[I] (a) K. Siegbahn. C. Nordlimg, %. Johansson, J. Heldman,. P.F. Heden, K_ Harmin, U. Gel& I. Bergmark. L-0. Werme, R. Manne and Y, Baer, ESCA applied to ...
ChemicalPhysics 51(1980) 459471 0 North-Holland Publishing Company
SCF DvM-xa WITH BASIS SET OF NUMERICAL HARTREE-FOCK AND ITS APPLICATIONS TO MoF6, WF,, AND UF,
FUNCTIONS
G.L. GUTSEV Department of Institute of Chemicai Physics of the USSR Academy
of Sciences, chernogolovhz
142432. USSR
and AA. LEVIN IRsritrzre of GeneraI nnd Inorganic Chemistry of the USSR Academy of Sciences, Moscow II 7071. USSR Received 4 July 1979; in f&
form 6 May 1980
A self-consistent version of the discrete variational method is descriied based on the use of numerical LCAO basis functions obtained S_ solutions of the Hartree-Fock equations for free atoms. Sets of single-zeta Stater functions are applied to approximate atomic densities further employed in the calculations of the Coulomb potential. The computer prOgm’IlS realizing this approach have been written and utilized to calculate electronic structures of molybdenum, tungsten and uranium hexafluorides. The ionization potentials catculated are in good quantitative agreement with experimental data. The deviations of the calculated valence state E’s from those determined by photoelectron spectroscopy do not exceed 1 eV_
1. Introduction
’
molecules, particularly those involving transition metal atoms. The Hartree-Fock-Roothaan (HFR)
Slater [3] suggested to approximate the non-local exchange potential by a local one. Slater’s approach irrvolveeaveraging over all HF spin orbitals followed by the replacement of the potential thus obtained with the exchange potential of free electron gas of the same density. Nunrerolls calculations have shown that the methods based on Slater’s local exchange (HFS equa-
method [2] is the traditional approach to ab initio
tions) as a rule give as accurate results as HFR calcula-
calculations on molecules. The corresponding procedure, however, rapidly grows cumbersome as the number of basis functions increases. On the other hand, the use of minimal basis sets constructed of analytical functions reduces reliability of the cahulation results. At present, HFR calculation schemes can hardly be applied to transition metal complexes. Treatment of these molecular systems by the semi-empirical methods successfully used with lighter atoms also generate obvious difficulties_ The difficulties of the J-IFR method calculation due to the use of non-tocal exchange potentials which vary from one HF spin orbital to another were successfully overcome by Slater.
tion schemes do while having the advantage of being fas less cumbersome. Among various methods of solving HFS equations, the Slater-Johnson method of scattered waves (SW-X$ [4J] is used most frequently. Efforts to improve its agreement with experiment have led to the development of the procedure of *‘overlapping spheres” [&I_ The result has, however, been the loss of method’s generality, as sphere radii play the part of semi-empirical adjustment T&meters. In addition to the SWXLYmethod, there are schemes based on EAO ex@&ons of the solutiorn~of HJ% equations, such as the LCAO-Xa method [7] -cmplying a basis constructed of analytical gaussiau functions, the Xo
Recent developments in experimental studies of molecular electronic structures [l] have stimulated
interest of chemists in methods of calculations of
459
460
G-L. Gutsev. AA. Levii / SCFDVM-&and
version of the “atoms-in-molecules” method [8], and also a few versions of the discrete variational Xar method (DVM-Xo) based on numerical integration of the l-lartree-Fock-Slater equations. Baerends et al. [9] have realized the DVM-& approach using. basis sets constructed of analytical SIater functions. These authors approximate the Coulomb potential by one-center distributions determined from the basis functions_ Within the LCAO-expansion approach, schemes involving comparatively small basis sets including “best” atomic functions are of considerable interest. These functions may, e.g_ be the HF [lo] or HFS [I I] functions or functions obtained from atomic type caiculations with model potentials [ 12]_ Numerical integration of the HFS equations makes it possible in principIe to solve the Iatter probIem_ However, the constructiou of the molecular potential then becomes an even more complex task than with basis sets of analytical functions. That have prompted Rosen et al. to develop the procedure of “self-consistent charges” (SCC) within the framework of the DVM-Xol approach (for non-relativistic HFS equations [13] and relativistic Dirac-Slater equations [14,15]). The SCC procedure, in combination with Muliiken’s popuiation analysis, generates spherically symmetrical charge distributions. Poisson’s equations are then solved by onedimensional numerical integration. This work describes a version of DVM-X@ involving the usual SCF procedure which provides the possibility to employ numerical, analytical and “mixed’ LCAO basis sets. The SCF procedure is based on approximating moIecular charge densities with sets of single-zeta functions centered on the atoms invohed [ 171 and independent of the basis functions_ This appears to be a more flexible and, at any rate, far less computation time-consuming procedure than the SCC one. Specific details of the numerical solution of HFS equations within the DVM approach are discussed in section 2. Section 3 describes the procedure for minim. i&ion of nonlinear functionals and its application to approximating atomic eIectronic densities by spherically symmetrical Slater functions_ Section 4 contains the results of calculations on MoFs, WFa and UFs obtained using computer programs realizing the DVM-Xa approach.
its applications
2. SCF DV-Xct method 2.1. One-electron HFS equations The non-relativistic one-electron equations solved are written in the usual form:
-30T[~~b’p(1)]“3
$i(l)=EiJ/i(l)
to be
-
I The molecular electronic density p(l) is defined in the MO basis, 6 i, by the expression
(2) where Iri is the occupation number of the ith orbital. The only parameter of the model is the exchange constant a! chosen within the range 213 Itau >6t,, >7ata>2tzg>4es, wasobtainedinah calculations. The fmt five of these levels are of predominantly &and nature; the 2t,, and 4es levels mainIy include molybdenum 4d and fluorine 2p AO’s. The orbital energies of the MOF, ground state underwent only negligible, not exceeding several hundredths au, variations from 1 to 3. The energy gap between the highest occupied tlg and lowest vacant tag Ieveis remained nearIy constant, of 3.8 to 3.9 eVThe optical transition energies calculated using SIater’s transition state concept, as a rule, deviated from the ground state orbitaI energy differences by not more than 0.1-0.5 eV (see aIs ref. [373)- The energy of the first optical transition calculated for MoFe is thus equal to 4.0 eV whereas the experimental vahre is 5.9 eV [38]. To explain this discrepancy, one shouId bear in mind that the caIcuIations of the transition energies with the ‘non-spin-polarized” version of DVM realized in our computer programs imply averaging over all moIecuIar terms arising from a given excited state. The choice of the starting atomic electronic configurations is an important problem with numerical atomic functions [13]_ This is, in particular, a critical problem with non-selfconsistent schemes where molecular eiectronic structures are calculated as superpositions of atomic densities and potentials. To study this problem, we performed caIcuIations with basis sets inchiding the wavefunctions of positive, negative MOions and the usul HF neutral MOground state functions. The electronic confiirations were chosen as follows: Mo*(~D, 4&‘5s’), MoO(~S, 4@5s’), MO-(%, 4ds5s2). The HF orbital energies for the corresponding atomic caIcuIations are given in table 3. In aII other respects, the calculations were identical: 2200 DVM integration points were used; the basis sets were complemented with the 5p MOSTO’s with
its applications
Table 3 OIW&C~IO~
HF SWI&
of MO in the ground ad
ionized
states(in Ry) Level Configuration
1s
2s
2P 3s
3P 36 4s 4P 4d 5s
6D term
7s term
6S tern
1443.2173 2065216 189.7096 37.9916 31.3951 19.3879 6.3256 4.2227 1.4546 0.9819
1442.4044 205.6971 188.8868 37.1683 30.5729 185686
1442.089
5.5257 3.4473 0.7158 0.4455
205.301 186571 36.8529 30.2579 18.2538 52118 3.1352 0.3917 0.0338
exponents equaI to 2.0. In each case, sets of singlezeta functions approximating the atomic densities for MO+,MO- and Moo were optimized. The results given in table 4 show that the molecular one-electron energies of the zeroth approximation which uses superposition of the HF densities and potentiaIs were significantly different_ After SCF iterations, however, the differences did not exceed 0.01 au_ The superposition of the electronic densities and potentials of the neutral atoms gave the one-electron energies nearest to the SCF Iimit. This conclusion agrees with the idea of only minor rearrangements of the atomic electronic densities upon the formation of chemical bonds [39]. On the other hand, MuIIiken’s population analysis [ 161 yielded the value of 1.2 to 1.6 for the charge on the central atom_ One might therefore expect that the basis constructed of the positive MOion wavefunctfons would provide a better description of the eleo tronic structure of MoF,_ Our calculations, however, show this conclusion to be unwarranted and provide one more evidence of a somewhat arbitrary nature of the population analysis. 4-Z. Valence IP’S and A0 populations In caIcuIations of valence level IP’s, the basis set of MOand F HF functions was complemented with the STO’s, 5p (MO) with the exponent of I .5 (approxirn-
G.L. Gutsev. AA
467
Levin / SCF D V&f-X@and its applications
Table 4
orbital eneIgieSfor the ground State of Level
[(Mo+)
5% 3t2g %g 7t1u 1t2u 6tru 7atg 2t2g
4% Charges MO onatoms F I-st IP in eV
MOF6
in the i-IF-basissupplemented with Sp(Mo) AO’s with exponent 2.0 (in au)
[(MO-IFc31’
F61-
zero+h appr.
MOF6
SCF
-0.2488
-0.0708
-0.4235 -0.6920
-0.2497 -0.3863
0.0917 -0.2845
-0.2528 -0.3890
-0.0852 -0.4335
-0.7024 -0.7086 -0.7332 -0.7471
-0.4025 -0.4058 -0.4332 -0.4583
-0.2920 -0.3000 -0.3120 -0.3357
-0.4071 -0.4073 -0.4373 -0.4606
-0.7704
-0.5205
-0.3499
-0.5262
-0.7902
-05326
-0.3644
-0.5360
-0.4416 -0.4498 -0.4625 -0.4852 -0.5011 -0.5164
-0.0777
0.2695
-
1.337 -0.222 14.64
-
-
ately a half sum of Clementi’s 5p exponents [20]) and 3s (F) with the diffuse exponent of 1.0. The fust cohmm of table 5 contains P’s calculated using Slater’s transition state concept, and the last column contains IF% found by Karlsson et al. [40] from the photoelectron spectra. Table 5 also includes the orbital energies of the valence levels obtained using the transition states for the ltlp and 4ep levels only. Comparison of the values cited shows that the orbital energies calculated for the highest and lowest transition states differ from each other and from the theoretical IF% by not more than 0.2 eV; the results obtained by Baerends and Ros [41] for ferrocene Iend a support to the latter conclusion_ It thus seems not necessary to calculate each IP value separately. Estimation
Table 5 IOnizatiOilpOteI’Iti&SfIOm Ytience levels for MoFs (in ev) Orbital Experim. enerses, TS energies,TS 1401 forlQg for4eg
Level
Ip’s
type
calculated
lh, 7t1.3 _-
15.17 15.62 15.63
14.96 15.44 15.43
15.07 15.8
%I
15.17 1551 15.60
6tlu 7alg 2t2g 4eg
16.13 16.85 18.54 18.89
16.24 17.05 18.56 19.03
16.06 16.88 18.39 18.89
16.55 17.61 18.52 19.076
Orbital
SCF
zeroth appr.
zerathappr.
1.261 -0.210 14.77
0.0917
-
.
SCF -0.0814 -0.2552 -0.3952 -0.4128 -0.4142 -0.4419 -0.4666 -0.5258 -0.5389 1.397 -0.238 14.68
of the valence P’s as orbital energies obtained by SCF calculations for one of the transition states, e.g. corresponding to the highest valence level, will suffice_ The population analysis (table 8) shows that populations of the outer s and p metal AO’s are not large.. Thus, the total population of the vacant molybdenum 5p orbital is =0.3. An analogy may be drawn with vacant non-transition element d-orbitals [42]. It ap pears that in both cases, vacant orbitals make only minor contributions to bonding and their importance is mostly overestimated (see also ref. 1431). A mere superposition of the atomic densities and potentials gives the correct IeGel ordering which does not change through the SCF iteration sequence_ The principal features revealed by the DVM-XCZ calculations of MoF6 are retained on going to WF,. The two lowest vacant orbitals in WF6 are also the eg and t2g orbitals which mostly include the metal 5d functions. The ordering of the valence levels is the same as with MoF6r It,, > 9t,, > 2tzU > 8t,, > Bar, > 3t28 > 5eg (which ahnost coincides with the level ordering obtained in ab initio calculations on the WOZ-cluster [44]). On the other hand, this ordering is somewhat different from that calcuIated by.Ellis and Rosen [45] using the SCC DVM-Xoz approach [13]. (According to ref. 1451, t&e 8alg level falls in between the 5eg and 3tzg levels.) The differences between our results and those obtained in ref. [46] (without overlapping of spheres)
G.L. Gutsev. AA. Levin ISCFDVM-Xm
468 Table 6
Ionization potentials and oneelcctron energies for \VFs Level
One-electron
Ionization potentials
energies
&VI
bu) DVM-Xa DVM-XII superpos. SCF
SW-Xa
TS for ltlg
-0D031 0.0273 -0.1733
0.0123 -0.0344 -0.2355
0.23
-0.4621
-0.3936
-0.4426
-0.4749 -0.4786
-0.4104 -0.4120
-0.4308 -0.4539
-0.4915 -05216 -0~441 -0.5678
-0.4315 -0.4647 -05303 -0.5393
-0.4678 -0.4624 -0.5392 -0.5597
experim. [401
[461 14.75
-
15.18
15.35 16.07
15.23 15.72 16.58
16.8 17.2
18.32
18.43
18.62
18.81
are much mare numerous which is indicative of inap-
plicability of the “muffii-tin” potential to WF,. The results for WF, are given in tables 6 and 8. The calculations were done with the basis set constructed of the HF tungsten and fluorine ground state atomic functions and complemented by the 6p (W) AC’s with the exponent of 1.5 and 3s (F) AO’s with the exponent of 1 .O. Spherically symmetrical distributions were only used to approximate the electronic density_ Table 6 contains
one-electron
energies of the WF,
ground state and the values obtained by the SW-X& method [46] _The IP values c&dated from the orbital energies using the transition state for the 1 tr, level are also given. For comparison, the experimental P’s [40] are included. (No correspondiig data are given in ref. [45] _) The agreement with experiment for WFe is not so good as for MoFe though in this case also, the values cafcuhrted deviate from the experimental ones by not more than 1 eV. The populations of basis AO’s and atomic charges calcrdated by Mull&en’s population analysis summarized in table 8 have almost the same values as for MoFe The energy gap between the lowest vacant, 6tas, and highest occupied, It r* levels is 4.7 eV_ This value remains practically the same irrespective of whether it is obtained as a difference between the ground state one-electron energies or by SCF calculations involving the transition state. The experimental energy of the
and itsappIications
fmt optical transition is 72 eV [38] _It is not clear at present to what degree the discrepancy is due to relativistic effects and/or averaging over the molecular terms. The caIcuIations on UF, were carried out on the assumption of octahedral geometry of this complex with the U-F bond length set equal to 1.99 8, [47]. The value of the uranium exchange parameter (0.6920) was also taken from ref. [47] _ The weighted mean value of the two parameters, (1/7)(ou + &Y~) = 0.7309, was used in the calculations. The basis set included fluorine HF functions and HF functions of uranium in the 5f36d17s2 configuration. This set was complemented by the 7p(U) STO’s with the exponent of I.5 to 2.0 and 3s(F) STO’s with the exponent of 1.0 (like in the calculations of MO and W hexafluorides). Spherically symmetrical adjustment functions were only used to approximate the electronic density distribution; these are’given in table 1. The ordering of valence and lowest vacant levels obtained (table 7), lot,, >9a,a> lt,>2t,, >9tr, >4t2s>6egt coincides with that calculated by the SCC DVM procedure [48]. It differs from the orderings obtained in SW-Xcr calculations both with [49] and without ]47] overlapping of spheres. Our results are in good quahtative agreement with those of Rosen’s non-relativistic calculations [48] _ The atomic charge values and the 6p and 5f uranium orbital populations calculated in this work by Mull&en’s population analysis fall in between the corresponding values obtained by nonrelativistic [48] and relativistic [37] SCC DVM-Xa cakulations. The charge on ‘uranium was found to be 1.554 [48], 1.610 (this work), and 1.705 [37]_ The 6p orbital populations were 5.71,5.84, and 5.90, and the 5f orbital populations 3_20,3.08, and 2.75, respectively_ Table 8 also contains the poprdation data on other basis AO’s. The calculated first IP value is equal to 13.6 eV against the experimental value of 14.05 eV [40], whereas Rosdn’s nonrelativistic calculations gave the fast IP value of 12.1 eV_ The application of the relativistic scheme improves the result to 12.9 eV; the latter value is, however, by l-25 eV lower than the experimental one_ The peak assignment that may be suggested on the basis of our results (if an assignment on a nonrelativistic level of approximation makes ‘sense) is as follows: the fmt five peaks correspond to the lot,,, 9ar, + It,,, 2ta,, 9tt,, + 4t,,, and eg levels, respectively. The energy of
G.L. Gutsev. A_A_ Leti
/
SCF D VM-Xa and its applimtions
469
Table 7.
Ionization potentials for UFe, transition state for uppermost molecular level (in eV) SCF DVM-Xa present paper
SCC DS ref. 1371
level
IP’S‘ TS for 10rlu
symmetry Oh
Experim. 1401 IP’s values
level symmetry-
Ip’s TS for 12~~
Oh [373
Oh’
alg
13.11 13.97
t%
14.39
fzu
15.24
‘*6g kg 377” llY8”
tiu
15.64
4% g76u
12.92 13.93 13.95 13.96 14.71 14.77 14.77 15.27 15.28
%z
15.90
15.28 15.37
%
16.26
10r8u 1078g gY8g
16.14
till
127gu %8g I%,"
the first optical transition,
lOtI,
-t2az,,
calculated
as the difference of orbital energies, is equal to 1.38 eV (1.46 and 3.78 eV in nonrelativistic [48] and relativistic [37] SCC DVM-Xor cakulations, respectively) which differs considerably from the experimental value of 3.04 eV [SO] _As mentioned, underestimation of the optical gap width is characteristic of all metal fluoride
assignment
calculations
performed
on the level of a
14.05
t1u
tl$Z
.15.35
algV tzu
15.98
tzll= t2g
1658
eg
17.30
“non-spin-polarized” approxixnation including MoF6 where relativistic effects can hardly be manifest_
5. Conclusion_ Additional comments Some more points should be mentioned in conelusion. The self-consistent DVM-XLYprocedure
Table 8 AO’s population for MoFe, WF6 and UF6 MOF,
WFfi
6F
2s 2pn 2PO 3s
MO
4s 4P 4%
.
4d.Y 5s 5P
Charges onatoms
MO F
11.734 21.199 10.350 -0.066
‘6~
1.924 6.039
W
UF6 2s 2PT 2PO 3s 5.3
5P
1.522 2.590 0.366
5de 5dr 6s
.0.331
6P
1.229 -0.205
W F
21.323
2s 11.769 ?p!pr 22.078
10.484 -0.085
2PO 3s
9.683 0.081
6s
2.066
6~ 6d 5f 7s 7P
5.841 1.466 3.078 -0.047 -0.005
11.559
6F
1.935 6.001 1.455 2.488 0.381 0.273
U
1.464 -0.244
U
1.610 -0.268
410
G-L. Cutsev, AA.
Levin /SCFDVM-XU
described in this work and based on approximating the atomic HF densities by “analytical functions” is fairly flexible. Within this approach, the amount of computations is reduced, on the one hand, by decreasing the number of the basis functions (by using the “best” numerical AG’s) and on the other, by only once calculating the Coulomb potentials arising from adjustment distributions_ “Exact” orthogonahzation of basis sets makes the results of calculations less dependent on the integration scheme applied which also reduces computer time expenses while providing their high accuracy_ it is worthwhile mentioning that the programs for approximating atomic electronic densities, oA, may also be used to approximate any pA component, e.g. squares of radial HF functions. The computations were carried out on a BESM-6 computer (32 k-bytes on the fast core memory, 0.8 pzs cycle time)_ The following example helps to form an idea of computer time expenses. The data given below refer to the calculations of uranium hexafluoride (96 basis functions). With 2200 integration points and 12 SCF iterations, the calculations took I h 40 min BESM-6 computer time. Each next iteration cycle after the first one took 5 min. The set of DVM Xor programs described above is thus fairly economical from the point of view of computation time. We believe it also provides the highest accuracy at present attaina5le with non-relativistic calculation schemes. References
[I] (a) K. Siegbahn. C.
Nordlimg, %. Johansson, J. Heldman, P.F. Heden, K_ Harmin, U. Gel& I. Bergmark. L-0. Werme, R. Manne and Y, Baer, ESCA applied to free molecules (North-HoBand, Amsterdam, 1969); (b) D_N’. l-mm, A.D. Baker, C. Baker and CR. Brundle, hlolecuhu photoelectron spectroscopy: a handbook of He 584 PE spectra (Wiley. London, 1970). [2] CCJ. Roothaan, Rev. hfod. Phys. 23 (1951) 69. [3] J-C. Slater, Phys. Rev. 81 (1951) 385. [4] K-H_ Johnson, J_ Chem_ Phys. 45 (1966) 3085_ [S] K.H. Johnson, Advan. Quantum Chem. 7 (1973) 143.
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[ 101 D.R. Hartree. The calculations of atomic structures (Wiley, New York, 1955). [ 1 I] F. Herman and S_ Skillman, Atomic structure calculitions (Prentice-Hall. En&wood Cliffs, 1963). [ 121 F-IV_Averill and D.E. Ellis, J_ Chem. Phys. 59 (1973) 6412. [13] A. Rosin, D-E. Ellis, H. Ada&i and F-IV_ Averill, J. Chem. Phys. 65 (1976) 3629. [ 141 A. R&n and DE_ Ellis, Chem. Phys. Letters 27 (1974) 595. [15] A. Ros& and D.E. Ellis, J. Chem. Phys. 62 (1975)
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itsapplicntions
[4& A: Roskn,Chem. Phys. Letters 55 (1978) 311. [49] M. &foxingand J.W. Moskowitz, Chem. Phys. Letters 38 (1976) 185. ISO] R. McJXarmid, J. Chem. Phys. 65 (1976) 168.
471
