Electrostatic interaction energies of independent aspherical atoms in molecules

Volume 180, number 6

CHEMICAL PHYSICS LETTERS

7 June 1991

Electrostatic interaction energies of independent aspherical atoms in molecules S.G. Wang, W.H.E. Schwarz and H.L. Lin ’ Theoretical

Chemistry,

The University,

P.O. Box 101240,

W-5900

Siegen. Germany

Received 26 September 1990; in final form 25 February 1991

Matter may be looked upon as consisting of superimposed independent atoms, which are spatially confined around specific positions by the chemical forces and which are weakly deformed thereby. Independent atoms with degenerate ground states are not fixed to be spherically symmetric as often supposed; it is more convenient for the interpretation of molecular and crystal charge distributions to refer to nonspherical atomic ground-state densities, the orientation of which is also determined by the chemical forces. In accordance with the virial theorem, one half of the quasiclassical electrostatic interaction energy,EE, of superimposed independent atoms is an approximation to the total bond energy, BE. The BE= EE/2 correlation also improves if oriented nonspherical instead of spherically averaged atoms are superimposed. This correlation does not mean that the bond energy has been explained electrostatically, because one must know the atomic positions (and orientations) in advance. Furthermore, density deformations due to the quantum-mechanical interactions contribute 2-3 eV to BE.

1. Introduction

Dalton [ 1 ] introduced the concept of atoms in chemistry as building blocks of matter, which retain their individuality and spherical shape to a large extent when they form compounds. It is often implicitly assumed or explicitly stated that independent atoms can only possessspherically symmetric charge distributions. A common practice is to compare the electron-density distribution in molecules and crystals with a reference density which is the superposition of densities of the independent atoms in their spherically symmetric ground states. The corresponding density difference is usually called the “deformation density”. It is believed that any deviation from sphericity of the charge distribution of atoms in molecules or crystals is an indication of the chemical forces between them. However, for atoms with degenerate (or nearly degenerate) ground states, the so-called “deformation density” does not correlate well with the covalent interaction strength (see fig. 1 and refs. [2,3]): Nz with a strong and short triple bond (fig. 1A) has, as expected, a pronounced bond charge, which results from strong quantum-mechanical covalent interference [ 41. The bond charge overlaps with the inner ends of dipolar charge shifts corresponding to lone pairs. FZ, however, with a weak and long single bond (fig. lB), exhibits much larger “deformation densities”. The electron density is even reduced in both the bond and lone-pair regions in comparison to the superposition of spherically averaged atoms used as the reference model. There is a strongly quadrupolar difference-density distribution around the nuclei, indicating that (in lowest-order approximation) the F atoms in F2 are not in a spherical, averaged ground-state ensemble but in a strongly quadrupolar state. The dominant part of the big “total difference density” in fig. 1B is the deviation of the atomic quadrupolar density from sphericity. Fig. 1C shows the “genuine chemical deformation density” of Fz with respect to oriented, quadrupolar groundstate F atoms. The density distribution of an independent atom in a degenerate ground state with angular mo’ Permanent address: Department of Chemistry, East China Normal University, Shanghai, People’s Republic of China. 0009-2614/91/S

03.50 0 1991 -El sevier Science Publishers B.V.

( North-Holland )

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N,-2N ~r$i:Si]

i-2

0

1 4 -2

Fig. 1. Deformation densities & =pmolecy,.-mndcpndcnt Lt0m,. Contour-line values: * n X 0.04 au= f n x 0.27 e/A’ with n = t, 1,2,3,4,5, .... (full lines) density increase; (dashed lines): density decrease. Length scale in au. (A) N2, molecule-spherical atoms; (6) Fz, molecule-spherically averaged F atoms; (C) F,, molecule-oriented quadrupolar ground-state atoms.

.

,

,

,

,

0

,

,

,

,

,

-2

+2

Fig. 2. Ground-state density of the F atom in the ‘P, state and in the spherically averaged 2Pxy.. ensemble (-). ( -) Density contour-line values: 0.67,2, 6.7,20, 67,200 e/A”.Length scale in au.

mentum 1 is not uniquely defined. It is a general linear combination of ( 2°)-monopolar, ( 22)-quadrupolar, .. .. and ( 22’)-po1ar distributions. A free independent atom may be in an aspherical ground state, e.g. F in the 2P, state with a ls22s22pZ2pj2pf population in the Hartree-Fock approximation; it may be in a sphericufgruundstate ensemble f (2PX+‘P,,+‘P,) corresponding to a ls22s22p~/32p~‘32p~/3 population; or it may be in some intermediate ground state which resembles the electron density in the respective molecule at best. Subtracting “optimally oriented” instead of spherically averaged independent F atoms from the F2 molecule, one obtains a small bond charge and well-separated lone-pair dipoles for the weak, long F2 bond. Here, the genuine “chemical deformation” is no longer buried under the big effect of orientation of independent quadrupolar atoms ISI. The difference between the most aspherical density and the spherically averaged density of a given degenerate atomic ground state is small (see fig. 2) but it is still up to an order of magnitude larger than typical bonddeformation densities. Accordingly, it is more suitable to look upon matter as a superposition of atoms at spec@edpositions, with the degenerate atomic ground-state components appropriately selected and oriented: atomic core positions and valence-orbital orientations and populations optimally adapted for chemical interaction in the sense of zeroth-order perturbation theory [ 3,6]. Just as the nuclear positions cannot be guessed, also the orientations and populations of the degenerate atomic valence shells must be determined from experiment [ 21 or from quantum-chemical calculations [ 31 by some optimization procedure. At higher-order perturbation level, the oriented valence shells become also weakly deformed (genuine chemical deformation). 510

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The total experimental or quantum-mechanical bond energy, BE [ 7 ] can be partitioned into components. Spackman and Maslen [ 8 ] have recently shown that the so-called quasiclassical [ 41 electrostatic energy, EE, of spherically averaged ground-state atoms A, B at their quantum-mechanical equilibrium positions,

(where pA is the electronic and nuclear charge distribution of atom A) represents a dominant part of the total bond energy; and that the difference between BE and EE correlates well with the difference in electronegativity (D-EN) [ 91 of the two atoms which form the bond. As noted above, the deformation density changes drastically if the independent-atom reference model is constructed from oriented ground-state atoms instead of spherically averaged ones. The questions, therefore, arise whether the EE also changes, and whether the EE of oriented atoms correlates better or worse with the BE than the EE of sphericalized atoms.

2. Calculations The electrostatic interaction energies, EE, of spherically averaged and of optimally oriented atoms in 30 molecules containing 33 different diatomic bonds were calculated by the Amsterdam Hartree-Fock-Slater method [lo] ( EEHFS). Triple-zeta ST0 basis sets were used for the atoms. The positions of the atomic nuclei in the molecules were taken from experiment. The populations of the degenerate oriented atomic valence shells in a given molecule were determined in two different ways (see tcfs. [ 3,111): first by Mulliken population analysis (“Mulliken orient”); second by minimizing the difference between the molecular density and the density of the superimposed independent degenerate atoms (“density orient”). The results are displayed in table 1, together with the total bond energies, BE, and the electrostatic energies between sphericalized Hat-tree-Fock atoms (EE3p:e’, Spa&man and Maslen [ 81) _The differences of a few tenths of an electron volt between the latter values in column 8 and the present ones (EE@? in column 7) are due to slightly different atomic densities in the Hartree-Fock ground state and Hartree-Fock-Slater ground configuration.

3. Analysis of data and discussion 3.1. BE-EE relations Spackman and Maslen [ 81 found linear correlations between BE and EE, and between the difference, BE-EE, and the absolute value of the electronegativity difference, 1D-EN 1, of the two bonded atoms. Overlap and, correspondingly, the EE correlate with covalent bond strength. According to Pauling [ 91, the square of D-EN is defined via polar bond stabilization. Therefore, we correlate BE with both EE and (D-EN)2. From the auxiliary material of ref. [S] on 148 molecules and the corresponding sphericalized ground-stateHartree-Fock atoms, we obtain BE? 1.1 eV = 1.1 +0.45 eV+ (0.5,?0.0,)EE~~ km

~0.8~

km=O.6,

+ 0.4,k0.09 eV(D-EN)2, /CD-EN

(la)

zo.27

where the ks are the total and individual coefficients of the linear correlation y= I,&

as defined by [ 121

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Table I Bond energy, BE, and electrostatic energy, EE, between spherically averaged or oriented atomic ground states for 33 bonds. D-EN is the difference of Pauling’s electronegativity between the bonded atoms. Energies in eV Molecule

Bond

BE”

EEnrs

AlF BF BH

Al-F B-F B-H B-H B-N B-O c-c c-c C-H C-H C-H Cl-H c-o F-F F-H F-K C-H C-N O-H S-H O-H O-H o-o Li-F Li-Li Mg-F Mg-0 Mg-S N-N N-O o-o P-P Be-Be

6.93 7.90 3.55 4.03 4.08 a.37 6.32 8.5 4.9 4.1 4.54 4.62 11.23 1.66 6.11 5.11 4.6 9.0 4.63 3.78 5.01 5.0 1.4 6.00 1.08 4.72 3.97 2.66 9.91 6.62 5.21 5.08 0.20

4.50 8.29 2.61 2.49 4.85 9.09 4.80 9.63 2.19 2.77 2.36 3.00 13.23 I .92 3.05 0.83 2.19 12.05 3.23 2.85 2.83 3.55 I .38 I.13 0.37 2.7 1 2.44 2.63 13.66 10.13 1.79 9.45 0.75

BH, BN BO CZ GHz CzH, CHZ CH4 CIH co F2 FH FK HCN HCN HO HS Hz0 Hz02 Hz& LiF Li2 MgF MgO MgS N2 NO 02 PZ Bel

b)

C)

EEHFS

6.68 11.68 2.6 I 2.49 7.76 14.46 6.37 16.60 3.28 2.82 2.36 3.36 19.65 1.92 3.07 0.64 3.47 19.3 1 3.18 3.2 1 3.01 3.50 1.33 1.33 0.37 2.7 1 2.44 2.45 22.92 12.69 8.84 13.68 0.75

a) Experimental, ref. [7]. ‘) Mulliken orient. ‘) Density orient. ‘) Spherical. r) Spherical, ref. [ 81. I) Ref. [9].

A=HFS

Ii)

2.39 4.13 1.11 0.98 0.44 5.56 -2.23 1.45 1.16 0.63 0.00 -0.47 1.62 -2.06 -0.59 -0.33 0.07 8.35 0.00 0.02 -0.34 0.27 -2.77 0.04 0.00 -0.03 -0.57 -1.16 9.26 0.40 -1.65 5.30 0.00

EEHFS

4.29 7.55 1.50 1.51 7.32 8.54 8.60 9.15 2.12 2.19 2.36 3.83 12.03 3.98 3.66 0.97 2.12 10.96 3.18 3.19 3.35 3.23 4. IO 1.29 0.37 2.74 3.01 3.61 13.66 12.29 10.49 8.38 0.75

e,

=HF

f)

D-EN r)

4.23 1.63 1.61

2.37 1.94 0.16 0.16 1.00 1.40 0.00 0.00 0.35 0.35 0.35 0.96 0.89 0.00 1.78 3.16 0.35 0.49 1.24 0.38 1.24 1.24 0.00 3.00 0.00 2.67 2.13 1.27 0.00 0.40 0.00 0.00 0.00

7.63 8.84 9.14

3.79 12.56 3.67 4.14 0.85

3.60 3.11

1.37 0.39 2.61 2.87 3.61 14.34 12.50 10.12 8.49

di Density orient minus spherical (column 5 -column

7).

where ( ) is the average over the data points. The accuracy limits are given in terms of the variances rr, estimated for subsets of 33 bonds, which is the number of bonds we have investigated in the present work. The correlation coefficient !~,,_a~is rather small: It does not even make a significant difference, if (D-EN)* or 1D-EN 1=d@%$ is used as the variable of linear regression. If the dependence of the BE on the D-EN is not accounted for at all, one obtains BE?1.5eV=2.2~0.4eV+(0.4,?O.I,)EE@,

k=kE,=0.65.

(lb)

For our 33 bonds and spherical ground-conJiguration Hartree-Fock-Slater atoms, we obtain similar results: BEfl.SeV=l.9+0.4,eV+ kt.,,=iM,

and 512

(0.S4+0.07)EE$~ kEE ZO.73

+ (0.33fO.lo)eV(D-EN)’ bEN=O.lz

(2a)

Volume 180,number 6

BE+ 1.7 eV=2.8?0.4eV+

CHEMICALPHYSICS LETTERS

(0.4a fO.O,)EEgz,

k=0.73.

7 June 1991

(2b)

First, in both cases roughly one half of the variation of the electrostatic interaction energies of sphericalized atoms represents the total bond-energy variation (factor of x 0.5 in front of EE). In the context of the virial theorem, according to which the energy of stationary states amounts to one half of the electrostatic energy of the exact charge distribution, this result seems very reasonable. Second, the factors for polar bond stabilization proportional to (D-EN) * are = j eV, which is much smaller than Pauling’s value of 1.0 eV. The classical electrostatic interaction of a simple superposition of neutral undeformed atoms at their correct positions already also accounts for a large part of the bond energy of polar bonds, while their bonding is conventionally ascribed to charge transfer (see also ref. [ 131). 3.2. Oriented versussphericalizedatoms in molecules The EEs of oriented atoms with populations according to Mulliken or density fitting are given in the upper and lower rows of the following formulae:

(3a) kEE=O&

kam=o.12

and

(3b) From eqs. (2a), (2b) to (3a), (3b), there is a 20% decrease in statistical error of BE from 1.5/1.7 eV to l.l/ 1.4 eV, respectively. The statistical accuracy of the constant and linear coefficients is also somewhat better for oriented atoms (0.25-0.3, eV instead of 0.4-0.4, eV for the constant term; ~0.045 instead of eO.075 for the term linear in EE). The correlation coefficients k,,, and kEEor k are ~0.1 better in eqs. (3). Density fitting yields smaller variances for the constant and linear coefficients than the Mulliken populations. Increasing thep,-populationof an atom in a molecule at the expense of its p,-population decreases the overlap of the electronic-charge clouds. Then, the electrons of one cloud undergo less nuclear attraction by the other atom. The electrostatic attractionenergy is thereby decreased. Changes of the EE between - 3 and + 9 eV are found upon atomic orientation (see column 6 of table 1). We give a classification into three cases: For one group of bonds, EE,,,< EE,,,,,,. When electron-rich atoms form a homopolar o-bond, their pmAOs are tilled with lone-pair electrons, i.e. p,-population> p,-population (F2, 02, HO-OH). In F1, for example, tbe F2 p,-population is 2, while the F2 pdpopulation is 1. The same situation happens in molecules with K- but no o-bonding ( CZ rr4). If an electron-rich atom forms a polar bond, its p-shell will become nearly closed so that the asphericity is only small. Then, EE,,,,, 5 EE,,,,, (MgS, MgO, ClH, FH, 0H2). For comparable p,, and pX populations, EEoAti x EE ,,,,.This occurs for highly symmetric bonding (CH,), for ionic bonding with closed-shell ions (LiF, RF, MgF), and for s-s bonds (Liz, Be*). EEorient2 EEsphe,and p=population 2 p,-population happen for homopolar H-bonds (R= CH, CH,, BH, BH,, SH, OH, H,O,) and for some other intermediate cases of single to double bonding (BN, NO). The other extreme case happens for short multiple bonds (&HZ, Nz: CO, BF, AlF, BO, P2, CN). Strong overlap enhances quantum-mechanical interference, leading to density accumulation on the bond path. Density fitting then favors p,,-population, resulting in EE,,d,,,, > EEsrhcr [ 111. So far we have based our discussion on energy arguments. Some people prefer to discuss electrostatic in513

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teractions with the help of Hellmann-Feynmanforces. For instance, F atoms oriented along the bond axis have parallel quadrupole moments which repel each other:

Upon integration, the corresponding additional repulsive Hellmann-Feynman force (i.e. that due to the atomic orientation) results in an additional repulsive antibonding energy contribution, as mentioned above. Although the conclusion is correct, the argument is irrelevant. The Hellmann-Feynman theorem holds for quantum-mechanically correct charge distributions and is very sensitive to density modifications near the atomic nuclei [ 141. It must not be applied to model densities such as that of superimposed independent atoms. For instance, the integral Hellmann-Feynman theorem always predicts an antibondingforce between neutral spherical atoms, while the electrostatic interaction energy of overlapping neutral spherical atoms is always bonding for medium and large distances [ 15 1.

4. Conclusion Concerning the electrondensitiesof molecules, the superposition of oriented independent ground-state atoms is a significantly better model than the superposition of spherically averaged atoms. Oriented independent atoms are also a more suitable reference for the definition of deformation densities if they are used in the discussion of chemical bonding. A striking example is F2 in fig. 1 (see also refs. [ 2,3] ). Concerning the electronic energies, Spackman and Maslen [ 81 have shown that the quasiclassical electrostatic interaction energies, EE, of spherically averaged independent atoms at their quantum-mechanically defined equilibrium positions represent a significant part of the chemical-bond energies. The latter cover the range from about 0 to 11 eV with an average around 5 f 24 eV. This relation has been investigated here in more detail.

I.“0 *.CI” 3.A.CN 4.Mgo d.B”,

4

514

8 0.5*EE

12 (eV)

Fig. 3. Bond energy, BE, of 33 bonds versus electrostatic interaction energy,EE, of oriented atoms in their Hartrce-Fock-Slatergroundconfigurations. The straight line BE= 3.1 eV+0.3, EE has u(BE)= f 1.4 eV and acorrelation coefficient k=O.&

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First, in accordance with the virial theorem, EE is approximately twice the bond energy. Therefore, the EE of undeformed atomic charge distributions, the superposition of which is a rough approximation to the true molecular charge distribution, correlates linearly with the bond energy with a slope of z 1 (see fig. 3). Second, since optimally oriented atoms yield a better approximation to the molecular charge distribution than the spherically averaged ones, oriented atoms also yield a better estimate of the molecular bond energy. Third, the deformations of (oriented) atomic ground states in the molecule by the chemical interactions yield only one contribution both to the molecular density (deformation density typically less than + 1 e/A3, see, e.g., figs. IA, 1C) as well as to the molecular bond energy (typically about 2 to 3 eV, see the section on the BE axis of fig. 3). Fourth, even in polar systems, the electrostatic interaction of neutral atoms yields a reasonable approximation to the “ionic bond energy”. Thus, BE-O.5 EEneutralatoms does only weakly correlate with the electronegativity difference, and D-EN “explains” just 0 to 2 eV of the BE. Many homopolar molecules lie x 1 eV below the straight line in fig. 3, while many heteropolar molecules lie x 1 eV above, corresponding to the term 0.3 eV (D-EN)* in eq. (3a). We also note the fact that both the superposition of neutral atoms as well as that of atomic ions reproduce molecular densities comparatively well (see refs. [ 13,161). The molecular energy consists of the electrostatic potential term and the quantum-mechanical kinetic term. Quantum-mechanical dynamics defines the equilibrium positions of the atomic cores and the optimal orientations and populations of open valence shells. If we know these from quantum mechanics or experiment, we can estimate the BE by taking just one half of the quasiclassical electrostatic interaction energy and adding the quantum and deformation corrections of about 2 to 3 eV. This yields the bond energy with an accuracy of &lf eV.

Acknowledgement We are grateful to Professor Baerends and his group for the Amsterdam DF program system and for their extensive support in adapting and using it, and to M.S. Liao for implementing the program on our VAX. We thank Professor Maslen for a copy of his auxiliary material. We thank Professor Baerends, Professor Maslen and Docent Jaquet for their constructive comments. We acknowledge financial support by DFG and by Fonds der Chemischen Industrie, and the services of the University Computer Center.

[ 1] J. Dalton, 1803,published in Mem. Lit. Philos. Manchester II I (1805)271. [ 21 W.H.E. Schwarz, K. Ruedenberg, L. Mensching, L.L.Miller,P. Valtazanos,W.VonNiessenand R. Jacobson,Angew. Chetn. Intern. Ed. EngJ. 28 (1989) 597; W.H.E. Schwarz, L. Mensching, K. Ruedenberg, R. Jacobson and L.L. Miller, Phys. Port, 19 ( 1988) 185. [3] W.H.E. Schwan, K. Ruedenberg and L. Mensching, J. Am. Chem. Sot. I 1I ( 1989) 6926; L. Mensching, W. Von Niessen, P. Valtazanos, K. Ruedenberg and W.H.E. Schwatz, J. Am. Chem. Sot. I1 1 ( 1989) 6933; K. Ruedenberg and W.H.E. Schwan, J. Chem. Phys. 92 ( 1990) 4956. [ 41 K. Ruedenberg, Rev. Mod. Phys. 34 ( 1962) 326; W. Kutzelnigg, in: The concept of the chemical bond, Vol. 2, ed. Z.B. MaksiC (Springer, Berlin, 1990) p. I. [ 51W.H.E. Schwarz, P. Valtazanos and K. Ruedenberg, Theoret. Chim. Acta 68 (1985) 471. [ 61 I.E. Niu, S.G. Wang, S. Irle, W.H.E. Schwan, H. Hayd and M. Dolg, J. Chem. Educ., submitted for publication. [ 71 G. Herzberg, Molecular spectra and molecular structure, Vol. 3 (Van Nostrand, Princeton, 1966); K.P. Huber and G. Her&erg, Molecular spectra and molecular structure, Vol. 4 (Van Nostrand Reinhold, New York, 1979); R.C. Weast, ed., Handbook of chemistry and physics, 55th Rd. (CRC Press, Boea Raton, 1974). [8] MA. Spackman and E.N. Maslen, J. Phys. Chem. 90 (1986) 2020. [ 91 L. Pauling, The nature of the chemical bond, 3rd Ed. (Cornell University Press, Ithaca, 1960); I. Mullay, Struct. Bonding 66 (1987) 1.

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[ IO] E.J. Baerends, D.E. Ellis and P. Ros, Chem. Phys. 2 (1973) 41; E.J. Baerends and P. Ros, Intern. J. Quantum Chem. Symp. 12 ( 1978) 169. [ I I ] W.H.E. Schwan and H.L. Lin, Theoret. Chim. Acta, submitted for publication. [ 121 J.B. Kennedy and A.M. Neville, Basic statistical methods for engineers and scientists, 2nd Ed. (Harper and Row, New York, 1976). [ 131 W.H.E. Schwarz, H.L. Lin, J.E. Niu, B. Hess and P. Seiler, Acta Cryst. A, submitted for publication. [ 141 MA. Spa&man and E.N. Maslen, Acta Cryst. A 41 (1985) 347. [ 151W.H.E. Schwan and H.E. Mons, Chem. Phys. Letters 156 (1989) 275. [ 161 P. Seiler and J.D. Dunitz, Helv. Chim. Acta 69 ( 1986) 1107.

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