ** I~aculrad de Quirnica. Uziversidad Nacimal Aut&orJra de bIp.Uico. Cd. lJJJirersi?aJia.Mkico. Received

D. I;: 04510.

Alcvico

2 M-?rch 1984; in final form 25 June 1984

A simple model based on screening effects is used to calculare ionizlltion and excitation energies of atoms using only ground-state information. The results obtained show better agreement with the espcrimental values than rhosc obrained through the USC:of Koopmnns’ npprosimation. spechlly for the inner shells.

1. Introduction The calculation of orbital ionization energies within the framework of the Hartree-Fock-Slater (HFS) method [I], is usually performed through the use of the transition-state (TS) concept. This approach makes use of a well known property of the Lagrange multipliers that appear in I-IFS theory [2], namely (a~~IFS/a~zi)r~j*,~i = ‘i 3

(1)

where E1,l-S is the total HFS energy of the system, and 4 and Jli are the Lagrange multiplier and occupation number of the ith orbital. This relation has been applied to calculate energy differences, since from the mean-value theorem, it is know that &!Y/Atli is equal to af$%zi evaluated at some point that lies in the interval ~zi- For simplicity, in the transition-state method the point selected is the * Permanent

address: Deparfamento de Quimica, Divisi6n de Ciencias Bisicas e Ingenieria, Unlversidad Aut6noma hletropolitana-lztapalapa. C.P. 55-534, M,idxico, D.F. 09340, Mcs ice. ** Present address: Department of Chemistry, University of Sort11 Caroline, Chapel-Hill, North Carolina, 27514, USA

394

one that corresponds to the electronic configuration that lies half way between the electron configuration of the initial and final states. Thus, for example, for the ionization energy of an electron in the ith orbital of a neutral atom, the TS is obtained by removing half an electron from the ith orbital, and the ionization energy is given, in view of eq. (l), by the eigenvalue of the ith orbital obtained from a HFS self-consistent calculation with such electron configuration. The good agreement with experimental results [3-91 suggests that the TS procedure takes care of relaxation and even correlation effects IlO]. Within the framework of Hartree-Fock theory, Goscinski et al. [ 1 I] have proposed a transition operator method which is similar in spirit to the TS method. The res_ults obtained through this approach are in good agreement with the experimental values. On the other hand, Brandi et al. [ 121 have, recently, applied the TS concept in connection with the hyper-HartreeFock method (HHF), and the results thus obtained are in very good agreement with those of Goscinski et al. Although the TS concept has been extensively used with success, particularly within the HFS theory, it

CHEMICM_ PHYSICS

Volume 109. number 4

LETTERS

implies a great computing effort, since it requires a different calculation for each orbital ionization energy. This situation may be avoided through the use of Koopmans’ approximation * which only requires information of the ground-state system to estimate orbital ionization energies. However, the absence of relaxation effects produces large errors, specially for the inner shell orbitals. In the present work we develop a simple model

to an approximate screening constant of the fomi IX I would lead in turn to S(ili) -:(fvl), w I‘1

based

we can rewrite

on screening

knowledge the system.

effects

to simulate

of some properties

the TS from a

of the ground

state

S(rrT)

- S($)

bital is given by i = [Z - S(lri)]/yj!

transition state

= ei(llp)

which

The starting point assumes that the Coulomb and exchange potentials of the HFS or Hartree-Fock methods may be replaced by an average screening constant, S, whose value will be a function of the occupation jzi_ Thus, the one-electron equations take the form -

[Z - S(ili)]/r)~i

= Ei~i

(2)

and therefore ~i(iZi) = -:

[Z - S(~Zi)] ‘/,~~ ,

(3

where r+ is the principal quantum number of the ith orbital. It follows from eq. (3) that the difference between the neutral atom eigenvalue ei($) and the eigenvalue corresponding to the TS, Ei(,rf)(,1p and jr: are the occupations of the it11 orbital in the ground and transition states respectively), is given by

= : {[Z - S(rr~)]2

-

[Z - S(n;)]‘]@

At this point, we focus on Hartree-Fock potential. One haves as -Z/r when r += 0 and r + =_ The arithmetic average treme points, namely, -[Z -

.

_

(6)

eq. (4) in the form

of - (,rf

{-TV’

(9

Using this relation, and recognizing that for a hydrogenie orbital, the average value of r-1 for the ith or-

e&)

2. Simulated

- 119j _

= $(,I:

- ,$Qz J&$

the properties of the may recall that it beas -(Z - N + 1)/r when between these two exi(N - l)]/r would lead

* In Hartree-Fock. Koopmans’ approximation gives the orbital ionization energy as -erHF [ 131. in Hartree-&cockSlater, Gopinathan (141 has shown that the equivalent of Koopmans’ approximation gives the orbital ionization , where J(i) is the self-repubion~ 1. eneray 0 as -eHFS + *J(i), integral for tie ith orbital (see also ref. [U]).

- ,r;>
.

(7)

gives the TS eigenvalue

in terms

of properties

of the ground state of the atom. Now, it is important to note that if the value of S(jzi) in eq. (3) were to be fned to give the HFS or the Hartree-Fock ei then the value of (r-l+ in eq. (6) would not be equal to the value of (r-t+corresponding to the HFS or Hartree-Fock methods. Moreover, one would not be able to adopt the value of S(lZi) that leads to cq. (5) However, since our main objective is to relate the transition-state eigenvalue with the ground-state eigenvalue, we assume that eqs. (3j, (5) and (6) are approximately valid for the same screening constant_ This would be probably a poor approsimation to calculate ei and i from a screening constant S(jzi) z $(N1). But here it is only used as a tool to arrive at eq. (7) Furthermore, we also assume that Ei and Cr-‘>i in eq. (7) correspond to the HFS or Hartree-Fock values. Thus, for the orbital ionization energy Ii, (JIM - ,ry) = -i, and using eqs. (1) and (7) one obtains I-I = -e;~=(,rq)

(4)

+ :(,I:

+ $ (r-t)~=

+ l/32$

for the HFS case. On the other Hartree-Fock method [I 61, (aEHHFla’li)ujf>ri

= ei + $ J(i)

hand,

(8) in the byper-

)

(9)

where J(i) is the self-repulsion integral for the ith orbital. Therefore, combining this relation with eq. (7) one has that f$HF(rrT) -

where

(2

I

= +tjF(ir~) -

-I- fJ(i)

+ i(nT

-

n~)O_-l)j

11o)2/8Y2 I 1 ’

we have assumed

(10)

that the self-repulsion

integral, 395

CHEMICAL

Voiume 109, number 4

PHYSICS

which should be evaluated with the TS orbitals, may be approximated by the value corresponding to the ground-state orbitals. Thus, using eqs. (9) and (10) we have that

24 riu_eust 19s4

LETTERS

Is”

zsb 2p= (initiaI)

___-

AE

then the TS is defined

19

2sb’ 2pc’ ,

as ls(Q+a’)/2 2s(b+b’)/2

.._ where a, b. c represent Although,

in principle,

this relation

the hyper-Hartree-Fock

method,

is only valid for in practice

be applied also to estimate Hartree-Fock because both methods give quitesimilar one uses canonical orbitals [ 171 +*_

3. Results

to eq. (l),

zp(C+C’)/l

the occupation numbers, the energy for such transition

it may

energies results when

and discussion

The results obtained from eqs. (8) and (11) are compared with the TS, Koopmans’, and experimental values in table I_ It may be seen that the simulated TS model agrees quite well with the TS, and gives better results than Koopmans’ approximation for the inner shell orbitals. The model presented here may also be applied to different excitation energies. That is, if the initial and final states of the atom for a given process are ri Compare the results of ref. [lS]

and according is given by

(12)

(final)

\vith phase of ref. 1171

L?&-= (U’ - L7)e(,r;s) + (c’-

c>e(iz~,>

+ @ - b)e(&) + ... )

(13

where .(,I:) are the transition-state Lagrange multipliers. Thus, substituting eq. (7) in the HFS case, and cq. (10) in the HHF case into eq. (13) one can estimate the excitation energy, using only information of the ground state of the atom. In table 2 we present the results obtained for multielectron X-ray transition energies. It may be seen that, in general, the simulated TS agrees better with the TS and with the results obtained through total energy differences than Koopmans’ approsimation. The overall situation indicates that the approach presented here might be very useful to determine ionization and the transition energies using only groundstate information. This may be understood in part, from the results obtained in relation with the hardness

Tabk 1 Orbital ionization encrpics for several atoms (au) At0111

iic Se

Orbital

IS IS IS

NJ

xr

IS

?s zp IS ‘s 2P 3s 31,

Fxp.“)

Ilartrcc-Fock Koopmans’ a)

TS b)

this work c,

S.9179 4.7316 0.3092 32.7724 1.9304 o.s.504 1 IS.6098 12.3221 9.5714 1.7774 0.5910

0.8575 4.5111 0.7919 31.9110 l.SO10 0.7239 117.4011 11.9303 9.2536 1.2133 0.5686

O.SSSl 4.5345 0.2761 32.2223 1.8354 0.715s 117.569 13.1207 9.1768 1.2037 0.5139

0.9037 4.45 0.343 31.98 1.78 0.7926 117.Sl 11.99 9.21 1.08 0.5792

Hxtrec-Fock-Slater

d)

Koopmans‘

TS

this work

1.091s 5.0673 0.3701 33.4298 1.8225 0.9696 119.572 11.924 9.7048 LlS36 0.6345

0.9824 4.7315 0.3376 32.2396 1.6601 0.8185 117.8065 11.4829 9.2146 1.0946 0.5705

1.0354 4SSS2 0.3398 32.8859 1.7314 0.8556 118.5360 11.7257 9.3591 1.1139 0.5719

b) Values taken from ref. [ 121. C) Eq. (lo), using the data of ref. [ 171. a) Values taken from ref. [ 171. d) The data was obtained from an Xu self-consistenr calculation, using the values of a reported in ref. [21]. For Koopmans’ approximation within the Xn method set ref. [ 13]_

Volume

109, number

Table 2 Comparison

4

CHEMICAL

of the two- and three-electron

_4tom

(Initial,

final state)

X-ray

transition

(ls-* 2p-’ , 2s-1 (ls-2 2p-*, 2s-’ (ls-2 2p-4 ,2s-’ (ls-2 zp-a, zs-t (Is-2 2p-a, ‘s-t (Is-’ 2p-I, zs-2) (Is-‘2p-I, 2s-2) (Is-’ 2p-t, 2s-‘) (Is-’ 2s-a, zp-3) (Is-’ 2s-*. 2p-3)

a) SW footnote a) of table 1. c) The data required \ws taken

cnergics

for several

21 Aupus1

atoms

zp-2)

=)

tiscI:

29.86 62.76 241.06 463.39 540.38 60.76 129.35 158.22 68.36 142.12

32.23 66.37 253.38 476.59 554.90 60.60 130.24 159.60 -

b) These values from ref. [17].

were taken from _ d) See footnote

2p-3) 2p-s) ap-3) 2p-3)

[19] $ through the simple model based on [TO] +$_ At present, WC are studying the extension of the model to molecular ionization lind excitation processes. effects

Acknowledgement It is a pleasure to acknowledge valuable discussions with A. Pisanty and the Departamento de C6mputo of Universidad Aut6noma Metropolitana for their technical assistance_ * Par: and Pearson have defined hardness as $(a’&-/ &V )IneutraI, that is, the second derivative of the energy with respect to the total number of electrons, evaluated for the neutral atom. ** In ref. [ 201 we have shown that the simple model based on screening effects leads to (a2~~a~v2)lneutraI z $(r-’ h, with i the highest occupied atomic orbital, for a second derivative that irtciudes relaxation effects.

References [ 1 ] J.C Slater, Quantum theory of molecules Vol. 4 (McGraw-Hill. New York, 1974). [2] J.F_ Janak, Phys. Rev. Bl8 (1978) 7165.

1954

(au) Hartrec-Fock-Shrter

of an atom screening

LETTERS

tlartree-Fock ~oo~mans’

N NC K Fe Ni Si Ca Ti Si Ca

PHYSICS

and solids,

b)

this work 32-25 65.27 240.93 468.60 551.03 i8.65 126.16 154.60 65.54 137.69

ref. [ 31. d) of table

c)

Roopmans’ 30.82 64.07 243.36 466.30 543.45 62.15 131.14 160.48 68.25 141.82

d)

TSb)

this work d)

32.19 66.22 254.19 477.55 555.82 61.59 130.97 160.11 58.83 120.57

33.2 0 66.53 243.17 47 1.33 555.01 60.19 128.28 156.95 65.32 137.37

1.

K.D. Sen, J. Phys. Bll (1978) L577. Scn. J. Chem. Phys. 71 (1979) 1035. Sen, J. Chem. Phys. 73 (1980) 4704. Sen, J. Chem. Phys. 75 (1981) 5971. Sen, P.C Schmidt and _A. \vciss, J. Chcm. Phys. 75 (1981) 1037. S.R. Gadre and R-G. Parr, J. Am_ Chem. L81 L.J. Bartolotti, Sot_ 102 (1980) 2945. K. Schwarz, J. Phys. Bll (1979) 1339. N.H. Beebe, Chem. Phys. Letters 19 (1973) 290. 0. Goscinski, B.T. Pickup and G. Purvis. Chcm. Phys. Letters 22 (1973) 167; 0. Goscinski, Intern. J. Quantum Chem. S9 (1975) 221. [I21 H.S. Brandi, MM. De Mates and R. Ferreira, Chem. Phys. Letters 73 (1980) 597. Physica 1 (1933) 104. [131 T-A. Koopmans, J. Phys. B12 (1979) 521. [ 141 M.-S. Gopinathan. 34s (1979) 901. LlSl K.D. Sen, 2. Naturforsch. [IhI J.C Shtcr, J.B. ,Mann, TM. Wilson and J.H. Wood, Phys. Rev. 184 (1969) 672. Calcuhrions I. Hartree[I71 J.B. Mann, Atomic Structure Fock Enegy Results for the Elcmtnts Hydropen to Lawrencium, Los Alamos Scientific Laboratory Report. LA-3690 (1967). The Hartree-Fock method for atoms [I81 C Froese-Fisher, (Wiley, New York, 1977). [I91 R.G. Parr and R.G. Pearson, J. Am. Chem. Sot. 105 (1983) 7512. J-I__ Gizquez and E. Ortiz, J. Chcm. Phys., submitted WI for publication. WI K. Schwarz, Phys. Rev. B5 (1972) 2466; I(. Schwarz, Theoret. Chim. Acta 34 (1974) 225. ]3]

L41 K.D. 151 K.D. [cl K.D. 171 K.D.

397