4496
.%RNOI,D
hf. KAROA N D LELAND c.
Vol. 80
ALLEN
LINCOLN LABORATORY, &fASSACHUSETTS INSTITUTE O F TECHNOLOGY, AXD THE MATERIALS RESEARCH LABORATORY, ORDNANCE MATERIALS RESEARCH OFFICE]
[CONTRIBUTIOS FROM T H E
Configuration Interaction in the Hydrogen Fluoride Molecule1 BY
ARNOLD hf. R A R O AN ~D LELAND c. L 4 r , L ~ N 3 RECEIVED JANUARY 30, 1958
A self-consistent field molecular orbital (SCF MO) calculation of the electronic structure of HF in the ground state and a t the observed internuclear separation has been made. The MO’s were approximated as linear combinations of atomic orbitals (LCAO’s), aiid all electrons and the exact Hamiltonian were used. Both a single determinant M O and a configuratiou interaction (CI) treatment were carried out with single exponential Slater AO’s as basis functions. The parameters for the single exponential basis set were chosen t o be identical to those used by 4.B. F. Duncan, and comparison is made with his work and that of others. The results including C I give 34.8% of the observed binding energy ilud a dipole moment of 4-0.6938 Debye unit (experimental value = +1.74 D).
Introduction Recently a number of simple diatomic molec u l e ~ ~ -have ’ been studied within the framework of the SCF LCAO-MO treatment formulated by Iioothaan.s A calculation of the HF molecule has been made by Kastlerg using a valence bond configuration to represent the ground state and superimposing on this a CI treatment. Discrepancies between the results of Duncan and Kastler, as well as other inconsistencies,1° led us to reconsider the single determinant SCF method and to re-evaluate all integrals in a consistent manner. This work has been extended to include CI and an electron distribution analysis. Atomic Orbitals and Method of Calculation.12T h e atomic orbitals are those used by Duncan
xi
s = ( 1 s ) ~= ( ~ ~ 3 / ~ ) exp( ~ / 2 -c,i.p) S E 1 . 0 2 9 5 1 ( 2 ~ )-~ 0.244726(l~)p (1) [ ( 2 S ) F = (c26/37r)1/z f‘F exp(-cCZf‘F)] (c25/7r)‘Iz f‘F CXp( -C2IF) PIo (COS 8) Z %E (2pO)F = (2p,)m = ( ~ ~ 5 / 2 5 ~ rF) ~exp( / 9 -cz~F)P1l(cos0) exp(+i+) ~1 8.7; ~1 = 2.6
These AO’s form the basis for an orthonormal set of MO’s, in particular, that set which minimizes the energy of the single Slater determinant representing the ground state molecular wave function. (1) T h e research reported in this document was supported by the U. S. Office of Naval Research, the Army, Navy and Air Force under contract with Massachusetts Institute of Technology, and the Ordnance Materials Research Office, Department of the Army. (2) Staff member, Lincoln Laboratory, Massachusetts Institute of Technology. (3) Visiting Fellow, Department of Physics, Massachusetts Institute of Technology, and Materials Research Laboratory, Ordnance Materials Research Office. (4) C. W. Scherr, J. Chem. Phys., 23, 569 (1955). (Nd. (5) R. C. Sahni, ibid., 25, 332 (1956), (BH). ( 6 ) J. E. Paulkner, ibid., 27, 369 (1957). (Lid. (7) A. B. F. Duncan, THIS J O U R N A L , 77, 2107 (1955), (€IF). ( 8 ) C. C. J. Roothaan, Rev. Mod. Phys., 2 3 , 69 (1951). (9) D. Kastler, J . chim. phrs.. 6 0 , 556 (1953). (IO) A. C. Hurley has pointed out in conversation that the molecular energy obtained b y Kastler is higher than that reported by Duncan, even though Kastler‘s results were based on a somewhat more complete calculation involving C I a n d essentially the same basis or-. bitnls as were used by Duncan. Roothaan“ also found t h a t with more variation parameters he obtaiued a higher F atom energy than that cnlculated by Duncan. (11) C. C. J. Roothaan, Laboratory of hlolecular Structure and Spectra, T R 1955 (University of Chicago, 1 Oct. 1954-30 Sept. 1955) p . 44. (12) Atomic units are used throughout. (10 unit of length E 0 . 5 2 9 3 A. Rydberg E unit of energy G 13.602 e.v.
=
All integrals were evaluated numerically on the MIT digital computer Whirlwind using programs written by F. J. C0rbat6.’~.’~The two-center oneand two-electron integrals were evaluated by the method of Barnett and Coulsonls using the standard method of expanding an exponential function about another center. As programmed by Corbat6 this expansion in general form is r h n , (k,r‘) =
where
and where the expansion functions, aj, are appropriate combinations of spherical Bessel functions of half-integer order and imaginary argument. The spherical polar coordinates ( r , e, 4 ) and (r’, e’, 4‘) refer, respectively, to the origin and to the center displaced by a distance a. The fluorine functions were generated using the limiting case a = 0, and the hydrogen 1s function was expanded about the fluorine center with a = I .7328 atomic units, the observed equilibrium separation for HF. Simpson’s rule integration was employed on a 101 point mesh. In general accuracy of the integrals was governed by the error in the integration method, so that the integrals are only tabulated to five places in the Appendix. The one-center kinetic energy integrals were evaluated analytically.16 The Roothaan procedure is an approach to a molecular HartreeFock solution in which the linear coefficients of a finite predetermined set of oneelectron basis orbitals are evaluated variationally. The scheme has been described in detail elsewhereR and yields a set of orthonormal MO’s expressed as a linear combination of the chosen basis -40’s
m,
=
-T “ipXp
P
(3)
(13) F. J. Corbat6, J . Che?n. Phys.. 24, 452 (1956). (14) F. J. Corbat6, Quarterly Progress Reporl, Solid-state and Molecular Theory Group, M.I.T., April 15, 1956, p. 33. ( I j j M. P. Barnett and C. A. Coulson, Phil. Trnizs., 8 2 4 3 , 221 (1951). (18) A comparison with Duncan’s integrals shows a rather systematic discrepancy in the second and third decimal place for those integrals involving ( 2 s ) ~and slightly larger differences in a few other cases.
CONFIGURATION INTERACTION IN THE HYDROGEN FLUORIDE MOLECULE
Sept. 5, 1958
where the linear coefficients, aip, are solutions of the matrix equation (15
+ G)ai = e,Sai
(4)
4497
Debye unit. The most recent experimental value has been obtained by G. A. Kuipers of the Eastman Kodak Company1gand is 4-1.74 D. The negatives of the one-electron parameters approximate the ionization potentials of the molecule (Koopmans’ theorem). From our results the first ionization potential would be about 12.7 e.v., but there are no experimental values with which to compare. Duncan’ has given a brief summary of existing experimental excitation energies. Configuration Interaction.-The single determinant molecular wave function can be only an approximation to the wave function representing the actual electron distribution in the ground state. A better approximation can be made by supplementing the ground state with a linear combination of configurations of appropriate ground state symmetry and multiplicity
A program has been written for this process by A. Meckler and R. K. Nesbet for the M.I.T. digital computer, Whirlwind, and requires for input the one- and two-electron integrals and an initial estimate of the density matrix, p, ( P A ” = 2 i a ~ a i , + ) . Equation 4 must be solved iteratively, and the program accomplishes this by successive density matrix alterations and Hamiltonian matrix diagonalizations until a stationary density matrix is realized. The final SCF alp, pxV, the one- and two-electron integrals transformed to the orthonormal SCF MO basis, and the one-electron energy parameters are obtained as output from the ~ 0 m p u t e r . l ~ The Best Single Determinant Wave Function.I n Mulliken’s notation the electronic configuration of the ground state of H F is (1a)2(2r)2(30)2= Cmr+r r (7) ( T + ) ~ ( T - ) * ,I 2 + . The orthonormal set of MO’s which solve eq. 4 are and applying to the undetermined coefficients, mr, lu = -1.000147~ - 0.0158415 - 0.002431~+ 0.004600h the standard variational treatment for minimizing 2u = -0.021903~ 0.9050203 4-0.089717~+ 0.1614921~ the energy. By choosing the SCF MO’s already determined, 3u = +0.028309~ + 0.4593255 - 0.6846952 + 0.579121h the leading term in the expression will be the single 46 -0.053023s - 0.543582s - 0.808494~+ 1.048445h determinant ground state function; and for this .+ = (2fit)F (5) choice the leading coefficient will also be maximized. The orbital energies, the total molecular energy, and The digital computer program for the Roothaan the resulting dissociation energy are given in Table scheme also constructs an unoccupied orthonormal MO (4r), and this is used in the CI. All configuraI. tions which can be formed from the set of MO’s a t TABLE I our disposal are included, with the restriction that CALCULATED ENERGIES (RYDDERGS) the l o orbital remains filled.2o One can show from Orbital energies E second-order perturbation theory that little interaction occurs between highly excited states, formed e(lu) - 62.290286 by promoting 1o electrons, and the ground state. e(2u) - 2.962522 The ground state and the six excited configura- 1.136526 e(3u) tions are listed in Table I1 with the coefficients, m,, 44u) 0.943008
+
-
0,934659 -209.355025 10.38‘7812 -198.967213 198.866503 - 0.100710 (1.3699 e.v.) Dissociation energy (exptl.)” - 6 . 0 8 e.v. A . C. Hurley, Proc. Phys. Soc. (London),A69,301(1956). C(R*:)
Total electronic energy Nuclear repulsion energy Total molecular energy - (H F atomic energy) Dissociation energy
+
The energy of the F atom was calculated using the same AO’s as were used in the molecular calculation. These results are in agreement with those of M. Krauss of the National Bureau of Standards,’B who has also essentially repeated Duncan’s single determinant calculation (the only difference being his use of 8.6s instead of 8.70 for the ( 1 s ) ~exponent). He obtains a total molecular energy of - 198.956 Rydbergs. With the fluorine nucleus taken a t the origin, the dipole moment is given by p =
- e f+*?+ d r
+ ero
(6)
where 4 is the antisymmetrized product of the occupied ground state MO’s and ro is the equilibrium internuclear separation. It is found to be +0.8494 (17) See L. C. Allen, Solid-state and Molecular Theory Group, M.I.T., January 16,1957,p. 10,for a description of the use and format of this program. (18) Private communication.
TABLEI1
MOLECULAR WAVEFUNCTION FROM CI Configurations 4, $1 (1u) 2 (2uj 2(3u)2 ( 7r +) 2(5T -) 2 42 (10 )y2u) * (40.)2 ( R +) 2 ( R -) 2
Coeffs. ,nr for seven configs.
BindinE energy from C I (Rydbergs)
0 990357 -0.10071 - .I25011 - ,14123 +3 (1u) (3u)* (4u) * (?r+) 2 ( R -) 2 - ,028254 ,14648 ~ ~ ( ~ u ) ~ ( ~ u ) ( ~ u ) ( ~ u-) ~,038659 ( ~ + ~ ~-( R .15313 - ) ~ ~ 6 ( 1 ~ ) 2 ( 2 ~ ) 2 ( 3 ~ ) ( 4 ~ ) ( R+ ) 2.017433 ( ~ - j 2 - ,15355 ~ p e ( 1 ~ ) ~ ( 2 ~ ) ( 3 ~ ) ~ ~ 4 u , (-a ’ ,003143 ) ~ ( ~ - ) ~ ,16364 ,030957 - ,15561 +,(lu)a(2u)2(3u)2(4u)* ( i T +) (7r -)
-
-
resulting from the complete CI treatment. The third column gives the binding energy resulting from the interaction of two through seven configurations added in order proceeding down the column. The first entry is the value obtained for the single determinant approximation. (19) Private communication. (20) Configurations $6 and Qa formed by excitation of a single electron from the ground state have no first-order interaction with the ground state, and. therefore, little effect in lowering the energy. They have been included so t h a t an exact comparison can be made with another calculation of H F (to be published) which has been carried out over a wide range of internuclear distances using HartrePFock AO’s on F. At larger internuclear distances second-order interactions become increasingly important.
ARNOLD M. KAROAND LELANDC. ALLEN
4498
Appendix
One-electron integrals Orbitals
J
XQ
XP
Overlap ~ ~ (
Kinetic energy ~
)
s~x 1 (~1 ) ((
1.00000 0.00000 1.00000 0 0 1.00000 0.05485 0.47227 0.29894 1.00000 1.00000
s s
ss ss z s 2s z z
h s h S h z
h h T T
Vol. 80
-~$)xa(l)dr ) ~ ~
37.84500
---
Potential energy----1 1 J"xD(l)Gxa(l)dT JxD(l)G~a(l)d~ moment JXD(1) (rFcos@)xQ(l)d+
8.70009
- 10.50364
- 1.20734
4.06860 0 0 3.38000 0.00524 .13199 .22198 .50000 3.38000
1.44775 0 0 I . 30001 0.23920 ,40702 ,23700 ,52782 1.30001
0.57711 . 00000 ,57274 .01618 ,17883 ,64933 ,03266 ,36812 ,37145 1.00000 0.53481
0 0 0 04858 ,55964 0
,00281 .26792 ,48681 1.73288 0
Two-electron integrals 1
SXP(l)XO(l)& xa(2)xt(2) ~ T I
XDXQY"XI 5
5 s
.cs s s s s s s S S S s s s
s s s s .c .s
2
z
s
z
sz
z .s z
s
s
.s
" S S
z z
.ss
.ss
z r z z z
2
h s s s h s ss h s h s z s hs2.S hS 2 h s h s h S S S h .? 5 h .y LS .c
ss
s
}$
sz
XDXuXrXr
Z
5.43754 -0.51635 0.08215 1 ,29513 -0,02149 ,91807 ,03104 ,04061 ,19954 1.29076 -0,01152 0.92945 1.01767 0.19482 - ,01213 ,06655 ,00120 ,00210 ,06836 ,00826 . 4 1525 ,00033 35848 ,00599
5
s
sz s
h h h h h
S S S z
It
z
/LZ
z h h s
0.04399 ,36903 ,02314 ,16847 ,23674 - ,00047 ,20940 . 0 1567 ,09370 ,22812 ,01372 ,11716 ,11218 ,52699 - .00062 .50052 ,01060 ,10289 ,52783 ,02940 ,27157 ,23059 ,62500
z s
.S s
.s s
ss
h z z s h z z .s h z z z h z h s h z h S h z h z h h s s h h S s h h S S h h z s h h z S h h z z h h h s 1zhh.S h h h z h h h h
The complete dipole moment is given by 1.1 = -eJ+*?$
dr
+ erg
(8)
and is found to be 4-0.693s Debye unit. Electron Population Analysis.-Mulliken21 has shown that qualitative infomation concerning the electronic charge distribution can be obtained from an analysis of the overlap and atomic populations defined by the relations m(i; P K , 41) 2X(i)aip,aiq1(xpk IxcI,) n(i) =
C C m(i; pk, 41) PI
Pk
pk, 91)
%($'k,
41) =
n =
C n(i) i
N(i;
pk)
N(i)aiPk2
i'k
..
~
aiPka'Ql(XPklXQl)
i
N ( i ; pk)
N(i) = N($'k)
populations (9)
=
+ N(i) k
l
lations
c n(%; i
_
'
1
1
J-Xmxd1)~ xd2) xt(2) driz
Pk)
_
(21) R . S . Mulliken, J . C h e m . P h y s . , 23, 1833 (1955).
(10)
sxP(l)x.(l)~z xa(2) ~ t ( 2~ ) T I Z
XrmXnX9X1
as
0.03104 ,04061 ,19954 ,05484 ,01255 ,06812 ,00955 ,02772
7rs
T S T S 7r
s
7r
s
a 3 a;
T i t 77s
7rh
7r.Y
7 r h
7rz
h
17
h
7i
x s .T
71
7r 7r
1 ,29076
-0.01152
ss
irirss
,92945 ,90798 ,06811 ,35825 ,20203 ,48868
z 7r7ih.s 7r 7r h 5 7r7rTi
P i r h Z P T
h h .
7riF 7r
7T
7r'- 7 r -
i
7r
0.10969
7r+
U . 96283
where aipkis the SCF LCAO coefficient of xPkill the ith filled MO, N ( i ) the number of electrons in the ith MO, (xpr1 xSJ the overlap integral for functions XPI Xs on centers k and 1- The bonding and antibonding nature of the ith MO is associated with the sign and magnitude of fi(i). The differin the ence between the number Of 2s free F atom and the effective number in the molecule is said to define the amount of 2s-2po promotion. The results of the electron population analysis for the single determinant wave functions are given in Tables I11 and IV. Discussion of Results The total energy found b y Duncan for the ground state configuration (- 199.4196 Ryd.) is considerably larger than our value (- 198.9672 Ryd.). The difference probably can be accounted for by the previously mentioned discrepancies in the integrals. The molecular energies reported here are consistent with the results of a number of other calculations also using simple exponential Slater AO's. 4-6,9 In particular, Kastler's total energy (- 199.0524 Ryd.) and dipole moment
Sept. 5 , 1958
CRYSTAL STRUCTURES OF
R.4RE
EARTHCARBIDES
4499
showed that with his basis set the energy of F- was higher than the F atom.22 Qualitatively this may explain the small dipole moment obtained, since W(i) N ( r ; 1s) N ( i ; 2s) N ( i ; 29) N(i; H) 2.00008 0.00043 0.00001 -0.00054 1,99998 the high total energy of F- implies inadequate 1u .19848 1,99998 charge on the fluorine center in the molecule. The 0.00057 1.77617 0.02476 2u 0.17071 1.17469 .65479 1,99999 fraction of the binding energy which is calculated 3u -0.00020 1.94731 1.19946 .85273 51.99995 and the amount of improvement contributed by N(pk) 2.00045 CI is very nearly the same as that found in similar s - p promotion = 0.0522 SCF calculations. From the SCF LCAO coefficients as well as the TABLE IV electron distribution analysis the 1u and 2u MO’s OVERLAP POPULATIOSS I N HF are seen to be formed almost completely from Is and n ( i ; Is, H) n ( i ; 2s, H) n(i; 29, H) n(i) 2s fluorine AO’s, respectively, and concentrated -0.00001 -0.00116 -0.00101 -0.00014 lo ,29263 about the fluorine nucleus. The overlap popula,27609 ,01732 2a - ,00078 ,47414 - .03196 tion is large for the 2u MO and i t is therefore bond3u - ,00360 - .50250 .49145 .25951 ing. The electron population of the 3 u MO is %($Jk,ql) - .00539 - .22655 distributed around both nuclei, with the overlap %(totaloverlap population) = 0.25951 population such that this MO is slightly antibonding. These results are in agreement with what has (+0.9261 D) are quite near the results of our CI been found for Nz, BH and treatment. (Kastler’s CI is equivalent to ours h TADDED ~ IN ~ PRoOF.-An ~ ionization potential of 12.6 since Is = 1u and because the omission of ~ $ 7 and the e.v. has been reported from the absorption spectrum of HF slight differences in exponential A 0 parameters in the vacuum ultraviolet in good agreement with our preproduce a negligible effect.) The calculated molecu- dicted value of 12.7 e.v. ( R . P. Iczkowski and J , L. Marlar energy is about two Rydbergs above the ob- grave, Symposium on Molecular Structure and Spectroserved value. An appreciable fraction of the dif- scopy, The Ohio State University, June 16-20, 1958). ference can be accounted for by the failure of Slater (22) Similar results have been obtained by L. C. Allen, ref. 17,p. 4. AO’s to represent the F- ion, with a resulting poor LEXINGTON, MASS. description of the F-H+ molecular state. Kastler WATERTOWN, MASS. TABLE I11 A 0 POPULATIONS IN HF
[CONTRIBUTION No. 570 FROM THE INSTITUTE FOR ATOMICRESEARCH, AND DEPARTMENT OF CHEMISTRY, IOWA STATE COLLEGE. WORK WAS PERFORMED I N THE AMESLABORATORY O F THE IT.s.ATOMIC ENERGY COMMISSION]
The Crystal Structures of Some of the Rare Earth Carbides BY F. H. SPEDDING, K. GSCHNEIDNER, JR.,’ A N D A. H. DAANE RECEIVED DECEMBER 13, 1957 A study of the rare earth-carbon systems has been made. The existence of the reported Lac*, CeC2, PrC2, NdC2 and SmCI, and the Ce& has been confirmed. The existence of the tetragonal CaC2 type structure for the other rare earth dicarbides has been shown. The lattice constants of these compounds decrease in a regular fashion, except for the YbC2, whose lattice parameters lie between those of Hoc2 and ErC2. The body-centered cubic PuzCs type structure has been found to exist in all of the rare earths from La to Ho. The lattice constants decrease in a normal manner, except for Ce2C3, whose lattice parameter is smaller than would be expected. A new rare earth carbide has been found, R&, which is similar to the face-centered cubic KaC1 type structure, except that it is deficient in carbon. This compound appears to exist over a to YCn.40. This compound has been found range of composition, i.~.,in the case of yttrium it was found to vary from Yc0.2~ to exist in the rare earth-carbon systems of Sm to Lu, and the lattice parameters decrease in a regular manner. No X-ray evidence was found for the existence of this lower carbide in the La-, Ce-, Pr- and Nd-C systems.
Introduction The crystal structures of several tetragonal rare earth dicarbides (LaC2, CeC2, PrC2, NdC2 and SmC2) and a hexagonal YC2 have been known for several years2 Recently the existence of other rare earth carbon compounds, CeC,a CezC33and CeC3,4 have been reported. CeC was reported to be face-centered cubic (NaC1 type) with a lattice constant of 5.130 A.; Ce2C3was found to be bodycentered cubic (plutonium sesquicarbide type5) with (1) Los Alamos Scientific Laboratory, Los Alamos, New Mexico. Based in part on a dissertation submitted by Karl Gschneidner, Jr., to the Graduate School, Iowa State College, in partial fulfillment of the requirements for the degree of Doctor of Philosophy, 1957. (2) M. von Stackelberg, 2. Elektvochem., 37, 542 (1931). (3) L.Brewer and 0. Krikorian, J . Electrochem. SOC..103,38 (1956) (4) J. C . Warf, paper presented a t the ACS Northwest Regional .Meeting, Seattle, Washington, 1956; Abstracted in Chem. Eng. News, 34, 3446 (1956). ( 5 ) W.H. Zachariasen, Acta Cryst.. 6 , 17 (1952).
a0 = 8.455 B.,while no structural data have been reported for CeC3. I n a study of the lanthanumcarbon systems only two compounds were found, La2C3 and LaC2. The structural details of these two carbides were determined using X-ray and neutron diffraction techniques.’
Experimental Materials.-The rare earth metals, 99.9% pure, were prepared by metallothermic reduction methods described by Spedding and Daane.8 In general, the metals were used in the form of filings or finely divided chips since it was easier to achieve homogeneous alloys by arc melting compressed cylinders of the metal filings and carbon powder. The carbon was prepared in the form of a powder by turning spectroscopic electrodes on a lathe.
-~
(6) F. H. Spedding, K . Gschneidner, J r . , and A. H Daane, “The Lanthanum-Carbon System,” to be published. (7) M . Atoji, et al., THIS JOURNAL, 80, 1804 (1958). ( 8 ) P . H.Spedding and A. H. Daane, J . Metals, 6, 504 (1954).
