Separation of hybridization, delocalization, and constructive

Arthur A. Low, and Michael B. Hall. J. Phys. Chem. , 1990, 94 (2), pp 628–637. DOI: 10.1021/j100365a025. Publication Date: January 1990. ACS Legacy ...
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J . Phys. Chem. 1990, 94, 628-637

628

effect upon the linking C-C bond (when the N-C-C-N and 0 - N - 0 planes are perpendicular) to the extent that it is broken and rearrangement to 1-nitro-3-aminocyclobutadienetakes place. This is not observed for the corresponding aza system. The C-C bond in nitroethane is not significantly weakened for any NO2 conformation, presumably because the analogous orbital interaction is too weak.

Acknowledgment. We appreciate very much the comments and suggestions of Dr. Jane S. Murray. We also thank the Air Force Office of Scientific Research for its support of this work through Grant No. AFOSR-88-0068. Registry No. 111, 120411-24-7; IV, 124177-20-4; V, 79-24-3; VI, 44397-83-3; VII, 25723-64-2; IX. 1241 77-21-5; tetrahedrane, 157-39-1.

Separation of Hybridizatlon, Delocalization, and Constructive Interference in the Electron Deformation Denstties of the First-Row Hydrides Arthur A. Low and Michael B. Hall* Department of Chemistry, Texas A & M University, College Station, Texas 77843 (Received: June 20, 1989)

The standard deformation density, the electron density of the molecule minus that of the spherical atom, may be considered to be the sum of the effects of hybridization (and polarization), charge delocalization (and charge transfer), and constructive interference. Sometimes these effects all contribute to the accumulation of electron density between the atoms, and at other times they oppose each other. By creating different promolecules, we have produced deformation density maps that isolate the effects of hybridization, delocalization, and constructive interference. Hybridization effects were isolated by substitution of the spherical promolecule by the GVB valence-state hybrid promolecule. Delocalization effects were isolated by using the GVB pair promolecule. What remains after removal of the hybridization and delocalization effects is the density difference due to constructive interference. This partitioning scheme is applied to the first-row hydrides, AH, (A = Li-F; x = 1-4) and Hz. For all the molecules studied, the internuclear accumulation due to charge transfer and delocalization is larger than that due to constructive interference.

Introduction Electron deformation density maps have been used widely to study the relationship between electron density and chemical bonding.' A deformation density map is defined as the difference between the total electron density of a molecule and the sum of the electron densities of the constituent atoms (or fragments) called the promolecule.2

AP = Pmolecule - Pprornolecule ppromoleculc = Cpi

i = atom or fragment i

(1) (2)

I

The most commonly used promolecule consists of a collection of the spherically averaged ground-state atoms. This promolecule is employed in the generation of what we shall call the "standard" deformation density. For the hydrogen molecule, the standard deformation density shows an accumulation of charge density in the "bonding" region between the nuclei and a deficit of charge density in the "antibonding" region outside of the hydrogen nuclei. This description is often cited as the prototype of the covalent b 0 n d . j ~ ~However, many studies, both theoretical and experimental, have questioned the suitability of the hydrogen molecule as the model for covalent b o n d i n g . ' ~ ~ , ~ ( I ) (a) Coppens, P.; Stevens, E. D. Adu. Quanr. Chem. 1977, I O , I . (b) Low, A. A,; Hall, M. B. Theoretical Models of Chemical Bonding, Part 2; Springer-Verlag: New York, in press. (2) Hirschfeld, F. L.; Rzotkiewicz, S. Mol. Phys. 1974, 27, 1319. (3) (a) Rudenberg, K. Rev. Mod. Phys. 1962,34, 326. (b) Feinberg, M. J.; Ruedenberg, K.; Mehler, E. L. Ado. Quantum Chem. 1970, 5, 28. (c) Feinber, M. J.; Ruedenberg, K . J . Chem. Phys. 1971, 54, 1495. (d) Kutzelnigg, W. Angew. Chem. 1973,85, 551; Angew. Chem., Int. Ed. Engl. 1973, 12, 546. (e) Levine, 1. N . Quantum Chemistry; 3rd ed.; Allyn and Bacon: Boston, MA, 1983; Chapter 13. (4) (a) Bader, R. F. W.; Chandra, A. K. Can. J . Chem. 1968.46.953. (b) Bader, R. F. W.; Preston, H. J . T. Int. J . Quantum Chem. 1969, 3, 327.

OQ22-3654/90/2094-0628$02.50/0

Theoretically, Hirshfeld and Rzotkiewicz2 found that, for the first-row diatomics, the electrostatic attraction between the two spherical atoms contributes more to the binding energy of the molecule than charge accumulation along the bond axis in all cases except for hydrogen. Bader and BeddalSbfound that the accumulation in the antibonding regions of the deformation density maps of the first-row diatomics was sometimes comparable to or even exceeded the accumulation in the bonding region. Both of these studies suggest that the distribution of charge density in the deformation density of H 2 is atypical and is unsatisfactory as a prototype for the covalent bond. Experimentally, Dunitz and Seiler6 found a lack of accumulation in the 0-0 bonds and weak accumulations in the N-N, C-0, and C-N bonds in the deformation density map obtained from the low-temperature X-ray analysis of 1,2,7$-tetraaza4,5,10,11-tetraoxatricyclo[6.4.1.lltetradecane. These observations also led these authors to question the suitability of the hydrogen molecule as a prototype for the description of covalent bonding. Smaller than expected or even a lack of accumulation between covalently bonded nuclei is common in the standard deformation density maps with atoms possessing greater than half-filled valence shells. Examples of maps possessing no accumulation between bonded nuclei include the 0-0bond in Hz088 and the F-F bond in F2.9 Weak accumulations in bonding regions have been observed for N-N,IO N-0," C-0,l2 and C-FI3 bonds in various ( 5 ) (a) Bader, R. F. W.; Henneker, W. H.; Cade, P. E. J . Cfiem. Phys. 1967, 46, 3341. (b) Bader, R. F. W.; Beddal, P. M. J . Chem. Phys. 1972, 56, 3320. (c) Bader, R. F. W.; Keaveney, I.; Cade, P. E. J . Cfiem.Phys. 1%7, 47, 3381. (d) Bader, R. F. W.; Bandrauk, A. D. J . Chem. Phys. 1968, 49, 1653. (e) Cade, P. E.; Bader, R. F. W.; Henneker, W. H.; Keaveney, I. J . Cfiem. Phys. 1969, 50, 5313. (6) Dunitz, J. D.; Seiler, P. J . Am. Chem. SOC.1983, 105, 7056. (7) Savariault, J.-M.; Lehmann, M. S. J. Am. Chem. SOC.1980, 102, 1298. (8) Breitensten, M.; Dannohl, H.; Meyer, H.; Schweig, A,; Seeger. R.; Seeger, U.; Zittlau, W. Int. Rev. Phys. Chem. 1983, 3, 355. (9) Kunze, K. L.; Hall, M. B. J . Am. Cfiem. SOC.1986, 108, 5122.

0 1990 American Chemical Society

Electron Deformation Density Maps organic molecules. All of these observations can be attributed to the relatively high population of the p orbitals of the spherically averaged N, 0, and F atoms with 1 .OOOO, 1.3333, and 1.6667 electrons per p orbital, respectively. In his early studies of the diatomic first-row hydrides, Bader et aI.& placed one electron in the p orbital pointing along the bond axis while averaging the remaining electrons over the pr orbitals. In their studies of the difference densities of diatomic molecules, Schwarz, Valtazanos, and Ruedenberg produce a chemical difference density using appropriately defined oriented atomic promolecules.14 In their study, they decomposed the difference density maps of F2 and N2 into u- and *-contributions through the use of natural orbitals from an MCSCF wave function. The concept of orbital hybridization, introduced by P a ~ l i n g ' ~ and Slater,16 has been utilized in producing hybrid atom promolecules which in turn generate hybrid deformation density m a p ~ . ~One J ~ example is a theoretical analysis of the F2molecule9 where the sum of the valence-state hybrid atoms was used as the promolecule. The hybrid atoms contained one electron in each bond hybrid orbital and two electrons in each of the lone pair orbitals. Subtraction of this promolecule from the molecular density revealed a deformation density more typical of what one would expect for a covalent bond with accumulation between the two F nuclei and deficit in the antibonding regions. This method was also utilized in an analysis of the deformation density of 1,2,7,8-tetraaza-4,5,10,1 I-tetraoxatricycl0[6.4.l.l]tetradecane.'~ In its hybrid deformation density map, there are accumulations in all the bonding regions between the nuclei in contrast to the standard deformation density map noted earlier. The use of valence-state hybrid atoms as the promolecule partitions the effects of atomic orientation, polarization, electron promotion, and orbital hybridization from the deformation density maps. We will refer to these effects collectively as hybridization effects. However, the effects of charge transfer and delocalization, which we will collectively refer to as delocalization effects, are still included in the hybrid deformation density maps in addition to constructive interference of bonding orbitals. In order to take this partitioning scheme one step further, it would be useful to produce a promolecule that removes the effects of charge transfer and delocalization leaving a deformation density which shows constructive interference only. Since the choice of promolecule is arbitrary, one is free to use different promolecules. Their use can reveal more information about the changes of the electron density upon bond formation than the standard deformation density. This partitioning scheme uses promolecules that have been derived from various manipulations of the generalized valence bond.19 In our partitioning scheme, various assumptions have been made about the nature of the wave function and the density. First, it is assumed that the GVB wave function is sufficiently accurate so that it includes all of the essential features of A-H bonding. That is, it will contain the effects of hybridization (including polarization), delocalization (IO) (a) Hope, H.; Otterson, T. Acta Crystallogr., Sect. B 1979,358,370. (b) Otterson, T.; Almlof, J.; Carle, J. Acra Chem. S c a d . Sect. A 1982,36A, Hope, H. Acta Crystallogr. Sect. B 1979,35B, 373. 63. (c) Otterson, T.; ( I I ) (a) Wang, Y.; Blessing, R. H.; Ross, F. K . ; Coppens, P. Acta Crystallogr., Sect. B 1976, 328, 572. (b) Coppens, P.; Lehmann, M. S. Acta Crystallogr., Sect. B 1976,328, 1777. (12) (a) van der Waal, H. R.; Vos, A. Acra Crystallogr., Sect. B 1979, 358, 1804. (b) Maverick, E.;Seiler, P.; Schweizer, W. B.;Dunitz, J. D. Acta Crystallogr., Sect. B 1980, 368, 615. (1 3) (a) Dunitz, J. D.; Schweizer, W.B.; Seiler, P. Helu. Chim. Acra 1983, 66, 123. (b) Dunitz, J. D.; Schweizer, W. B.; Seiler, P. Helu. Chim. Acta 1983,66. 134. (14)Schwarz, W. H. E.; Valtazanos, P.; Ruedenberg, K. Theor. Chim. Acta 1985,68,471. (IS) Pauling, L. J . Am. Chem. SOC.1931, 53, 1367. (16) Slater, J. C.Phys. Reu. 1931,37, 481. (17) (a) Figgis, B. N.; Forsyth, J. B.; Reynolds, P. A. Inorg. Chem. 1987, 26, 101. (b) Figgis, B. N.; Reynolds, P. A. J. Chem. Soc.,Dalton Trans. 1986, 125. (c) Figgis, B. N.; Reynolds, P. A.; White, A. H. Inorg. Chem. 1985,24, 1864. (d) Figgis, B. N.; Reynolds, P. A,; Wright, S. J. Am. Chem. Soc. 1983, 105,434. ( 1 8 ) Kunze, K. L.; Hall, M. B. J. Am. Chem. SOC.1987, 109, 7617. (19) Goddard, W. A,; Dunning, T. H.; Hunt, W. J.; Hay, P. J. Acc. Chem. Res. 1973, 6, 368.

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 629

TABLE I: LiH BH3 NH3 HF

Equilibrium Geometriesa

R(LiH) R(BH) R(NH) L(HNH) R(HF)

1.6034 1.1880 0.9982 108.88 0.8983

BeH2 CH4 H20

R(BeH) R(CH) R(OH) L(H0H)

H2

R(HH)

1.3308 1.0828 0.9404 107.03 0.7341

4Bond lengths are expressed in angstroms and bond angles in degrees.

(including charge transfer), and constructive interference. Second, it is assumed that the GVB pair orbitals represent the atoms after the effects of hybridization and delocalization have taken place but before the electrons are spin coupled to make the covalent bond. Therefore, subtraction of the density of the singly occupied GVB pair orbitals from the total GVB molecular density will show the changes in density due to constructive interference only. Third, it is assumed that truncation of the GVB pair orbitals removes the effect of delocalization and results in GVB hybrid orbitals. Therefore, the GVB hybrid orbitals only contain the effects of hybridization and polarization. In this paper, we introduce and apply our partitioning scheme to produce deformation density maps of the first-row hydrides AH, (A = Li-F, x = 1-4) and H2 with the goal of separation of the effects of hybridization, delocalization, and constructive interference. Additionally, the densities of the respective promolecules will be subtracted from one another to produce difference density maps which show how each effect changes the charge density. It also should be noted that the boundaries between these different effects are not totally clear. Although it is not possible to cleanly separate them, the use of this partitioning scheme is useful to see the general pattern of charge movement due to these effects. The first-row hydrides were chosen for a systematic study of this partitioning scheme not only because of the ease of the calculations but also, in this series, the type of bonding between the A and H atoms changes from ionic to covalent to polar covalent. Thus, one can see how these different contributions to the density change as the type of bonding changes. Finally, this partitioning scheme is a totally theoretical approach and cannot be used in the analysis of an experimental density but it is useful in understanding certain features that appear in experimental deformation density s t ~ d i e s . ~ , ~ ~ Theoretical Methods The molecular and atomic orbitals were generated by using as basis functions the Dunning triple-{ [5s,Sp] contraction20 of the Huzinaga (10s6p) primitive Gaussian basis2' for Be to F with one d polarization function added.22 For Li, the Dunning [Ss] contractionZoof the Huzinaga (10s) primitive Gaussian basis2' with four p polarization functions23was used. For H, the Dunning [3s] contraction20of the Huzinaga (5s) primitive Gaussian basis2' with one p polarizationZ2was used. Molecular orbital calculations were performed via the closed-shell Hartree-Fock-Roothaan (HFR) method.24 The geometries of the molecules were optimized by full gradient methods.25 The optimized geometric parameters are listed in Table I. Compared to other theoretically optimized geometric parameters, these parameters are fairly close to those of near-Hartree-Fock calculatiomZ6 The parameters obtained from Pople's 6-31G* basis set27are closer to experimental parameters but not to those of near-Hartree-Fock calculations. The canonical HFR valence molecular orbitals were localized with the Boys criteriaZ8to generate orbitals for bond densities and lone pair orbital densities. Bond orbitals and lone pair orbitals (20) Dunning, T.H. J . Chem. Phys. 1971, 55, 716. (21)Huzinaga, S. J . Chem. Phys. 1965,42, 1293. (22) Ahlrichs, R.;Taylor, P. R. J . Chim. Phys. 1981, 78, 315. (23) Lie, G. C.;Clementi, E. J. Chem. Phys. 1974.60, 1275. (24)Roothaan, C. C.J. Reo. Mod. Phys. 1951, 23, 69. (25)Pulay, P.Mol. Phys. 1969,17, 197. (26) For example, rOH = 0.941 A, LHOH = 106.6O for [6s5p2d/3slp] contraction of ( I ls7p2d/5slp)Gaussian basis set in: Dunning, T. H.; Pitzer, R. M.; Aung, S. J. Chem. Phys. 1972.57,5044. (27) Hariharan, P.C.;Pople, J. A. Mol. Phys. 1974,27, 209. (28) Foster, J. M.;Boys, S. F. Reu. Mod. Phys. 1960,32, 300.

630 The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 I

0650

0750

0850

-

charge t r a n s f e r and

Low and Hall

-

singly-occupied GVB pair o r b i t a l s

dalence-state hybrid atoms

hybridization and polarization deformation density

0 950

spherical atoms I050

I 150

Energy (hartrees)

density molecule

Figure 1. Flow chart showing the partitioning scheme used in this study. The energy scale shown on the left is based on the respective energies calculated for the hydrogen molecule and its different promolecules. The spherical atom energy is twice the energy of a spherical H atom. The energy of the valence-state hybrid atoms was twice the energy of the first S C F cycle of an atomic calculation of H starting from the valence-state hybrid orbital. The energy of the singly occupied GVB pair orbitals was twice the energy of the first S C F cycle of a molecular calculation on H2+starting from a singly occupied GVB pair orbital.

were also generated by generalized valence bond (GVB) calculations performed at the optimized HFR geometries. In the GVB calculation, each A-H bond was treated as a GVB pair and the lone pairs were kept doubly occupied. The GVB pair orbitals were obtained from the two GVB natural orbitals of each bond pair through the following equation:

c I and c2 are CI coefficients of the GVB natural orbitals ( c l > 0, c2 < 0), 4, and d 2 are the strongly and weakly occupied GVB natural orbitals, and +a and 4 b are the GVB pair orbitals. Atomic orbitals were generated by calculations on the same basis sets using the spin-restricted, symmetry-equivalent Hartree-Fock-Roothaan method.29 Hybrid valence-state atomic orbitals were generated in two ways. First, the localized HFR molecular orbitals were truncated of the functions on the other atoms in a procedure similar to the method of Newton and S ~ i t k e s . ~The ~ atomic core orbitals and the hybrid valence orbitals were then renormalized and symmetrically orthogonalized by using the Lowdin procedure31 to form the LMO hybrids. Second, the GVB valence-state hybrids were generated by truncation of the functions of the other atoms on the GVB pair orbital centered on A (or H). These truncated orbitals were then renormalized and symmetrically orthogonalized with the atomic core orbitals as before. The truncation, renormalization, and orthogonalization were performed by a modified version of the A T M O L ~package of prog r a m ~ .The ~ ~ rest of the above procedures were performed by using the GAMESS package of programs.33 All density plots were generated using the program MOP LOT.^^ They are contoured geometrically with each contour differing by (29) Guest, M. F.: Saunders, V. R. Mol. Phys. 1974, 28, 819. (30) Newton. M. D.: Switkes, E.; Lipscomb, W. N. J . Chem. Phys. 1970, 53, 2645. (31) Lowdin, P.-0. J . Chem. Phys. 1950, 18, 365. (32) Hillier, 1. H.; Saunders, V. R.; Guest. M. F. A T M O L ~System, Chemistry Department, University of Manchester, U.K., and SERC Laboratory, Daresbury, U.K. (33) M. F. Guest, Daresbury Laboratory, U. K. provided the FPS versions Of G A M E S .

(34) Lichtenberger, D. L. Ph.D. Dissertation, University of Wisconsin, Madison, W1. 1974. Program available from: Quantum Chemistry Program Exchange, Indiana University, Bloomington, IN 47401 : Program 284.

TABLE 11: How the Different Density Maps in This Partitioning Scheme Were Computed density of spherical atom promolecule LMO hybrid deformation LMO hybrid density promolecule GVB hybrid deformation GVB hybrid density promolecule constructive interference GVB pair deformation density promolecule GVB hybrid difference density spherical atom promolecule delocalization difference GVB hybrid density promolecule map standard deformation density

was subtracted from total HFR molecular electron density total HFR molecular electron density total GVB molecular electron density total GVB molecular electron density GVB hybrid promolecule GVB pair promolecule

a factor of 2. The smallest positive and negative contour has a value of f2-12 (2.4414 X lo-”) electrons (au)”. Negative contours are dashed. In the orbital plots of the GVB pair orbitals, the smallest contours have a value of fTS(3.125 X 10-2)[electrons ( a ~ ) -l J~2 ]. In this paper, a deformation density will be defined as the difference between the total molecular density (either HFR or GVB) and the density of a selected promolecule. A difference density will be defined as the difference between the densities of two different promolecules. How the various deformation and difference density maps in this paper were computed is illustrated in Table 11. The spherical atom promolecule consisted of the sum of densities of the spherically averaged ground-state atoms. The LMO hybrid promolecule consisted of the sum of the densities of the LMO valence-state hybrid atoms. The GVB hybrid promolecule consisted of the sum of the GVB valence-state hybrid atoms. For the valence-state hybrid atoms, one electron was placed in each hybrid orbital pointing along the A-H bond direction and two electrons were placed in each lone pair orbital. Finally, the GVB pair promolecule consisted of the density of the singly occupied GVB pair orbitals plus the molecular core orbitals. All of the computations were performed on the Texas A & M University Chemistry Department VAX 11/780 and on the FPS164 array processor. Results Hydrogen Molecule. In order to visualize the partitioning scheme used in this paper more clearly, a flow chart showing the different promolecules and their relative energies is shown in Figure 1. The partitioning scheme forms two closed cycles; therefore if the maps in each cycle are added together, they should

Electron Deformation Density Maps

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 631

result in the standard deformation density map. This can be seen when the constituent parts of the difference density maps in the outer cycle are added together: hybrid difference density = GVB hybrids - spherical atoms delocalized difference density = GVB pairs - GVB hybrids constructive interference deformation density = molecule - GVB pairs net: standard deformation density = molecule - spherical atoms (4) For the hydrogen molecule, the GVB wave function, ignoring the spin part, is written in terms of the GVB pair orbitals as ~ G V = B

N[a(l) b(2)

+ b(1)

(5)

where N is the normalization constant and a and b are GVB pair orbitals originating on HI and H2, respectively. At the equilibrium bond distance, the GVB pair orbital originating on HI, a, will also contain a significant amount of character on HZ.Likewise, the GVB pair orbital originating on H2, b, will contain a significant amount of character on HI. The GVB valence-state hybrid orbitals are calculated from the GVB pair orbitals by truncating the functions on the other atoms off. The truncated orbitals, a’ and b’, are then renormalized to give the hybrid orbitals.

6hyb = ”(a’)

(6)

6hyb = N’(b’)

(7)

The GVB pair orbitals may be expressed as consisting of the truncated orbital on the main atom plus a fraction of an orbital on the other atom. The orbital on the other atom is denoted as a” for the orbital on H I and b” for the orbital on H2 (with N” as a normalization constant): a = ”’(a’ Ab”) (8)

+ b = N”(b’ + Xa”)

(9)

The density of the H2 molecule from the GVB wave function may be derived from the following integral: PGvB = 2

J 1qGvBl2dT2

t 10)

Putting eq 5 into IO and integrating yields the following expression: PGVB= 2fl[a2(1)

+ bz(l) + SabIa(1) b(1) + b(1) a(l)ll

(11)

Then, by substituting eq 8 and 9 into 1 1 , one obtains pGVB

+ b’* + (Xa”)2 + + X(a’b” + + b’a” + a”b’) + Sab(a’b’+ b’a’ + X(a’b’’ + b”a’ + b’a” + a”b’) + X2(a”b” + b”a”)]] ( 1 2)

= 2flN”2[a’2

b”a’

The densities of the valence-state hybrid and the GVB pair promolecule can be expressed as the sum of the squares of the respective orbitals:

+ b’2) ppairs= a2 + bZ = N”Z[a’2+ b’2 + (Aa”)2 + + Phyb

= N’2(a’2

X(a’b”

+ b”a’ + b’a” + a”b’)]

(13)

(14)

Comparing the expressions for the densities in eq 12, 13, and 14, it can be observed that the density of the molecule from the GVB wave function can be split into four parts. In eq 12, the first two terms correspond, with a change in normalization constants, to the density of the GVB hybrid orbitals. The next pair of terms, (Xa”)2and correspond to the density change due to charge transfer. The next term, Xta’b’’ + b”a’ + b’”’ + a”b’), corresponds to the density change due to charge delocalization. The charge delocalization term consists of the cross product of

I

I

@

Figure 2. (a) Standard deformation density map for H,; (b) GVB hybrid difference density map for H,; (c) hybrid deformation density map for H2; (d) delocalization difference density map for H,; (e) constructive interference difference density map for H,. Density maps are contoured geometrically with each contour differing by a factor of 2. The smallest positive and negative contours has a value of f2-I2 electrons (au)-’ (2.4414 X Negative contours are dashed.

atomic based orbitals; thus, it could be viewed as part of the overall constructive interference of the molecule. However, with a very large basis set, it would be possible to achieve this degree of delocalization without using functions from the other atoms. Now, one would not view any of the terms in eq 14 as contributing to the constructive interference. We will take the latter view in the remainder of this paper. The sum of the first five terms in eq 12 corresponds, again with a change in normalization constants, to the density of the GVB pair orbitals. The last term in eq 12 corresponds to the density change due to constructive interference of the GVB pair orbitals. The series of difference density maps which make up our partitioning scheme are shown for the hydrogen atom in Figure 2. In Figure 2a, the standard deformation density map shows the expected features of accumulation in the internuclear region and deficit in the outer nonbonding regions. The bond accumulation completely surrounds the hydrogen nuclei. The GVB hybrid difference density map, shown in Figure 2b, shows that the GVB valence-state hybrid orbitals have contracted relative to the spherical atom. This is seen by the accumulation regions near the nuclei surrounded by deficit regions in the outer region. The accumulation regions about the nuclei are extended toward the center of the molecule due to some H pz mixing into the GVB hybrid. The effect of this contraction can be seen in the GVB hybrid deformation density map shown in Figure 2c. The internuclear accumulation region has contracted considerably toward the center of the molecule. The deficit region has also become more compact in size and is now located adjacent to the hydrogen nucleus.

632 The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 The charge delocalization difference density map, shown in Figure 2d, looks quite similar to the standard deformation density map. However, both the accumulation and deficit regions of this map contain one less contour than their respective counterparts in the standard deformation density map. The constructive interference deformation density map, shown in Figure 2e, contains an accumulation region located between the two hydrogen nuclei which is the smallest of all the accumulation regions in all the maps. The deficit regions are adjacent to the hydrogen nuclei. One would expect that the deformation density map of the hydrogen molecule should consist mainly of constructive interference and delocalization effects. However, the constructive interference deformation density map makes up only a minor part of the standard deformation density. The delocalization term in eq 12 represents the major contribution to the standard deformation density and is responsible for about half the accumulation at the bond midpoint. The remaining accumulation consist of two nearly equal contributions from hybridization and polarization (mainly contraction of the hybrid orbitals) and the constructive interference. The use of this partitioning scheme to produce various difference density maps for the hydrogen molecule clearly shows the different effects contributing to the standard deformation density map. Application of this scheme to the first-row hydrides follows. Deformation Densities. The standard deformation density maps of the first-row hydrides are shown in Figure 3. In this series of deformation density maps, the map of LiH (Figure 3a) is the most radically different from the others. The largest accumulation region is nearly centered on the H atom with a polarization toward the Li atom. The features about the Li atom are what differentiates this map from the others. There is a dipolar pattern of charge density about the Li atom with an accumulation region on the side of the Li atom pointing away from the H atom and a deficit region on the side of the Li atom closer to the H atom. Surrounding this region is another dipolar pattern of charge density with loss on the side opposite the hydrogen and gain on the other side which combines with the accumulation region about the H atom. These patterns, also seen in Bader's maps,& can be attributed to polarization of the Li 1s core away from the H atom in response to the negative charge about the H atom, and transfer of the Li 2s electron density closer to the H atom, respectively. The transfer of charge from one nucleus to another and the polarization of the density increase on both the anionic nucleus and of the density left on the cationic nucleus has been defined by Bader as characteristic of ionic bonding.& In the rest of the maps, a progressive series of changes occur from BeH, to HF. In the map of BeH2, the internuclear bond accumulation is centered much closer to the H atom than the Be atom. As one proceeds from the maps of BeHz to HF, the center of this bond accumulation moves from the H atom to the central atom, A. This can be attributed to changes in the charge transfer caused by the increased electronegativity of the A atom. In the maps of BeH,, BH3, and CH4, the largest accumulation occurs along the A-H bond axis. In the maps of NH3, H 2 0 , and H F , however, the largest accumulation in the maps are associated with the lone pair(s) and not the A-H bond. The internuclear accumulation region extends to the central atom A in the maps of BeH,, BH,, and CHI. However, starting with the map of NH,, a small region of density deficit appears along the bond direction adjacent to the N nucleus. The deficit region near the central A atom grows larger in the maps of H 2 0 and HF. These deficit regions can be attributed to the increasingly large occupation of the p orbitals of spherically symmetric N, 0, and F atoms and the polarization of the 2s electron pair away from the A-H bonds (a major component of the lone pair). The p, population is so large in HF that it produces a region of density deficit on the back side of the F atom. In the maps of NH,, H 2 0 , and HF, the largest accumulation regions correspond to the lone pair regions. These features can also be attributed to the choice of the spherical atom promolecule since the p orbitals in the spherical N, 0, and F atoms do not contain enough charge density to compensate for the doubly

Low and Hall

8

I Figure 3. Standard deformation density of (a) LiH in a plane containing the molecule; (b) BeH, in a plane containing the molecule; (c) BH3 in a plane perpendicular to the molecular plane containing the boron and one of the hydrogen atoms; (d) CH4 in a plane containing the carbon atom and two of the hydrogen atoms; (e) NH, in a plane containing the nitrogen, one of the hydrogen atoms, and the nitrogen lone pair; (f) H20 in the plane containing all of the atoms; (9) H20in the plane perpendicular to the molecular plane containing the oxygen lone pairs; (h) HF in a plane containing the molecule. Density plots are contoured as in Figure 2.

occupied lone pair orbitals and the polarization of the 2s2 density away from the A-H bond. A related feature to the lone pair accumulation regions is the large deficit regions perpendicular to the bond axis in the maps of BH3 and BeH,. These are due to vacant p orbitals present in the molecules BeH, and BH,. The relation to the lone pair accumulations is that the accumulations are due to doubly occupied hybrid orbitals where the deficits are due to vacant p orbitals so they can be considered as opposites. In the map of BeH,, a small region of density deficit appears on the nonbonding side of the H atom, a significant distance from the H atom. In the maps from BH, to HF, this region grows larger in area and value. Also, it moves closer to the H atom. In the map of NH,, the deficit region begins to bend around the back of the H atom. This bend becomes more exaggerated in the maps of H 2 0 and HF. In the map of HF, this deficit region is found adjacent to the H atom. These deficit regions, again observed

Electron Deformation Density Maps

in Bader’s diatomic maps,“ can be attributed to movement of the charge density from the back side of the H atom to form the A-H bond. As the electronegativity of atom A increases, the deficit region on the back side of H grows. In the standard deformation density maps, there are many features which serve to mask the actual internuclear accumulation due to bonding of the A and H atoms. These added features can enhance or deplete the accumulation in the bonding region. With the use of the GVB valence-state hybrid promolecule, we can remove the effects of hybridization from the standard deformation density maps. Much of our description above is based on knowledge gained by examining these hybrid deformation densities. Hybridization of the Atoms. The GVB hybrid deformation density maps show the deformation density of a molecule with hybridized atoms as the promolecule. As stated in the Introduction, in the interest of time and space, the effects of atomic orientation, polarization, electron promotion, and orbital hybridization are collectively referred to as the effects of hybridization. In order to more clearly see what this density difference is, the density of the spherical atom promolecule was subtracted from the density fo the GVB hybrid promolecule to produce GVB hybrid difference density maps. These maps for the first-row hydrides are shown in Figure 4. In the GVB hybrid difference density map of LiH (Figure 4a), there is a region of deficit on the nonbonding side of the Li atom with a corresponding accumulation on the opposite side which merges with the accumulation region about the hydrogen atom. This can be attributed to 2s-2pZ mixing upon formation of the GVB hybrid of the Li atom. The region closest to the Li atom is one of deficit with a small accumulation region on the back side of the Li atom. As was the case with H,, the GVB hybrid orbital on the H atom in all these difference density maps has contracted relative to the spherical H atom as indicated by an accumulation region about the H atom. Comparison of the GVB valence-state hybrid orbitals on all the H atoms from LiH to H F to that of the spherical H Is orbital confirms that the GVB hybrid orbital has indeed contracted relative to the spherical H atom. The degree of contraction increases as one proceeds from LiH to HF. The accumulation region about the H atom is quite polarized toward the A atom in LiH. This polarization diminishes in the maps of BeH2, BH3, and CH4. In the maps of H 2 0 and HF, the accumulation region about the H atom is now polarized away from the A atom. This is caused by the fact that these maps show the sum of the densities of two atoms. In the maps of LiH, BeH,, and BH3, the valence-state hybrids of the central A atom all possess more charge density between the A-H atoms than the spherical atom because the central A atom has less than a half-filled shell. This causes the polarization of the accumulation about the H atom toward the A atom. However, for HzO and HF, the GVB valence-state hybrids possess less charge density between the A-H atoms than the spherical atom because the central A atom now has a shell which is more than half-filled. This causes the polarization of the accumulation region about the H atom to be directed away from the A atom. Since the hybrid orbital pointing along the A-H bond axis contains one electron, the GVB hybrid promolecule will contain more density than the spherical atom promolecule in the internuclear A-H region for BeH, and BH3. Conversely, the GVB hybrid promolecule for H,O and H F will contain less density along the AH bond axis than the spherical atom promolecule. Thus, the AH internuclear region is now a region of deficit in the difference density maps. The hybrid promolecule possesses vacant p orbitals in Be and B atoms and filled lone pair orbitals for N, 0, and F. Therefore, the large deficit regions perpendicular to the bond axis appear in the hybrid difference density maps of BeHz and BH3 and the large lone pair accumulation regions appear in the difference density maps of NH,, H 2 0 , and HF. From this series of difference density maps, a number of conclusions may be derived. First, due to the low population of the p orbitals in the spherical Be and B atoms, the bond accumulation

The Journal of Physical Chemistry, Vol, 94, No. 2, 1990 633

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0 Figure 4. GVB hybrid difference density maps of (a) LiH, (b) BeH2,(c) BH3,(d) CH4,(e) NH3, (f) H20, (g) lone pairs, and (h) HF, in the same planes as described in Figure 3. Contouring is the same as Figure 2.

observed in the standard deformation density maps of BeH2 and BH, will be enhanced because the effects of hybridization add density to this region. Conversely, due to the large population of the p orbitals in the spherical N, 0, and F atoms, the internuclear accumulation in the standard deformation density maps of NH3, H20,and H F will be depleted because the effects of hybridization now subtract density from this region. Second, the large deficit regions due to vacant p orbitals and the large accumulation regions due to the lone pairs should be accounted for in the valence-state hybrid promolecule. Therefore, these features should not dominate the GVB valence-state hybrid deformation density maps as they do in the standard deformation density maps. GVB Hybrid Deformation Density. The GVB hybrid deformation density maps of the first-row hydrides are shown in Figure 5. The map of LiH (Figure Sa) shows a similar pattern about the Li atom similar to that in the standard deformation density map with an inner polarization of the Li 1s core away from the H atom and an outer polarization of the Li 2s valence density toward the H atom. The accumulation about the H atom is more polarized toward the Li atom than before with the largest contour no longer containing the H atom. This is the result of the contraction of the GVB hybrid orbital of the H atom. There is a notch

634

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990

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Figure 5. GVB hybrid deformation density maps of (a) LiH, (b) BeH2, (c) BH,, (d) CH,, (e) NH,, (f) H 2 0 , ( 9 ) H20lone pairs, and (h) HF, in the same planes as described in Figure 3. Contouring is the same as Figure 2.

in the contours of accumulation on the back of the H atom which could be the beginnings of the formation of the charge deficit regions found at the back of the H atoms in the maps of the other first-row hydrides. When compared to the standard deformation density maps, the amount of accumulation in the bonding region in the hybrid deformation density maps is smaller for BeH, and BH3, roughly equal for CH, and NH,, and larger for H 2 0 and HF, as expected from the hybrid difference density maps. Unlike the standard deformation density, the accumulation region between the A and H atoms in the maps of BeH,, BH,, and C H does not extend toward the central A atom. The area close to the A atom, just outside the A atom's core, is now one of charge deficit. This deficit region can be attributed to the mixing of 2s character into the 2p orbitals upon formation of the GVB valence-state hybrid atoms. Addition of s character into a p orbital will serve to increase the density near the nucleus. Similar to the standard deformation density, the maximum contour in the internuclear accumulation region shifts in position

Low and Hall relative to the A and H atoms as A changes from Be to F. In the map of BeH,, the maximum in the accumulation is almost adjacent to the H atom. As one proceeds from BeH, to HF, the accumulation peak shifts toward the A atom. In the map of HF, the maximum in the bond accumulation is much closer to the F atom than the H atom. This migration of charge density in the internuclear region can be attributed to charge-transfer effects. The deficit region on the nonbonding side of the H atom in the GVB hybrid deformation density map is adjacent to the H atom in the mps from BeH, to HF. This is in contrast to the standard deformation density maps where for BeH, and BH3 this deficit region is located a significant distance away from the H atom. In addition, in all the maps, this deficit region is larger in size and value than the similar region in the standard deformtion densities. The region of accumulation adjacent to the H atom facing the A atom is smaller in magnitude relative to the similar region in the standard deformation density maps. All of these observations are due to the contraction of the H atom's valence-state hybrid orbital relative to the spherical orbital as observed in the GVB hybrid difference density maps. In the map of NH3 (Figure 5e), the lone pair accumulation in the standard deformation density map has been replaced by a region of deficit near the N atom and a diffuse accumulation region in the outer regions. The origin of this change can be found by comparing the lone pair regions of the standard deformation density and the GVB hybrid difference density maps of NH, (Figures 3e and 4e). Because the GVB lone pair orbital contracts when it is renormalized after truncation, the lone pair region in the GVB hybrid has a larger accumulation close to the N atom and is of a smaller overall size than the lone pair accumulation in the standard deformation density map. Another way of viewing this effect would be the atomic lone pair orbital expanding when the bonds to the H atoms are formed. Similar patterns are observed in the lone pair regions of H 2 0 and HF. Overall, the partitioning of hybridization and polarization from the standard deformation density maps produced maps with substantial accumulations between the A and H atoms. The size of the accumulation region grows as one proceeds from BeH, to H F in contrast to the standard deformation density where the size of the accumulation region stayed fairly constant throughout the series. Delocalization of Charge. The effects of charge transfer can clearly be seen in the GVB hybrid deformation density maps by the migration of the maxima in the internuclear accumulation region from H to A as one proceeds from LiH to HF. In order to isolate the density changes due to charge transfer and delocalization, the density of the GVB hybrid promolecule was subtracted from the density of the GVB pair promolecule. These delocalization difference density maps are shown for the first-row hydrides in Figure 6. In the map of LiH (Figure 6a), a large accumulation region is nearly centered on the hydrogen atom. Near the Li atom, there is the now familiar dipolar pattern of accumulation on the back side of the Li atom and a deficit region facing the H atom surrounded by an opposite charge flow in the outer regions. As one proceeds from the maps of BeH, to HF, the accumulation region migrates further from the H atom. In the map of N H 3 (Figure 6e), the largest contour of the internuclear accumulation region no longer contains the H atom. Furthermore, in the map of H F (Figure 6h), the largest contour in the internuclear accumulation region is almost adjacent to the F atom. This can be attributed to the effects of the charge-transfer contribution to the delocalization shown in these maps. Starting with the map of BeH,, areas of charge deficit appear at the back of the hydrogen atom. This region grows larger as A becomes increasingly electronegative as in the earlier maps. The lone pair regions in the maps of NH,, H,O, and H F are similar in appearance to the GVB hybrid deformation density maps. This is not too surprising since, in the delocalization difference density maps, the atomic lone pair orbitals are subtracted from the molecular lone pair orbitals as in the hybrid deformation density maps; Le., there is little constructive inter-

T h e Journal of Physical Chemistry, Vol. 94,No. 2, 1990 635

Electron Deformation Density Maps

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Figure 6. Delocalization difference density maps of (a) LiH, (b) BeH2, (c) BH,, (d) CH,, (e) ”3, (f) H20, (g) H 2 0 lone pairs, and (h) HF, in the same planes as described in Figure 3. Contouring is the same as Figure 2.

Figure 7. Constructive interference deformation density maps of (a) LiH, (b) BeH,, (c) BH,, (d) CH4, (e) NH,, (f) H20, (g) H20lone pairs, and (h) HF, in the same planes as described in Figure 3. Contouring is the same as Figure 2.

ference in the lone pair region with this GVB model of the bonding. Constructive Interference. The density difference due to constructive interference of the GVB pair orbitals are shown for the first-row hydrides in Figure 7. In the map of LiH (Figure 7a), there is just one contour of accumulation at the back side of the Li atom. The rest of the features in the map are nearly centered about the H atom with a lopsided ringlike accumulation region surrounding a deficit region near the H nucleus. This accumulation ring is larger on the side closer to the Li atom due to Li character in both GVB pair orbitals (vide infra). A clear progression of changes can be observed in the internuclear accumulation regions of all the maps from LiH to HF. This is in contrast to the earlier maps where the maps of LiH seemed to be different than those of the rest of the series. In the map of LiH, the internuclear accumulation region bends back around the H atom until it meets itself on the back side of the H atom. This bend of the accumulation region around the H atom becomes much less pronounced in the maps of BeH, (Figure 7b) and BH, (Figure 7c). In the map of CH4 (Figure 7d), the bend

of the internuclear accumulation region is almost gone. Starting with the map of NH3 (Figure 7e), the accumulation region begins to bend back around the central A atom. This bend becomes more pronounced in the maps of NH3 and HF. Throughout this series of deformation density maps, the position of the maximum contour of the accumulation region relative to the A and H atom stays fairly constant. The deficit region near the H nucleus, which was centered about the H nucleus in the map of LiH, moves out toward the back side of the H atom replacing the “retreating” accumulation region. The deficit region spreads out further to the back side of the H atom as one progresses from BeH2 to HF. In the maps of BeH2, BH,, and CH,, the only significant accumulation regions are associated with the A-H bond. The area about both the A and H atoms is one of charge deficit. In the map of NH,, there is a very small accumulation in the lone pair region. This accumulation in the lone pair region grows larger in the maps of H 2 0 and even larger still in the map of HF. This pattern of accumulation and the changes in the internuclear accumulation region may be rationalized through examination of

Low and Hall

636 The Journal of Physical Chemistry, Vol. 94, No. 2, 1990

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Figure 8. Orbital plots of the GVB pair orbitals: for LiH, (a) GVB pair orbital originating on H and (b) GVB pair orbital originating on Li; for BH,, (c) GVB pair orbital originating on H and (d) GVB pair orbital originating on B; for NH3, (e) GVB pair orbital originating on H and (f) GVB pair orbital originating on N; and for HF, (9) GVB pair orbital originating on H and (h) GVB pair orbital originating on F. Plots are contoured geometrically with each contour differing by a factor of 2. The smallest contours have a value of A2-5 [electrons ( a ~ j - ~ (0.031 ] ” ~ 25).

the GVB pair orbitals associated with the A-H bond. The GVB pair orbitals for LiH, BH,, NH3, and H F are shown in Figure 8. Remember, from the analysis of the hydrogen molecule, the GVB pair orbital possesses character on both A and H atoms. For LiH, the GVB pair orbital on H (Figure 8a) consists mainly of a H Is orbital with a much smaller part on Li. The GVB pair orbital on Li (Figure 8b), in contrast, contains a relatively large amount of H s character. When these two orbitals interact with each other, most of the constructive interference would be expected to be near the hydrogen atom. Thus, the constructive interference is bent around the H atom because it represents the interaction of two mainly hydrogen s-like orbitals. This will be characteristic of the constructive interference of an ionic bond. Looking at the GVB pair orbitals for BH, (Figure 8, c and d) and N H 3 (Figure 8, e and f), a steady change can be observed in the orbitals. In the orbital on the H atom, the relative amount

of A character steadily increases as shown by the distortion of the contour levels toward the A atom and by the increased number of contours on the back side of the A atom. The orbital on A contains an increasing amount of A p character relative to the A s and H s character as one progresses from BH3 to NH3. These observations can explain the differences in the bend of the constructive interference in the maps of BH, and NH,. For BH3, the orbital on the B atom still contains a significant amount of H s character while the orbital on the H atom almost entirely consists of H s character. Therefore, when these two orbitals interact, the constructive interference should still be bent back around the H atom since it still involves the interaction of two orbitals with a large amount of H s character. However, for NH,, the GVB pair orbital on H contains a significant amount of N p character while the orbital on N consists mainly of N p character. When these two orbitals interact, the constructive interference will be bent back around the N atom since it involves the interaction of two orbitals with a significant amount of N p character. The GVB pair orbitals of H F (Figure 8, g and h) both contain large amounts of F pz character. Even the orbital on H looks quite like a F pz orbital with some H s character mixed in. When these two orbitals interact, the internuclear constructive interference will be bent back around the F atom. However, that is not the only place these orbitals will constructively interfere. On the back side of the F atom, there is a large orbital lobe for both GVB pair orbitals. These orbital lobes will also constructively interfere producing the constructive interference on the back side of the F atom in the constructive interference deformation density map of HF. Thus the accumulation region on the nonbonding side of the F atom is due to the interaction of two GVB pair orbitals which consist of mostly F p character. A similar argument can be applied to rationalize the accumulation in the lone pair region of the constructive interference deformation density map of H 2 0 . Localized Orbital Hybrids. In the earlier studies of the valence-state hybrid deformation density maps, Hartree-Fock localized molecular orbitals were truncated and renormalized to produce the valence-state hybrid atoms. This method could be criticized for several reasons. Central among them is that in truncation of the LMO’s a significant portion of the orbital is thrown away.35 In order to try to produce hybrid orbitals without removing a significant portion of an orbital, the GVB pair orbitals were truncated to produce the GVB valence-state hybrid orbitals. This difference is caused by the fact that the GVB pair orbitals are located mostly on one of the atoms in the bond pair whereas the LMO’s are located between the bond pairs. In order to compare the two methods of producing hybrid orbitals, the LMO hybrid deformation density maps of the first-row hydrides are shown in Figure 9. Comparison of these maps to the GVB hybrid deformation density maps (Figure 5) shows that the maps are qualitatively similar. There are differences in the shapes of the deficit regions in the nonbonding regions and of the accumulation regions. One example is the difference in the amount of polarization of the accumulation about the H atom in the maps of LiH. However, the size of the accumulation regions and the main features are quite similar. Discussion

According to the Hohenberg-Kohn theory,36the energy of the system is directly linked to the total electron density, p(r). This applies only to systems such as the total molecular density or separated atomic densities in their ground states. Accordingly, one might expect the standard deformation density to show a relationship between the accumulation in the bonding region and the A-H bond energy. However, it does not because the spherical atom promolecule does not accurately represent the ground state of the atomic system. Furthermore, there may be significant density changes in regions removed from the accumulation region which are important in determining the bond energy. It would (35) Palke, W . E . Croat Chem. Acta 1984, 57, 779. (36) Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136B, 864

The Journal of Physical Chemistry, Vol. 94, No. 2, 1990 637

Electron Deformation Density Maps

LiH, the bond accumulation grows steadily in the maps of BeH, and BH,, stays constant from BH3 to NH,, and then grows again for H 2 0 and HF. This roughly follows the pattern of AH bond energies with a BH bond energy larger than that of LiH; the BH, CH, and N H bond energies all within 25 kJ mol-' of each other and progressively larger bond energies for O H and HF. The bond accumulation for the GVB hybrid deformation density map of H 2 also fits in with these bond energies. For the constructive interference deformation density maps, a pattern similar to that observed in the hybrid maps can be observed from LiH to HzO. However, the largest contour in the map of H F has the same value as that of H20. This is probably due to a larger contribution of charge transfer to the bonding of H F ( H F H'F) which has already been accounted for by the GVB pair promolecule. Referring to the schematic diagram of the partitioning scheme in Figure 1, it can be observed that the deformation density is made up of the sum of various effects. Using our partitioning scheme, we have divided the standard deformation density into three parts: hybridization (and polarization), delocalization (and charge transfer), and constructive interference. Hybridization requires a certain amount of energy in order to promote the electrons and to mix the s and p orbitals. The amount of hybridization will change due to different bonding requirements and differences in the energy of the 2s and 2p orbitals as one proceeds from Li to F. The energy required for hybridization will be compensated for by the better overlap of the hybrid orbitals as compared to the spherical atoms. The valence-state deformation density maps are representative of the energy difference between the valence-state hybrid atoms and the molecule. As noted earlier, the energy difference seems to be qualitatively correlated to the A-H bond energy. The effects of delocalization are always attractive. This is due to the Coulombic attraction between the electron density and both nuclei since delocalization, by our definition, contributes to the charge density accumulation between the two nuclei. For all of the first-row hydrides, the constructive interference deformation density maps possess the smallest internuclear accumulation of all the deformation density maps. Thus, according to our model, major contributions to the deformation density are made by terms other than constructive interference.

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Figure 9. LMO hybrid deformation density maps of (a) LiH, (b) BeHz, (g) HzO lone pairs, and (h) HF, (c) BH,, (d) CHI, (e) NH,, (f) H20, in the same planes as described in Figure 3. Contouring is the same as Figure 2 . be conceptually useful to have a deformation density which did correspond to the bond energies. From the standard deformation density maps, no relationship between the A-H bond energy and the largest contour value in the internuclear accumulation region can be observed. The smallest accumulation does, in fact, occur for LiH, but the other maps, with the exception of HF, have the same value for the largest contour in the internuclear accumulation region. This is not too surprising since, as noted before, the use of the spherical atom promolecule leads to an enhancement of the bond accumulation for BeH2 and BH3 and to a depletion of the bond accumulation for H 2 0 and HF. The enhancement for the weaker A-H bonds and the depletion for the stronger A-H bonds lead to standard deformation density maps with relatively similar amounts of accumulation between all A-H bonds. In the GVB hybrid deformation density maps, there is a qualitative relationship between the value of the largest contour in the bond accumulation and the AH bond energy. Starting with

Conclusion The standard deformation density can be thought of as the sum of the density differences caused by hybridization and polarization, charge transfer, and delocalization and constructive interference. Using our partitioning scheme, we have been able to observe the density differences due to each of the effects. Use of the GVB hybrid promolecule effectively removes the effects of hybridization from the standard deformation density map. The GVB hybrid deformation density maps have their major accumulation region occurring between the A and H atoms. The GVB pair promolecule produces deformation density maps in which the migration of the bond accumulation to the more electronegative A atoms has been removed. These maps also show a more continuous change from LiH (ionic bond) to CH4 (covalent bond) to H F (polar covalent bond). For the hydrogen molecule, the pattern of charge accumulation between the hydrogen nuclei and charge deficit outside the nuclei in the standard deformation density map has been taken as the model for covalent bonding. However, for some covalently bonded molecules, there is little or no charge accumulation between the bonded atoms. Using our partitioning scheme, we have been able to produce deformation density maps which possess internuclear accumulation regions between the atoms and charge depletion in the outer regions. Acknowledgment. We thank the National Science Foundation, Grant CHE 86-19420, for support, and I. H. Hillier, V. R. Saunder, and M. F. Guest for providing the ATMOL3 and GAM= programs.