A FLOATING SPHERICAL GAUSSIAE ORBITALMODELOF MOLECULAR STRUCTURE
1289
A Floating Spherical Gaussian Orbital Model of Molecular
Structure. 111. First-Row Atom Hydrides by Arthur A. Frost Department of Chemistry, Northwestern University, Evanston, Illinois
(Received September $6,1967)
The FSGO model is applied to the first-row hydrides and several isoelectronic ionic species as follows: BeHz, CH3-, HzO, NH2-, H,F+, H F , LiH2-, BH,+, BH3, BeH3-, CH3+, CHI, BHd-, PI;Hd+, Ha02+, "3, OH-. The trends in the bond lengths and electronegativities are correctly predicted. Furthermore, the bond lengths are predicted quaiititatively within an accuracy of 6%. Bond angles in NH3 and HzO are low by about 17y0,calculated angle in NH3 being 87.6" compared with the experimental value of 106.6', while for HzO the values are 88.4' compared with 104.5'.
&of,
Introduction The floating spherical Gaussian orbital model (FSGO) as presently used applies to singlet ground states of molecules having localizable orbitals. Each localized orbital is represented by a spherical Gaussian function
with variable "orbital-radius" pi and with variable position of its center. The total electronic wave function is a single Slater determinant of closed shells. The separate orbitals are nonorthogonal with overlapmatrix 8. The inverse overlap matrix, T = 8-l) is involved in the energy formula
By minimizing the total energy with respect to all parameters, the orbital radii and positions and the nuclear coordinates, the electronic and geometrical structure is predicted. Paper I' of this series details the calculations and discusses the result for the LiH molecule at length. Paper IIz presents results of oneand two-orbital systems, including among others the He and Be atorns and the molecules and molecule ions H2, He22+,HeH+, H3+, H42+,HeH-, and BeH+, and the Hez interaction. The present paper contains results of calculations on systems of more general chemical interest, including compounds of the first-row elements from Li to F. Possible compounds of interest are the homonuclear diatoms Li2-Fz and the diatomic hydrides LiH, BeH, BH, CH, NH, OH, and HF. The first series raises complications because of the rapidly increasing number of electron pairs and orbitals as one proceeds toward Fz with its 18 electrons. Also there is the particular difficulty with Bz and 0 2 , which have ground-state triplets. The present calculations which involve only a single closed-shell Slater determinant would presumably
provide approximations to the lowest singlet states of such molecules. The diatomic hydrides ReH, BH, CH, NH, and OH can be calculated on this model, but are chemically unstable with respect to the higher valence species BeH2, BH3, CHI, NH3, and HzO. It is this latter series including also H F and the previously discussed' LiH that will be described now. The increasing number of hydrogen atoms as CH, is approached adds little to the computation time, as this depends primarily on the number of electrons which in the molecules from CH, to H F is constant at ten.
Computational Procedure and Results Preliminary results on this series of molecules have previously been reported.3 LiH with its four electrons and two orbitals has been discussed in papers I and 11. ReHz was assumed to be linear and symmetrical. Of the three localized orbitals for the six electrons, one is an inner shell centered on the Be nucleus while the other two are bonding orbitals located on the molecular axis and equidistant from the Be. The energy was minimized with respect to variations in all the possible parameters consistent with the symmetry, namely four: the orbital radius of the inner shell, the orbital radius for each bonding orbital both considered equal, the position of a bonding orbital, and the bond distance Be-H. Results for the BeHz isoelectronic series are in Table I. Figure 1 shows the orbitals and nuclear positions schematically. BHBwith four orbitals was assumed to be planar with trigonal symmetry, therefore D3h. There were four (1) Paper I: A . A. Frost, J. Chem. Phys., 47, 3707 (1967). (2) Paper 11: A. A. Frost, ibid., 47, 3714 (1967). (3) A. A . Frost, B. H. Prentice, 111, and R . A. Rouse, J . Amer. Chem. SOC.,89, 3064 (1967). Also presented in part a t t h e Symposium on Computers in Chemistry, San Diego, Calif., June 1967, and a t t h e 154th National Meeting of t h e American Chemical Society, Chicago, Ill., Sept 1967.
Volume 78, Number 4 April 1068
1290
ARTHURA. FROST
Table I1: FSGO Model Calculations for BHa and Related Ions"
H I bohr
Figure 1. Schematic diagram of localized orbitals in the BeHz sequence. Each spherical Gaussian orbital is represented by a circle drawn with its radius equal to the orbital radius p.
Molecule
-E, hsrtrees
BeHs-
13.655
BHs
22.297
CHa +
33.300
Orbital radii, Bohrs
0.511 2.199 0.399 1.851 0.327 1.590
Bond orbits1 distance t rom center, Bohr8
Bond length, Bohrs
2,462
2.878
1.672
2.353
1.150
2. 062b
a Atomic units. * Compare this with the value 2.05 au calculated by the SCF-MO-LC(LCG0) method: G. von Bunau, G. Diercksen, and H. Preuss, Intern. J. Quantum Chem., 1, 645 (1967).
Table I11 : FSGO Model Calculations for CHI and Its Isoelectronic Ions" Table I : FSGO Model Calculations for BeHt Isoelectronic Series"
Molecule
-E, hartrees,
LiHz-
7.026
BeHz
13,214
BHz +
21.563
Orbital radii,b Bohrs
0,707 2.524 0.510 2.108 0.398 1.794
Bond orbital distance from center, Bohrs
Bond length, Bohrs
3.287
Molecule
-E, hartrees
BH4-
22.692
3.514'
CHI
33.992
2.107
2.669
NH4+
47.893
1.438
2.292
H4O2+
64.527
'
'
First value for inner shell, second for bondAtomic units. ing orbital. ' A calculation by H. Preuss and G. Diercksen, Intern. J. Quantum Chem., 1, 631 (1967), with their SCF-MOLC(LCG0) method yields 3.5 =t0.2 au. a
parameters varied as in the case of BeH2, namely: the orbital radius of the inner shell, the orbital radius of the bonding orbitals, the distance of the bonding orbitals from the B nucleus, and the B-H bond distance. Table I1 presents the calculated results for BH3 and the isoelectronic CH3+and BeH3-. CHI with five orbitals was assumed to be tetrahedral. Again there were just four parameters defined in a similar manner to those used for BeHz and BH3. The isoelectronic ions BH4-, NH4+, and H402+ were calculated in the same manner by merely changing the charge on the central nucleus. Table I11 shows the results. The NH3 molecule and its isoelectronically related ions CH3- and H30+ constitute a much more involved calculation because of the lowered symmetry. Assuming CBVsymmetry, let the heavy nucleus be at the origin of a set of Cartesian coordinates with the x axis The Journal of Physical Chemistry
Orbital radius, Bohrs
0,400 1.970 0.328 1.694 0.278 1.469 0.240 1.276
Bond orbital distanae from center, Bohrs
Bond length, Bohrs
1.905
2,493
1.256
2.107
0.795
1.876
0.473
1.756
" Atomic units. being the threefold axis. Let one H nucleus and the corresponding NH bonding orbital be at arbitrary positions in the yx plane and the other nuclei and bonding orbitals at equivalent positions out of the plane. The inner shell and lone-pair orbitals are assumed to be on the x axis with variable x coordinates. All together there are then three parameters for orbital radii, four parameters for orbital positions, and two parameters for nuclear positions, making a total of nine parameters in the variation problem. Results are shown in Table IV. The HzO molecule and related ionic species are assumed to have Czv symmetry with the z axis as the Cz axis of the molecule and the three nuclei in the yz plane. The heavy nucleus is at the origin. Variable parameters would then include the y and z coordinates of one H nucleus, the y and z coordinates of a bonding orbital, the x coordinate of the inner shell orbital, the z and z coordinates of a lone-pair orbital assumed to be in the zx plane, and the orbital radii of each of the three different kinds of orbitals. When the compu-
A FLOATING SPHERICAL GAUSSIAN ORBITALMODELOF MOLECULAR STRUCTURE Table IV : FSGO Model Calculations for NHa and Related Ionsafb CHa-
Negative total energy Orbital radii Inner shell Bonding Lone pair Orbital positions Inner-shell z coordinate Lone-pair z coordinate Bonding in yz plane y coordinatle z coordinate H nuclei positions H in yz plane y coordinate z coordinate AH bond length HAH bond angle
Table V: FSGO Model Calculations for H20and Related Ionsagb NHs
Ha0 +
33.315
47.568
64.647
0.327 1,899 2.032
0.277 1.554 1.627
0.240 1.297 1.332
0.0008 -0.068
0.0007 0.160
0.0006 0.157
1.079 0.824
0.611 0.515
0.368 0.329
1.823 1.374 2.283 87.5'
1.527 1.147 1.910 87.6'
1.390 0.979 1.700 90.2'
' Axis of molecule is z; heavy atom at origin; one H atom and bonding orbital in yz plane. distances in Bohrs.
1291
Atomic units: energy in hartrees,
Negative total energy Orbital radii Inner shell Bonding Lone pair Orbital positions Inner shell z coordinate Lone pairs x coordinate z coordinate Bonding y coordinate z coordinate H nuclei positions y coordinate z coordinate AH bond length HAH bond angle
NHz-
H20
HaF *
46.791
64,288
84.874
0.277 1.618 1.660
0.240 1,308 1.331
0.212 1.097 1.104
0,0006
0.0005
0,0004
(f0.100) - 0.0039
( f0,100)
( f O .100) 0.020
A0.656 0.726
AO. 369
0.426
f0, 207 0.249
11.345 1.431 1.964 86.4'
f l .161
f l . 103
1,195 1.666 88.4"
1.061 1,531 92.2'
0.022
a Axis of molecule is t; heavy atom is a t origin with molecule Atomic units: energy in hartrees, distances in in yz plane. Bohrs; values in parentheses held constant.
'
tation is allowed to proceed toward a minimum in the total energy by the variation of all ten parameters, the calculation soon breaks down owing to the tendency of the two lone-pair orbitals to coalesce. As these orbitals approach each other their overlap integral tends toward unity, causing the overlap matrix to become nearly singular. The inverse overlap matrix needed in the energy calculation1 then "blows up" causing too great a loss in significant figures in the energy. To get around this difficulty, the x coordinates of the lone-pair orbitals were held constant at values such as k0.4,1 0 . 2 , and 10.1, while the nine other parameters were varied. I n this manner it was found that the energy and the geometrical parameters approached limiting values rapidly enough so that with x = hO.1 sufficient accuracy was attained. At this pair of values such that the centers of the lone-pair orbitals were separated by 0.2, the calculation proceeded satisfactorily, even though the overlap integral between these orbitals was approximately 0.99. The actual limiting case of the coalescence of the two like orbitals was handled by replacing the two orbitals by their sum and difference. In the limit the sum is an s orbital and the difference a p orbital. Details are given in the Appendix. Such a linear transformation of basis orbitals in a closed-shell Slater determinantal wave function causes no change in the many-electron wave function, but does make possible a more accurate calculation of the energy since the s and p orbitals are orthogonal. The energy for a fixed set of parameters was calculated by the POLYATOM program4 confirming the results obtained with spherical lone-pair orbitals at a small finite separation. Results for these molecular species are shown in Table V.
The final member of the hydride series is HF. This molecule and the isoelectronic species OH- were calculated assuming the inner shell and bonding orbitals were on the internuclear x axis and that the three lone-pair orbitals were located about this axis with C ~ Vsymmetry. The complete minimization problem, while preserving this orbital symmetry, would involve the variation of eight parameters. There are three different orbital radii, three different z coordinates for orbital positions, one off-axis y coordinate for a lone-pair orbital arbitrarily restricted to the yx plane, and the x coordinate of the H nucleus or bond length taking the 0 or F nucleus at the origin. Again there was a tendency for the lone-pair orbitals to coalesce on the molecular axis. It was necessary to go through a limiting procedure where the off-axis distance of the three lone-pair orbitals was held constant at successively smaller values, all other parameters being varied in each case to minimize the energy. The calculation was still valid with the orbitals off the axis by 0.2 Bohr where the overlap integrals were 0.95. At this distance the limiting structure appeared to be obtained with sufficient accuracy. Results are in Table VI. The computations were carried out on a CDC3400 computer. I n a typical calculation the first variation involved 10% changes in the starting values of the parameters. The step size was diminished successively (4) J. G. Czismadia, M. C. Harrison, J. W. Moskowitz, and B. T. Sutcliffe, Theor. Chim. Acta, 6 , 191 (1966). The POLYATOM program was obtained from the Quantum Chemistry Program Exchange, Indiana University, Bloomington, Ind.
Volume 73, Number 4
April 1968
ARTHURA. FROST
1292
to the order of O . O O l ~ owith the energy becoming constant to eight significant figures. The most time-consuming calculation was that of HzO, where a single Table VI : FSGO Model Calculations for H F and OH
Kegative total energy Orbital radii Inner shell Bonding Lone pair Orbital positions Inner shell z coordinate Lone pairs y coordinate z coordinate Bonding z coordinate Bond length
-"'
OH-
HF
63.488
84.571
0.241 1.350 1.362
0.212
0.0004
0.0003
(0.200) 0.0067
(0.200) 0.018
0.678 1.679
0.345 1.482
1,111
1.107
a Axis of molecule is z; heavy atom a t origin; H in $ 2 direction. Atomic units: energy in hartrees, distances in Bohrs; values in parentheses held constant.
energy calculation for fixed parameters required about 0.2 sec and the total energy minimization involving several hundred such energy calculations took about 2 min. This time requirement is, of course, far less than would be used in a typical SCF-RIO calculation, even with a minimal basis set.
Figure 2. Schematic diagram of the localized orbitals and the predicted geometric structure of HnO. The center of each bonding orbital is slightly displaced from its internuclear axis. The lone-pair orbitals are indicated by the dashed circle and are displaced above and below the plane of the nuclei.
Table VI1 : FSGO Model. Summary of Results for Neutral Hydrides
Molecule
LiH BeHz BHs
-Bond C&lcdQ
length, 1-y Obsdb
1.712 1.412 1.245 1.115
1.595 1.093
1.011
1,012
0.784
0.917
Percentage distance of bond orbitals toward H nuclei
89.3 78.9 71.2 59.7
Discussion of Results The calculated molecular structures are all qualitative, as one would expect. The bond lengths decrease regularly as the atomic number of the first-row atom increases. This is obviously a shrinkage effect due to the increased electrostatic attraction of the bonding electrons. There is even a semiquantitative agreement between calculated and observed bond lengths, as shown in Table VII, with an average error of 6% for the five molecules where experimental data are available. Also shown in the table are the relative positions of the bond orbitals with respect to the A and H nuclei. As the central atom A varies from Li to F, the bond orbital shifts from a position near the H nucleus to a position near A. This effect obviously relates to the increasing electronegativity of A. KH3 and HzO are special cases, in that the bond orbitals are not centered on the internuclear lines but rather slightly displaced toward the molecular axis forming what might be considered as bonds bent inward. Figure 2 shows in schematic fashion the situation for HzO. H,O and NH3 are properly predicted to be nonlinear and pyramidal, respectively. However, the HOH and HNH bond angles, which were calculated to be 88.4 and 87.6", are rather low compared to the experimental The Journal of Physical Chemistry
HF
23.3
a A few of the values quoted here differ slightly from the corresponding values presented in ref 3, owing to improved accuracy of calculation. Observed values from L. E. Sutton, Ed., "Interatomic Distances," Special Publication No. 18, The Chemical Society, London, 1965. Along axis. Perpendicular to axis.
values of 104.5 and 106.6", having an average error of 1701,. A large error in bond angles as compared with bond lengths might be expected on account of the smaller force constant for the bending distortion, as compared with that for a stretching distortion. That the bond angle of NH3is too small can be related to the extremely high inversion barrier3 predicted by this simple model. It may be observed in Tables IV-VI that the x coordinates of the lone-pair orbitals in NHs, HzO, and HF are positive. This would appear to be quite contrary to expectations, in view of the general result of more conventional LCAO theory according to which the lone pairs would occupy orbitals aimed generally in the opposite direction. However, the present orbitals are nonorthogonal, a situation which seriously complicates
A FLOATING SPHERICAL GAUSSIAN ORBITALMODELOF MOLECULAR STRUCTURE the interpretation. If the basis floating orbitals were orthogonalized by a suitable linear transformation (with no effect on the total many-electron antisymmetric wave function) a more familiar picture might be obtained. Unfortunately, there is no unique way to carry out the orthogonalization. The energies of the various molecular species as calculated with this model may be compared with Hartree-Fock-Roothaan SCF-LCAO-MO energies, inasmuch as in each type of calculation the energy is minimized for the wave function expressed as a single Slater determinant. Naturally, the present model has a comparatively crude wave function, so it is expected that the calculated energy will be well above the Hartree-Fock limit. A comparison of available results3 shows that the present model provides almost universally about 85y0 of the Hartree-Fock energy. The 15y0loss is primarily due to the lack of suitable cusps for the inner-shell wave functions. This error in total energy would not necessarily mean a drastic error in the shape of the energy hypersurface, but might be primarily just a displacement upward in energy in which case geometrical predictions by the present model could conceivably still be reasonably good. Bond distances and bond angles as predicted by SCF-LCAO-MO theory5 appear to be somewhat better however.
Acknowledgments. The National Science Foundation supported this research. Mr. Robert A. Rouse provided assistttnce.
Appendix Coalescence of T w o Orbitals. Consider two normalized spherical Gaussian orbitals, cp1 and 9 2 , with equal radii, p, and centered on the x axis at =tR/2, respectively. These are nonorthogonal and have an overlap integral, S, approaching unity as R approaches zero. A convenient pair of orthogonalized orbitals is the sum and difference. When normalized, these are 1 *l
=
4-s
(Pl
+
1293
When the bracketed quantity is expanded it becomes +2xR/p2 plus terms higher order in R. By expanding
S = exp[-$(R/~)~]
-- 2S,
the expression for X,the normalizing factor 1 / 4 2 becomes 1
1
(if)' +
cv
2 --
d2-zs-4 P
E - +
R Therefore, the limit of
$2
*
.
a
*..
as R approaches zero is
which is a normalized pz Gaussian orbital centered at the origin. Coalescence of Three Orbitals in Equilateral Triangular Array. Consider three normalized spherical Gaussian orbitals with equal radii and positioned at the corners of an equilateral triangle: 91centered at ( 5 , y) = (0, R) ; PZ centered at ( + 4 3 R / 2 , -R/2); p3 centered at (- d ? R / 2 , -R/2). These may be replaced by the orthogonal linear combinations 1
cpz)
1
I n the limit $1 = cp1 = pZ and is just a Gaussian s orbital. J12 becomes a Gaussian pLorbital as follows
where S is the overlap integral for any pair of tals. I n going to the limit (R = 0) $1
=
cp1
cp
orbi-
= cp2 = 9 3
and is just an s orbital. J/z and #3 both have nodal surfaces and in the limit become pz and py Gaussian orbitals, respectively. (5) See, for example, M. Krauss, J. Res. Nat. Bur. Stand., 68A, 635 (1964),and S. D.Peyerimhoff, R. J. Buenker, and L. C. Allen, J . Chem. Phys., 45, 734 (1966).
Volume 78, Number 4 April 1968
