Correlation of 1s binding energy with the average quantum mechanical potential at A nucleus

Volume 6. number 6

CHEMICAL

CORRELATION QUANTUM

OF

1s

PHYSICS LETTERS

BINDING

MECHANICAL

ENERGY POTENTIAL

I6

WITH AT

TEE

September 1970

AVERAGE

A NUCLEUS

MAURICE E. SCHWARTZ Department of Chemistry and The Radiation L&oratory *, University of Notre Dame du Lac, Notre Dame, Indkana 46556. USh

Received 3 August 1970 Changes in binding energies of ls core electrons for C. N. 0. and F atoms in different moIecular environments have been studied by examination of negatives of inner-shell orbital energies (-Els) Krom ab initio “double-[ ” quality LCAO SCF MO wavefunctions. As contrasted with many previous correlatioos of binding energies with point charge models. changes of orbital energies have been related to the potential at the nucleus on which the 1s orbital sits. properly calculated as a quantummechanical ex-

pectation value. The contribution of the 1s to the potential is insensitive to environment_ and consideration focuses on the “external potential” ‘Pext due to the remainder of the system outside of the Is. Shifts

in -El6 are in all cases nearly the same as shifts in %ed (using atomic unitsj. the average oE eleven cases giving A(-Els)/A@ext) = 1.11. An analysis of the equations. based on localized valence hIO% and supported by numerical examples. shows this to be a generally expected result. with some possible exceptions being outlined.

1. INTRODUCTION Measurements of binding energies (BE) of innet-shell (core) electrons by the X-ray photoelectron spectroscopy method called ESCA promises to be a powerful probe of molecular structure [l-3]. In this paper we consider ab initio quantum mechanical studies of 1s BE’s for a number of first-row atoms in different molecular environments. Special emphasis will be placed on a quantum mechanical generalization of the simple charged a-tom potential model (ref. [l], section V.2; see also ref. [4]) for changes in BE’s (to be denoted by ABE). According to this model, changes in the inner-shell BE of atom n in different molecular environments are the same as changes in the potential at the nucleus ft. First applications found linear correlations oJ BE with the charge associated with the atom, and more recently, with the charges at all atoms in the molecule. A good summary of this approach can be Iound in the two books cl, 21 by the Uppsala worke?s, with several other papers [5-81 furnishing examples. The problem of associating a point charge with an atom in a molecule is not trivial (from a strict quantum mechanical viewpoint it cannot be done), and one migbl define charge by -many ‘.. non-unique c?iteria. It was pointed out recently [F] that linear c?rreiations.of BE’s with charge : ;

are known for several different inner-shell BE’s and several different atomic charge definitions, and refs. (21 and [5-8] contain relevant examples. Thus, even if a linear correIation is found, the physical situation is not necessarily well-rcpresented; This does not, of course, preclude the usefulness of such approaches as semi-empiricai methods for extrapolation and interpretation of ABE’s 181, but at the same time the correlations found do not show how the BE varies with the potential at the nucleus which arises from the other nuclei and from the proper electronic distribution given by the electronic wavefunction. It is the purpose of this paper to consider the relationships of Is BE’s with the potential at nuclei as found by rigorous quantum mechanical descriptions. t To my knowledge there have been’just two previous considerations, both brief, of the quantum mechanical potential as related to BE-s. Ha and O’Konski [9] have noted the near equality of changes -in 1s orbital energies and the potential due to the valence electrons in many neutral and ionic states of first and second row atoms. This striking result is somewhat unsatisfactory, * The Radiation Laboratory

of the University

of Notre

Dame is operated under contract with the U.S. Atomic

Energy Commission. COCB-38-735.

This is AEC Document No.

631

Volume 6. number 6

CH&ICAL PHYSICSLETTERS

however, since .orbitaI energies from the restricted Iiartree-Fock wavefunctions employed do not always &rrespond to ionization energies by the usual Koopmans’ analysic+ On the other hand, in a very recent @per Basch [lOl exmined C and F Is BE’s in the ~uorome~nes and %Guriaa. direct linear correlation (slope = 1) with the quantum mechanical potent*&1at the nuclei, We &1X comment later on the connection with our jtre’sent ‘work. Although considerable el@ronic reorganization occurs with. core ionization [ 111, complicating in principle the interpretation of ABE’s, evidence is accumulating [2,6,?, 12-141 that the use of, negatives of orbital energies (-6) for core MO’s fiom.accurate closed-shell LCAO SCF’MO wavefilnctions [X5] gives reasonably good prediction of the chges ABE. In this paper we assume ,this optimistic attitude that one can predict and understand &BE’s on the basis of ground state MO electronic structure. 2. -CALCULATIONS ANO RESjLTS

_

’

;,

__I

632 -: .:‘ .

i..,. :_

.. .

Table 1

Calculated potentials and 1s orbital edergies at tbe nuclei Lnau

c:-

-.

._

’

-%?

-*ext

-%s

3.4454

3.3972

HCN

11.3168 11.3180 lf.3l.65

11.2018 11.2591 1X.3122

CH4 WC2

3.364j 3.3239 3.2778 3.2703

3.1.3167 11.3692 11.4102

CH.9

11.3185

&CO

u.wra

CO

11.3217

1QI3 HCN

13.2991 15.3023

5.0850

5.0091

15.5355 15.6283

%?0 H$O co

15.2769

‘7.0651

20.5525

15.2783 15,277L

7.0196 6,954s

20.6046 26.6782

HOF

25.2785

6.9441

20.6833

flF CIi#

17.2496 17.2496

9.3411 9.3629

26.2812 26.2606

HOI?

17.2508

9.2843

26.3443

N:

0:

F:

We consider IS BE’S in fifteen different mole&es containing C; N, 0, and F, using the’ab initio IXAO SCF MO model [lS]. Basis func_tions.for the LCAO expansions were taken fram accurate atomic calculations using gaussian type .drbi_tals (GTO). The groups of s functions of Whitten [IS] were used, the long-range component of the Zs-like gfoup being split out [Xi] to give a ttital of four s groups. The five-terms 2p atomic orbit& of Htuziega 1171 were simihrly split i+to four-term and one- term (most diffuse) gro.ups. At hydrogen a five-term [16] GTO expansion of Is-was split into four-term and oneterm (most diffuse)-groups. This is a “doublezeta” qtility basis set, quite similar to that . used by Ba&h-and Snyder [?I. This basis has -been found [2,13,14] to give shifts of inner-shell orbital eneigies in accord with experimental values; Ground state exgrimental geometries and 3 hydrogen Is scale of 42 are used thoughout f ll]. Atomic &its %re used unless otherwise specified. Table, 1 summarizes. the orbital energids-of MO’s which are e~s~~t:&lly the atomic 1s orbit-, ais as judged.b$ ejgenvalue and basis fuliction constitution. -Also included are quantities from -the quantum m.%hanical g.enera.li%tiG.nal the p&e&la1 at nutil~eus?zarising from tie dogbiy . r _ ’ or;cu@ed y_O!s @j and the,other nuclei.Z$: . ._ . .’

15 SejMmber 1970

3 =-2 C
r: Zm/Rmn. m+tz

We have separated the contribution of the 1s MO at the nucleus, #ls, leaving the Vxternal potential”, *-t. The potential is of course not equal to the orbital energy, and we have yet to make a connection between the two quantiticts. (Since we work in atomic units, we can speak of the %quaLity” of potential and energy quantities.) From a variety of expectation value calculations we have found that intrinsically the IS MO at a particular atom is insensitive to molecular environment, even though very small contributions from other basis functions.may contribute to the MO. Thus we see in table 1 that #Is is essentially constant f0$ a given atom, and that non-trivial. changes in the -potential at a nucleus . are’ due to changes i;l @e&. {We-have considered the ext$me case of re$acirig the-1s yO’s.by free atG&~,lg ACM: no essential difference oEcw&. . . ,. : Qtiali&veIy ihe changes ‘iti i&is and *at go .+as’anticipated: the less &?ga&+e @e pbtenfial’_ -_: ___. _ : _, .. ‘9.: ~I _. . ‘., ._ ’ ‘. -1 1 _ , .. y ‘. _. : .‘. .,, .__ ,_ :\ , I _,‘.I -1. .‘..

Volume 6. dumber 6

CHEMICAL PHYSICS LEXTERS

Table 2 Shift.9a) in la orbitni energies in eV ad external potentials in eV/e+ (1 au = 27.21eV)

c:

Czrr2

1:s

1.91

BCN. .. - 3.00

0:

2.49

1.20

33

5.31

0.96

H$O

c55

4.56

1.00

co

5.67

4.76

1.19

HZCO

1.42

1.24

115

co

5.43

3.00

1.14

FOH

3.56

3.29

1.08

FOH

1.72

1.55

1.11

-6.56

-6.59

0.95

2.53

2.07

the average *(-~l~)/*(*~fi) being 1.11 for the eleven shifts of the table. Fig. 1 iILustrates this with a plot of A(-Eis) against A(@e&. This is an

important result: without any calibration at all we find a direct connection of Is BE (~1s) with environment (*at), showing expLicitLy the monitoring of core BE’s by valence etectrons and nuclei. It is striking that all of the 1st ABE’s should fall on essentialLy the same pLot against A(*&, since the different atoms C, N, 0, ax’ F, in different environments are involved. But we do expect other cases wadd aLso fall into this generat reLationship for reasons ta be discussed in the next section. ALSO, Basch [lOI found r’eYults like ours for both C and F 1s in the fluoro-

1.19

CIbF

The shifts in both quantities are very nearly the same for all atomsand environments considered,

methanes.

F: CH3F N:

HCN

a) C shifts from CQ,

from NH3.

-I

0

0

I

1.22 .

from H20, F frcm HF. N

2

3

6

Fig. 1. Change of ~EIorbital energy, A(+, versus change in external potential, A(@&). The.numbers wfer tc the molecules aa ordered in table 2: 1-5 arc QH2 to CO, etc. /*

(Le., &e

zaiP?e

pc#itiTe

*i

%Wrn@

atmic

en-

vironment) the greater is the 1s BE as~measured by -e15. Tl$s point is examined quantitatively in hbIe O-where A@E’s as changes in -els are corn-pared to shifts in the exterG.1~ potentiaL *,=t_

3. ANALYSIS

AND DISCUSSION

We now anaLyze these results in some detaiL to see how why the relationship shouLd hoLd, and to see what it may’ suggest for further studies of inner-shell BE%. Basch [IO) has made a different analysis of the problem in which he COTLapsed the 1s into the nucleus and considered the subsequent equations. We do not make such an approximation. For reasons to become obvious we assume that the 1s MO% are fixed as they emerge from the LCAO SCF MO wavefunction determinations but that the vaLence MO% have been transformed to localized orbitaL form (LMO’s) [IS]. This means we can stiIL use Koopmans’ theorem and -Q to measure 1s BE’s and make connections between Is BE’s and a valence electron distribution partitioned into highly l$caLized portions. We denote the 1s -MO considered as simply ls, and the LMO’s as Q (as far as a given 1s is concerned, 1s at other nuclei are considered in the set %I. As the anaLysis proceeds, we look at numerical examples taken from C and F 1s BE’s in our CH4, CH3F, and HF calculations. The orbital energy of Is at nucleus 2, is 1153 ~1s = hlsls+C(2

J~si-Klsij

,

i

where.lilsls

is the he-electron

part and Jlsi

aadK~,g~ethausuaLcatitiand~

contributionti to the two-electron convenieqtly re-written as

part. This is

633

Volume 6, number 6

CHEMICAL PHYSICSLETTERS

15 Sepkmber 1970

tively; and -35. 0565 au for F 1s in b&h HF and CH3F). Now we-&n compare the negative of the orbital energy to the potential, which we partition just as -E is: iht -Eext = - El8 +6 is where we.have divided the two-electroa interaction energy of Is %ith the valence LMO’s into “local” contributions (due to Li which connect. atom n) and “distant” contributions (due to Lj not connected to atom n). Because of the high lomlization of the Is-electron distribution (from far away it.is effectively a delta function density), the distant LMO’s 5‘ should have vanishingly small exchange-integrals with Is, Bnd we expect as a very good approximation-that Klsj = 0 0’ = distant), as the summary of interaction integrals in table 3 shows. We expect, further, that the distant coulomb integr$Z should have the same .magnitude as an electron-nuclear attraction inbgral of the distant Li fqr nucleus n, as also ghown in table 3.. Similarly the attraction of the 18 density t0 other nuclei should be a pointcharge value: -Gs(!)l%&Imlls(l)) fi: Gn/Rnm a iksult we haye verified to four decimal for all such interactions here. Lastly,

, places

is the localized internal energy of Is, and should be insensitive to envitonment (tint - 14.4250 and -14.4247 au for Cl8 in CH& and. IT-H3F, respec-

+ c

2,/R,

- c

-2

c

(L.(l)(

j=ciist ' '

-I-c m+n

(2 Jlsi-$si)

Molecule

%a

Li

CH4

1%

C-H

CHQF

lsc

:

lS* _

C-H C-F BF F lone pair C-H C-F 1% F lone p#r

:

i=loc

m*n

Z,/Rmn

:/‘,,IL+l))

-2 c {Li(l)\b'+i(l)) i=loc

-

The problem now simplifies: the first two terms on the tight of these equations are identical and-contribute e&ally to A(-E~~) and A( +&). So, with changing environment comparison of changes in -cl8 and & requires just consideration of changes in the electronic interaction of 1s with adjacent LMO’s -

c (2 Jisii=loc

and the interaction the nucleus -

Klsi)

(1)

of these adjacent

c 2(Li(l)j i=loc

l/?ljALiU))

LMO’s with

-

Table 3 Interaction integrab of Is and LMOQ in au

.’

-

= -2 c (Lj(l)ll/QnlL+l)) j =dist

JlBi

Klsi

0.6557

0.0179

0.6746

0.6680

0.0195

0.6891

0.584i

0.0103

0.5945

0.3'321

0.0000

.0.3820

0.3517

0.0002

0.3518

0.2980 1.0390 0.3821 1.2388

O.QOOl 0.0345 &OOOO 0.0631

0.2981 1.0678 0.3021 1.2912 L.

(wl)ImIlIwl))

(2)

Volume 6. number .S

CHEMICAL PHYSICS LETTERS

These *ll not be identical, since 1s is not completely localized as far as adjacent L&IO% are concerned. But they should be similar, and their changes with environment should be nearly equal. In fact, the near equality of a(-~) and A(@ext) requires these assumptions. The data of table 3 can be used to verify this. Ratios of individual terms of eqs. (1) and (2) are in the range 0.94 to 0.97. Changes in eqs. (1) and (2) are +0.0666 and +0.0732 au, respectively, for C 1s in CH4 to CH3F, and +0.0356 and +0.0360 au for F 1s in HF to CH3F. The analysis thus seems substantially correct, but it does merit a few more remarks. Our numerical examples in the analysis here involved no dramatic changes in the general nature of the chemical bonding of the atom whose 1s BE is considered (e.g., C is roughly “s~3~ in both CH4 and CH3F), and one reasonably expects eqs. (1) and (2) to change similarly. Gn the other hand, from CH4 to HCN, for example, represents a qualitative change in the bonding nature of C, so relative changes would not necessarily be expected to agree so well. That is, the reference state for 6BE.s affects the relative agreement of A(-~1s) and A(*ext). For the molecules considered in this paper, the overall effect is not too important, however, as table 2 and fig. 1 show. There is thus good reason to expect, as a general feature of the problem, that similar relationships will exist between core orbital energies and external potentials, at least for first row atoms. For second-row and larger atoms two possible breakdowns of this analysis may be reasonable speculated upon. First, rather more dramatic changes in local atomic environment (“oxidation state”) can occur (e.g., S in H2S and SF6) than in these first-row atoms, and perhaps cause changes in quantities (1) and (2) to be different. Seccndly, the anisotropic nature of the core 2p orbitals might cause changes in quantities (1) and (2) to deviate more from one another than for spherical 1s (and 2s) orbitals. The facts [ 1,2,6] that one is able to find good pointcharge correlations for S in a wide variety of environments, and also that experimental shifts of 2s and 2p BE’s with changing environment are about the same, suggest that these two effects are relatively minor. It would be good to have the average potential discussed here examined for other atoms and other environments of the same atoms. Since the same molecultir integrals necessary for ab initio LCACSCF MO calculations are sufficient for computing tfie external potential, this would be a :

15 sepfember 1970

simple matter, and it could become a routine part of such theoretical studies of core BE’s,. just as simple point-charge correlations are now sought. For example, Gelius et al. [S] already have ab initio results for orbital energies and a calibrated point charge potential for S 2p BE’s. Although they felt that the complete charge distribution is too complicated to be included in a simple and perspicuous model for the interpretation of chemical shifts, the ease of calculation of 9 and its concomitant correlation with -E would suggest otherwise. Their sulfur-containing molecules offer the two possible breakdowns mentioned in the previous paragraph, SO a study of A(-E2p) and A(&& would be especially worthwhile. ACKNOWLEDGEMENTS This work has been supported in part by a grant from the Petroleum Research Fund administered by the American Chemical Society. Generous amount of Univac 1107 computer time were given by the University of Notre Dame. REFERENCES [l]

[2]

K. Siegbahn, C. Nordling, A. Fablman, R. No&berg, K.Hamrin, J.Hedman, G. Jolxmsson, T. Bergmark, S. -E. Karlseon, I. Lindgren and B. Lfudherg, “ESCA - atomic, molecular, and solid state structure studied by means of electron spectroscow” (Almavist and W&sells. Uutwala. 1967).

K.&$&n. C. Nordung, G. Jo&&on, J.Hecl&n, P. F. Heden. K.Hamriu. H. Geiius. T. Bergmark. L.O. Werme, R.Manne and Y.Baer, ESCA applied tc free molecules (North-Holland. Amsterdam,

1969). [3] D. M: Hercules, Anal, Chem. 42 (19’70) ZOA. [4] A. Fahlman, K.Hamrin, J.Hedman. R.Nordherg, C. NordHng and K. Siegbahu. Nature 210 (l966) 4. ; (51 D. N. Hendrickson, J. M.Hollander and W. L. Jolly, Inorg. Chem. 8 (1369) 2642; M. Pelavin. D. N.Hendrickson. J.M.HoIlander and W. L. Jolly; J. Phys. Chem. 74 (lS70) 1116. [S] U. G&us. B.Hoos and P. Sfegkhh, Chem. Phys. Letters 4 (1970) 471. [7] H. Basch &d L:C. Snyder. Chem. Phya. Letters s (1969) 333. [a] M. E.Schwartz. C. A.CcmIson and L.C.Allen, J. Am. Chem. Sac. 92 (l.970) 447. [S] T. K.Ha and C. T. O’KonsM. Chem. Phya. Letters

3 (1969) 603. _&I H. Has&, Chem. PFiys.Letters 5 fl970) 337 [ll] M. E. S&warts, C&n. Phys. Letters 5 (197;) 50. [12] T.D. Tiramae, J. AXE. Chem. Sot., to be published. [131D. W. Da+, J.M. HoIIander, D.A. SbIrley and T. D. Thomas, J. Clam. Phys. 52 (1970) 3296. [14FM. E. ~bsvartz, unpublished reauIfs.

-

635

..

J;:&&;

&c;‘c;

Rev; nmd.. a&.

3qiasi)

6ai

G;'G.Hal&Pm.-Roy.So& A205'(1951).541. ': (l6i J.~.W~~%I,-,T.,~U%IL.P~~S~ 44 (1966)_S?9., -. . . /'_.~ . Y

..

Ii71s.Huh&i,‘&

cbxxl:

h

Php.

(1965) i393.

ps] C;+l&ie and Kprtedenberg, Rev. Mod. Phpa. -350993)457., ' _,: .:, ,_ . :. ^ . . : - _-

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)

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... ‘.

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:656 _ ; .I__ _, ~

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::

1. -.

_I_ ._..

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:. ..

’ 1

._ .. -

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