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Atomic orbital deformation in bond formation: energy effects
Volume 131, number 3
CHEMICAL PHYSICS LETTERS
ATOMIC ORBITAL DEFORMATION
7 November 1986
IN BOND FORMATION: ENERGY EFFECTS
Eric MAGNUSSON ’ Research School of Chemutry, Australian National University, G.P.O. Box 4, Canberra, A.C.T. 2601, Australia
Received 16 April 1986; in final form 27 August 1986
Changes in the radial dependence of atomic orbitals which accompany electronegativity equalization during molecule formation may be monitored by the parallel use of flexible and constrained basis sets in molecular orbital calculations. The stabilization associated with orbital deformation in molecules like BH, or CH.,, which contain many bonding MOs in the valence shell, is due to an increase in the attractive, one-electron term in the molecular energy expression relative to the electron repulsion term. The stabilization which occurs in molecules with an excess of non-bonding electrons in the valence shell is due to reduced interelectronic repulsion.
1. Introduction Pauling’s electroneutrality principle was an early example of the recognition that charge transfer between two different atoms must take place during bond formation and that it is driven by the electronegativity difference between them [ 11. An even earlier discovery was the fact that molecule formation was accompanied by, and even dependent on, an intra-atomic redistribution of charge [2]. Now that the concept of electronegativity equalization has been conceptualized in terms of density functional theory, the roles of these two ways of redeploying charge during molecule formation are easier to understand [ 31. Contributions are made to the molecular electronegativity not only by the potential due to the electronic charge cloud and the nuclei, but by nonCoulombic potentials related to kinetic energy, correlation energy, and exchange energy [4]. All four sources of potential will affect the molecular wavefunction, which, if constructed on an atomic orbital basis, might be expected to exhibit different effects in different orbitals. In particular, bonding MOs should be clearly distinguished from non-bonding MOs by exchange effects. ’ Present address: Department of Chemistry, University of New South Wales, ADFA, Northcott Drive, Campbell, A.C.T. 2600, Australia.
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The importance of the effect of the molecular environment on atomic wavefunctions has been estimated elsewhere [ 51 by comparing the results of Hartree-Fock MO calculations employing constrained, atom-optimized basis sets with calculations using flexible basis sets. The results, qualitative because of the basis-set level of the calculations and the restriction to the Hartree-Fock formalism, nevertheless suggest that molecules of main group elements owe 25-50% of their binding energies to the intra-atomic part of the redeployment of charge which occurs when molecules are formed. In this report, the reasons for the effect on molecular energy of the changes in radial profile of atomic orbitals are explored. For ease of interpretation the calculations are restricted to hydrides in which the major part of the molecular charge density is associated with the central atom. The molecular wavefunctions are calculated with the well-tested 6-3 1G “split valence” basis developed by the Pople group and recalculated at the same geometries with an atomic orbital basis optimized for the ground states of the atoms and constrained to retain that form. Because the primitive Gaussian functions are the same, a particularly ready comparison is possible. The binding energies calculated with the 6-3 1G basis set are often far from the experimental values, a fact which is mainly due to the absence of configuration interaction, but this does not interfere with conclusions 0 009-2614/86/$03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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CHEMICAL PHYSICS LETTERS
drawn about the importance of intra-atomic relaxation effects relative to Hat-tree-Fock binding energies.
2. Method of calculation Molecular orbital calculations conforming to the above prescription have been carried out on a range of molecules containing first- and second-row elements with the “flexible” 6-3 IG basis set recommended by the Pople group [6] and the “constrained” basis set (6-4G) derived from it by combining the primitive Gaussian functions of the valence shell in single four-Gaussian contractions for the valence shell s and p orbitals with contraction coefficients chosen to minimize the ground-state energies of the atoms in question. A description of the technique used in constructing the constrained basis set will be published elsewhere [ 71. Inner shell contractions were the same for both basis sets. Calculations were also performed at the level of the “supplemented” 6-3 1G* basis set and the geometry used for all calculations of each species was obtained by optimization at this level [ 8 1. Calculations using the “flexible” 3-2 1G basis set and the “constrained” 3-3G basis derived from it were also carried out. The results were similar to those obtained from the larger basis and are not reported here; they make it reasonable to argue that conclusions about orbital deformation effects are independent of basis-set size. For atoms the calculations followed the unrestricted Hartree-Fock formalism and for molecules the restricted HF procedure was used. Convergence was deemed to have been reached in the SCF procedure when the rms change in the density matrix between successive cycles was less than 10p9; inconsistencies in atomic wavefunctions became serious when the convergence condition was less rigorous. The GAUSSIAN 80 suite of programs was used for all calculations [ 91. Hartree-Fock molecular energies obtained at the “flexible” and “constrained” basis-set levels ( Ef and EC) yield energy differences (A&_) which may be ascribed to the effect of changes in the radial profile of atomic orbitals in the molecular wavefunction. For first- and second-row hydrides bE,,, takes values between 0.1 and 0.3 hartree. When calculated with basis sets constructed in this way, A&,, is almost
7 November 1986
entirely due to valence-shell effects [ 71. Intra-atomic effects are also apparent in calculations of atomic states with the flexible and constrained basis sets and any analysis of the relative importance of intra-atomic and interatomic interactions in molecule formation must include a control of this variable. Calculations with a wide range of molecules show that the stabilizations due to intraatomic orbital relaxation, appropriately corrected, generally account for one quarter to one half of the Hartree-Fock binding energy [ 5 1.
3. Basis set flexibility effect on molecular energies Electronegativity equalization in the first-and second-row hydrides is expected to lead to contraction of the atomic orbitals used to represent the valence shell electrons of atoms like carbon or silicon; for highly electronegative elements orbital expansion is expected. Table 1 shows the extent to which this expectation is met in flexible basis-set wavefunctions, using mean orbital radii ((r) ) as an indication of contraction/expansion of the AOs in the molecules relative to the free atoms.
Table 1 Mean radii (pm) of valence shell s and p atomic orbitals in LCAO wavefunctions of first- and second-row hydrides a) Molecule XH,
Electronegativity difference
Atomic wavefunction
Molecular wavefunction
Cr.)
Cr.)
(rp)
(rp)
0.0
84
-
73
-
RHx CH, NH, Hz0 HF
-0.1 0.4 0.9 1.4 1.9
103 82 69 60 53
171 132 107 91 73
96 83 70 61 53
128 123 103 94 74
SM, PHI HS HCl
-0.3 0.0 0.4 0.9
110 98 86 77
228 193 166 141
106 97 86 77
173 160 156 129
Hz
‘) Mean radii in atomic wavefunctions refer to the lowest energy UHF s or p orbital in the valence shell. For the molecular case. ( r, > and ( r,,) are given for the bonding orbitals to which the valence shell AOs of the heavy atom make the largest contribution (3-21G level UHF and RHF calculations).
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Volume 13 1, number 3
7 November 1986
The magnitudes of both one- and two-electron integrals are increased when the orbital exponent is increased, but to different degrees, and the result depends on the nature of the electron distribution. For example, one-centre repulsion integrals are much more strongly affected by a change in exponent than two-centre integrals. Therefore, reduction of exponent might be favoured to lower the total energy in electron-rich atoms and molecules with similar structures and increase of the exponent to raise the total energy in compounds with few non-bonding electrons and many bonds. , Results of the parallel calculations are given in tables 2 and 3. The one-electron energy component (C 2H,,) is obtained from the total molecular energy and the sum of the eigenvalues (2 2H,, = 2E,,, - 2 1 t,) and the two-electron component by the relation 1 ( W-K) =2 IXE,-Et,,. Although the values of A&,, (the lowering in total molecular energy between results at constrained and flexible basis-set levels) for the compounds of first- and second-row elements in tables 2 and 3 are modest (0.13 + 0.07 and 0.09 f 0.03 hartree) these values are the result of changes in opposite directions of quantities which are individually quite large.
Mean p orbital radii in table 1 refer to the MO to which the central atom valence p orbital makes the greatest contribution and for which, therefore, “exchange potential” effects are expected to be greatest. s orbitals are generally little involved in bonding and the ( rS) are almost indistinguishable from the atomic values. As expected BH3, CH4 and SiH4 are exceptions to this rule, since in these compounds topological considerations enforce a high level of s orbital bonding [ lo]. The radial dependence of valence AOs across the full MO spectrum varies over a wide range, responding to the changing constraints of the electrostatic, exchange, kinetic and correlation terms in the potential from one part of the molecule to another. For example, in HCl the MO with the largest p orbital contribution has (r,) =407 pm, while ( rP) = 461 pm in the non-bonding px electron pairs on the chlorine atom. Compared with the atomic mean radius ( (r,) =441 pm), the orbital is contracted when involved in bonding, expanded when nonbonding. The effects of orbital contraction and expansion may be obtained by analyzing the one- and two-electron energy components of the molecular energies.
Table 2 Effect of basis-set flexibility on attractive and repulsive terms in the molecular energy: first-row hydrides a)
BH,
CH,
NH,
-26.3768 (-26.2678) -0.1090
-40.1804 ( - 39.9820) -0.1984
-56.1615 ( - 56.0405) -0.1210
-75.8659 (-75.9843) -0.1184
-99.9830 (-99.8939) -0.0890
-9.3102 (-9.7176)
- 13.7863 (- 14.2132)
-18.3537 ( - 18.4910)
-23.6976 (-23.5322)
-29.8812 (-29.4765)
-34.1332 ( - 33.1003) - 1.0329
-52.7882 (-51.5377) - 1.2504
-75.6156 (-75.0989) -0.5167
- 104.5735 ( - 104.6675) 0.0941
- 140.2036 (-140.8349) 0.6313
A(2J-Wnex
7.7565 (6.8325) 0.9240
12.6078 (11.5555) 1.0521
19.4541 (19.0584) 0.3957
28.5892 (28.8016) -0.2124
40.2207 (40.9410) - 0.7203
supplemented-basis results A@,.. AI+%,, A(2J-K)%,.
-0.1223 - 1.0675 0.9453
-0.2132 - 1.2765 1.0636
-0.1439 - 0.5705 0.4266
-0.1448 0.0205 -0.1654
-0.1088 0.5835 -0.6923
Dff,,
Hz0
HF
a) See text for meanings of symbols. Results are given for 6-3 1G calculations and, in parentheses, 6-4G level calculations. The supple mented basis is the 6-3 1G* basis, and results obtained by its use are distinguished by an asterisk [ 61.
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CHEMICAL PHYSICS LETTERS
7 November 1986
Table 3 Effect of basis-set flexibility on attractive and repulsive terms in the molecular energy: second-row hydrides ‘) SiH,
PH,
H,S
HCl
-291.1735 (-291.0807) -0.0929
- 342.3960 ( - 342.3207) -0.0753
- 398.6267 ( - 398.5632) -0.0635
-460.0341 (-459.9178) -0.1163
Xe,
-89.8610 (-90.9251)
- 105.9488 ( - 106.6769)
- 123.3780 (-123.6331)
- 142.1066 (-141.9375)
x2&
-402.6251 (-400.3111) -2.3140
-472.8944 (-471.2877) - 1.6068
- 550.4973 ( - 549.8602) -0.6371
-635.8549 ( - 635.9606) 0.1056
130.4985 (128.9669) 1.5315
151.8706 (152.2970) 0.5736
175.8209 (176.0428) -0.2219
-0.1272 -1.8930 1.7657
-0.1041 - 0.8262 0.7217
-0.1422 0.2765 -0.4187
E total
A&er
We. 1(2J-K) 6(2J-J&e,
111.4515 (109.2305) 2.2211
supplemented-basis results Wk. -0.1445 AH%,, -2.6122 A(2J-K)%,, 2.4678
a) See text for meanings of symbols. Results are given for 6-31G calculations and, in parentheses, 3-3G level calculations. The supplemented basis is the 6-3 lG* basis [ 61.
When the one- and two-electron components of the molecular energy are separated, the compounds from the first and second rows behave similarly. For molecules like BH3, CH, and SiH4 the extra stabilization resulting from contraction of the atomic orbitals involved in bonding (observable only with the flexible basis set) is gained through the one-electron term, in spite of the concurrent disadvantage of increased electron repulsion. H2 falls into the same category. For the hydrides of elements to the right of the periodic table, the opposite situation prevails. For H20, HF and HCI the resultant negative A&,, is achieved, because the reduced one-electron term is more than offset by the reduced interelectronic repulsion term. HZ0 and HZ!3are both close to the crossover.
4. Basis-set effects on eigenvahes In atoms, the effect of basis-set flexibility (AC& is uniformly negative for both core and valence shell eigenvalues. The magnitudes rise from left to right as expected for systems in which interelectronic repulsion rises in importance as the p shell is tilled. The effects in molecules contrast sharply with this. (For data, see table 4.)
In molecules, the energies of individual molecular orbitals (t,) are profoundly affected by the constraint of using atom-optimized atomic orbitals. Across the first-row hydrides, BH3, CH4 and NH3 show a rise in core orbital energies associated with removal of the constraint of the atom-optimized 64G basis set, i.e. de flexis positive. For the core orbitals of Hz0 and HF, AtfleXis negative. In the secondTable 4 Core orbital energy differences for atoms and molecules calculated with flexible and constrained basis sets Molecule
RH3 CH, NH3 HzG HF SiH, PH3
HzS HCl
Aen,, atom
molecule
0.023 -0.188 -0.067 -0.133 -0.392 - 0.046 - 0.036 -0.081 -0.142
0.228 0.185 0.034 -0.101 -0.189 0.176 0.118 0.028 -0.038
” Energy differences calculated from lowest UHF eigenvalues (atomic ground states) and lowest RHF eigenvalues (molecules), each calculated at 6-31G and 6-4G levels.
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row hydrides, the behaviour is similar, but only HCl shows a negative AC,,_for the innermost orbital. For orbitals outside the core, the eigenvalue differences between the flexible and constrained level calculations are as follows. For BHJ, CH, and SiH4, Atflex is positive for all MOs including the core orbital, but across the periodic table the other compounds exhibit negative Aeflexvalues for progressively more and more MOs until HF and HCl are reached, when A+,,, becomes negative for all MOs. (There is a single exception for HCl, the HOMO.) The magnitude of the eigenvalue differences is generally much smaller for the outer orbitals than for the core orbitals. Attention to the individual MO energies shows how important the redistribution of charge around atoms can be, when the split valence basis makes it possible. Normally, when results of calculations at a single basis-set level on different molecules are compared, an increase in the core eigenvalue is taken to indicate raised core/valence interelectronic repulsion and a lowering in the effective charge of the atom [ 111. By analogy, a positive AC,,_for the core eigenvalue may also be interpreted as due to a lowered effective charge, this time as a result of the changes in valence shell radial distribution accompanying electronegativity equalization. Analysis of this behaviour corroborates the hypothesis above that orbital deformation is responsible for an enhanced one-electron energy term in compounds like CH4 and BHs and a reduced interelectronic repulsion term in molecules like HF and H20. The core orbital energy of CH, is useful as a first example. The density of the C,, function, the principal component of this MO, is very high. It changes from 0.991 to 0.996 when the 6-4G set is replaced by the flexible 6-3 1G set. By itself, this change should lower the MO energy by about 0.06 hartree, this tigure being based on the presumed relationship between C,, density and orbital energy. Instead, Acflex= + 0.19 hartree, showing that electron repulsion between the carbon core electrons and valence shell electrons must have risen to offset this stabilization by about 0.25 hartree, a rather large quantity. Altogether, repulsion rises by 1.05 hartree as the AOs contributing to the valence shell become more compact. However, the overall stabilization (AEfleX = -0.20 hartree) shows that contraction has increased the attractive term even more. 228
7 November 1986
CHEMICAL PHYSICS LETTERS
With
HF,
the
core
eigenvalue
is reduced
( Aeaex = - 0.19 hartree), and again this movement must be attributed to a change in repulsion between core and valence electrons, since the density of the F,, A0 in the core MO is reduced by changing from the 6-4G to the 6-3 1G basis; by itself this reduction should cause the MO energy to rise. In this case, the overall stabilization (A&,, = - 0.09 hartree) is due to a large reduction in total repulsion (0.72 hartree) offsetting the reduction in the attractive term (0.63 hartree)
.
Analysis of the one- and two-electron components of the molecular energies calculated at the 6-31G* basis-set level reveals the effects of supplementation of the s,p basis. In CH4, supplementation increases both the attractive and the repulsive energy terms, but the former more than the latter, leading to a total energy increment of - 0.0 13 hartree. Contrariwise, in HCl the overall energy response to supplementation (0.026 hartree) occurs, because the attractive energy rises by only 0.17 1 hartree while the electron repulsion term drops by 0.197 hartree. The effect of the higher-order functions is simply to enhance the two kinds of effects referred to above - increased attractive terms in the energies of CH4 and similar molecules, and reduced repulsive terms in the energies of HF, HCl, etc. Acknowledgement I thank Dr. Ross Nobes for computational assistance and the referee for suggesting the extension of the analysis to the supplemented basis set results. References [1] L. Pauling, J. Chem. Sot. (1948) 1461. [2] S. Wang, Phys. Rev. 31 (1928) 579. [ 31 R.G. Parr, R.A. Donnelly, M. Levy and W.E. Palke, J. Chem. Phys. 68 (1978) 3801; R.G. Parr, Ann. Rev. Phys.Chem. (1983) 631; N.H. March, Roy. Sot. Chem. Spec. Per. Rept. (Theoret. Chem.) 4 (1984) 92; R.G. ParrandR.G.Pearson, J.Am. Chem. Sot. 105 (1983) 7512; R.T. Sanderson, Science 114 (195 1) 670; Chemical bonds and bond energy, 2nd ed. (Academic Press, New York, 1976); P. Politzer and H. Weinstein, J. Chem. Phys. 71 (1979) 4218.
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[4] P. Politzer, R.G. Parr and D. Murphy, J. Chem. Phys. 79 (1983) 3859. [ 51 E. Magnusson, to be published. [6] W.J. Hehre, R. Ditchtield and J.A. Pople, J. Chem. Phys. 56 (1972) 2257; J.D. Dill and J.A. Pople, J. Chem. Phys. 62 (1975) 2921; M.M. Francl, W.J. Pietro, W.J. Hehre, J.S. Binkley, D.J. DeFrees and J.A. Pople, J. Chem. Phys. 77 (1983) 3654; J.S. Binkley, J.A. Pople and W.J. Hehre, J. Am. Chem. Sot. 102 (1980) 939; M.S. Gordon, J.S. Binkley, J.A. Pople, W.J. Pietroand W.J. Hehre, J. Am. Chem. Sot. 104 (1982) 2797; W.J. Pietro, M.M. Francl, W.J. Hehre, D.J. DeFrees, J.A. Pople and J.S. Binkley, J. Am. Chem. Sot. 104 (1982) 5039; R. Ditchfield, W.J. Hehre and J.A. Pople, J. Chem. Phys. 54 (1971) 724.
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[ 71 E. Magnusson, to be published. [8] R.A. Whiteside, M.J. Frisch and J.A. Pople, eds., Camegie-Mellon Quantum Chemistry Archive, 3rd ed., Department of Chemistry, Carnegie-Mellon University, Pittsburgh (1983). [ 91 J.S. Binkley, R.A. Whiteside, R. Krishnan, R. Seeger, D.J. DeFrees, H.B. Schlegel, S. Topiol, L.R. Kahn and J.A. Pople, GAUSSIAN 80, QCPE 12 (1980) 406. [lo] E.A. Magnusson, J. Am. Chem. Sot. 106 (1984) 1177. [ 1 I] W.L. Jolly and W.B. Perry, J. Am. Chem. Sot. 95 (1973) 5442; P. Politzer and A. Politzer, J. Am. Chem. Sot. 95 (1973) 5450.
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CHEMICAL PHYSICS LETTERS
ATOMIC ORBITAL DEFORMATION
7 November 1986
IN BOND FORMATION: ENERGY EFFECTS
Eric MAGNUSSON ’ Research School of Chemutry, Australian National University, G.P.O. Box 4, Canberra, A.C.T. 2601, Australia
Received 16 April 1986; in final form 27 August 1986
Changes in the radial dependence of atomic orbitals which accompany electronegativity equalization during molecule formation may be monitored by the parallel use of flexible and constrained basis sets in molecular orbital calculations. The stabilization associated with orbital deformation in molecules like BH, or CH.,, which contain many bonding MOs in the valence shell, is due to an increase in the attractive, one-electron term in the molecular energy expression relative to the electron repulsion term. The stabilization which occurs in molecules with an excess of non-bonding electrons in the valence shell is due to reduced interelectronic repulsion.
1. Introduction Pauling’s electroneutrality principle was an early example of the recognition that charge transfer between two different atoms must take place during bond formation and that it is driven by the electronegativity difference between them [ 11. An even earlier discovery was the fact that molecule formation was accompanied by, and even dependent on, an intra-atomic redistribution of charge [2]. Now that the concept of electronegativity equalization has been conceptualized in terms of density functional theory, the roles of these two ways of redeploying charge during molecule formation are easier to understand [ 31. Contributions are made to the molecular electronegativity not only by the potential due to the electronic charge cloud and the nuclei, but by nonCoulombic potentials related to kinetic energy, correlation energy, and exchange energy [4]. All four sources of potential will affect the molecular wavefunction, which, if constructed on an atomic orbital basis, might be expected to exhibit different effects in different orbitals. In particular, bonding MOs should be clearly distinguished from non-bonding MOs by exchange effects. ’ Present address: Department of Chemistry, University of New South Wales, ADFA, Northcott Drive, Campbell, A.C.T. 2600, Australia.
224
The importance of the effect of the molecular environment on atomic wavefunctions has been estimated elsewhere [ 51 by comparing the results of Hartree-Fock MO calculations employing constrained, atom-optimized basis sets with calculations using flexible basis sets. The results, qualitative because of the basis-set level of the calculations and the restriction to the Hartree-Fock formalism, nevertheless suggest that molecules of main group elements owe 25-50% of their binding energies to the intra-atomic part of the redeployment of charge which occurs when molecules are formed. In this report, the reasons for the effect on molecular energy of the changes in radial profile of atomic orbitals are explored. For ease of interpretation the calculations are restricted to hydrides in which the major part of the molecular charge density is associated with the central atom. The molecular wavefunctions are calculated with the well-tested 6-3 1G “split valence” basis developed by the Pople group and recalculated at the same geometries with an atomic orbital basis optimized for the ground states of the atoms and constrained to retain that form. Because the primitive Gaussian functions are the same, a particularly ready comparison is possible. The binding energies calculated with the 6-3 1G basis set are often far from the experimental values, a fact which is mainly due to the absence of configuration interaction, but this does not interfere with conclusions 0 009-2614/86/$03.50 Q Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 13 1, number 3
CHEMICAL PHYSICS LETTERS
drawn about the importance of intra-atomic relaxation effects relative to Hat-tree-Fock binding energies.
2. Method of calculation Molecular orbital calculations conforming to the above prescription have been carried out on a range of molecules containing first- and second-row elements with the “flexible” 6-3 IG basis set recommended by the Pople group [6] and the “constrained” basis set (6-4G) derived from it by combining the primitive Gaussian functions of the valence shell in single four-Gaussian contractions for the valence shell s and p orbitals with contraction coefficients chosen to minimize the ground-state energies of the atoms in question. A description of the technique used in constructing the constrained basis set will be published elsewhere [ 71. Inner shell contractions were the same for both basis sets. Calculations were also performed at the level of the “supplemented” 6-3 1G* basis set and the geometry used for all calculations of each species was obtained by optimization at this level [ 8 1. Calculations using the “flexible” 3-2 1G basis set and the “constrained” 3-3G basis derived from it were also carried out. The results were similar to those obtained from the larger basis and are not reported here; they make it reasonable to argue that conclusions about orbital deformation effects are independent of basis-set size. For atoms the calculations followed the unrestricted Hartree-Fock formalism and for molecules the restricted HF procedure was used. Convergence was deemed to have been reached in the SCF procedure when the rms change in the density matrix between successive cycles was less than 10p9; inconsistencies in atomic wavefunctions became serious when the convergence condition was less rigorous. The GAUSSIAN 80 suite of programs was used for all calculations [ 91. Hartree-Fock molecular energies obtained at the “flexible” and “constrained” basis-set levels ( Ef and EC) yield energy differences (A&_) which may be ascribed to the effect of changes in the radial profile of atomic orbitals in the molecular wavefunction. For first- and second-row hydrides bE,,, takes values between 0.1 and 0.3 hartree. When calculated with basis sets constructed in this way, A&,, is almost
7 November 1986
entirely due to valence-shell effects [ 71. Intra-atomic effects are also apparent in calculations of atomic states with the flexible and constrained basis sets and any analysis of the relative importance of intra-atomic and interatomic interactions in molecule formation must include a control of this variable. Calculations with a wide range of molecules show that the stabilizations due to intraatomic orbital relaxation, appropriately corrected, generally account for one quarter to one half of the Hartree-Fock binding energy [ 5 1.
3. Basis set flexibility effect on molecular energies Electronegativity equalization in the first-and second-row hydrides is expected to lead to contraction of the atomic orbitals used to represent the valence shell electrons of atoms like carbon or silicon; for highly electronegative elements orbital expansion is expected. Table 1 shows the extent to which this expectation is met in flexible basis-set wavefunctions, using mean orbital radii ((r) ) as an indication of contraction/expansion of the AOs in the molecules relative to the free atoms.
Table 1 Mean radii (pm) of valence shell s and p atomic orbitals in LCAO wavefunctions of first- and second-row hydrides a) Molecule XH,
Electronegativity difference
Atomic wavefunction
Molecular wavefunction
Cr.)
Cr.)
(rp)
(rp)
0.0
84
-
73
-
RHx CH, NH, Hz0 HF
-0.1 0.4 0.9 1.4 1.9
103 82 69 60 53
171 132 107 91 73
96 83 70 61 53
128 123 103 94 74
SM, PHI HS HCl
-0.3 0.0 0.4 0.9
110 98 86 77
228 193 166 141
106 97 86 77
173 160 156 129
Hz
‘) Mean radii in atomic wavefunctions refer to the lowest energy UHF s or p orbital in the valence shell. For the molecular case. ( r, > and ( r,,) are given for the bonding orbitals to which the valence shell AOs of the heavy atom make the largest contribution (3-21G level UHF and RHF calculations).
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7 November 1986
The magnitudes of both one- and two-electron integrals are increased when the orbital exponent is increased, but to different degrees, and the result depends on the nature of the electron distribution. For example, one-centre repulsion integrals are much more strongly affected by a change in exponent than two-centre integrals. Therefore, reduction of exponent might be favoured to lower the total energy in electron-rich atoms and molecules with similar structures and increase of the exponent to raise the total energy in compounds with few non-bonding electrons and many bonds. , Results of the parallel calculations are given in tables 2 and 3. The one-electron energy component (C 2H,,) is obtained from the total molecular energy and the sum of the eigenvalues (2 2H,, = 2E,,, - 2 1 t,) and the two-electron component by the relation 1 ( W-K) =2 IXE,-Et,,. Although the values of A&,, (the lowering in total molecular energy between results at constrained and flexible basis-set levels) for the compounds of first- and second-row elements in tables 2 and 3 are modest (0.13 + 0.07 and 0.09 f 0.03 hartree) these values are the result of changes in opposite directions of quantities which are individually quite large.
Mean p orbital radii in table 1 refer to the MO to which the central atom valence p orbital makes the greatest contribution and for which, therefore, “exchange potential” effects are expected to be greatest. s orbitals are generally little involved in bonding and the ( rS) are almost indistinguishable from the atomic values. As expected BH3, CH4 and SiH4 are exceptions to this rule, since in these compounds topological considerations enforce a high level of s orbital bonding [ lo]. The radial dependence of valence AOs across the full MO spectrum varies over a wide range, responding to the changing constraints of the electrostatic, exchange, kinetic and correlation terms in the potential from one part of the molecule to another. For example, in HCl the MO with the largest p orbital contribution has (r,) =407 pm, while ( rP) = 461 pm in the non-bonding px electron pairs on the chlorine atom. Compared with the atomic mean radius ( (r,) =441 pm), the orbital is contracted when involved in bonding, expanded when nonbonding. The effects of orbital contraction and expansion may be obtained by analyzing the one- and two-electron energy components of the molecular energies.
Table 2 Effect of basis-set flexibility on attractive and repulsive terms in the molecular energy: first-row hydrides a)
BH,
CH,
NH,
-26.3768 (-26.2678) -0.1090
-40.1804 ( - 39.9820) -0.1984
-56.1615 ( - 56.0405) -0.1210
-75.8659 (-75.9843) -0.1184
-99.9830 (-99.8939) -0.0890
-9.3102 (-9.7176)
- 13.7863 (- 14.2132)
-18.3537 ( - 18.4910)
-23.6976 (-23.5322)
-29.8812 (-29.4765)
-34.1332 ( - 33.1003) - 1.0329
-52.7882 (-51.5377) - 1.2504
-75.6156 (-75.0989) -0.5167
- 104.5735 ( - 104.6675) 0.0941
- 140.2036 (-140.8349) 0.6313
A(2J-Wnex
7.7565 (6.8325) 0.9240
12.6078 (11.5555) 1.0521
19.4541 (19.0584) 0.3957
28.5892 (28.8016) -0.2124
40.2207 (40.9410) - 0.7203
supplemented-basis results A@,.. AI+%,, A(2J-K)%,.
-0.1223 - 1.0675 0.9453
-0.2132 - 1.2765 1.0636
-0.1439 - 0.5705 0.4266
-0.1448 0.0205 -0.1654
-0.1088 0.5835 -0.6923
Dff,,
Hz0
HF
a) See text for meanings of symbols. Results are given for 6-3 1G calculations and, in parentheses, 6-4G level calculations. The supple mented basis is the 6-3 1G* basis, and results obtained by its use are distinguished by an asterisk [ 61.
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7 November 1986
Table 3 Effect of basis-set flexibility on attractive and repulsive terms in the molecular energy: second-row hydrides ‘) SiH,
PH,
H,S
HCl
-291.1735 (-291.0807) -0.0929
- 342.3960 ( - 342.3207) -0.0753
- 398.6267 ( - 398.5632) -0.0635
-460.0341 (-459.9178) -0.1163
Xe,
-89.8610 (-90.9251)
- 105.9488 ( - 106.6769)
- 123.3780 (-123.6331)
- 142.1066 (-141.9375)
x2&
-402.6251 (-400.3111) -2.3140
-472.8944 (-471.2877) - 1.6068
- 550.4973 ( - 549.8602) -0.6371
-635.8549 ( - 635.9606) 0.1056
130.4985 (128.9669) 1.5315
151.8706 (152.2970) 0.5736
175.8209 (176.0428) -0.2219
-0.1272 -1.8930 1.7657
-0.1041 - 0.8262 0.7217
-0.1422 0.2765 -0.4187
E total
A&er
We. 1(2J-K) 6(2J-J&e,
111.4515 (109.2305) 2.2211
supplemented-basis results Wk. -0.1445 AH%,, -2.6122 A(2J-K)%,, 2.4678
a) See text for meanings of symbols. Results are given for 6-31G calculations and, in parentheses, 3-3G level calculations. The supplemented basis is the 6-3 lG* basis [ 61.
When the one- and two-electron components of the molecular energy are separated, the compounds from the first and second rows behave similarly. For molecules like BH3, CH, and SiH4 the extra stabilization resulting from contraction of the atomic orbitals involved in bonding (observable only with the flexible basis set) is gained through the one-electron term, in spite of the concurrent disadvantage of increased electron repulsion. H2 falls into the same category. For the hydrides of elements to the right of the periodic table, the opposite situation prevails. For H20, HF and HCI the resultant negative A&,, is achieved, because the reduced one-electron term is more than offset by the reduced interelectronic repulsion term. HZ0 and HZ!3are both close to the crossover.
4. Basis-set effects on eigenvahes In atoms, the effect of basis-set flexibility (AC& is uniformly negative for both core and valence shell eigenvalues. The magnitudes rise from left to right as expected for systems in which interelectronic repulsion rises in importance as the p shell is tilled. The effects in molecules contrast sharply with this. (For data, see table 4.)
In molecules, the energies of individual molecular orbitals (t,) are profoundly affected by the constraint of using atom-optimized atomic orbitals. Across the first-row hydrides, BH3, CH4 and NH3 show a rise in core orbital energies associated with removal of the constraint of the atom-optimized 64G basis set, i.e. de flexis positive. For the core orbitals of Hz0 and HF, AtfleXis negative. In the secondTable 4 Core orbital energy differences for atoms and molecules calculated with flexible and constrained basis sets Molecule
RH3 CH, NH3 HzG HF SiH, PH3
HzS HCl
Aen,, atom
molecule
0.023 -0.188 -0.067 -0.133 -0.392 - 0.046 - 0.036 -0.081 -0.142
0.228 0.185 0.034 -0.101 -0.189 0.176 0.118 0.028 -0.038
” Energy differences calculated from lowest UHF eigenvalues (atomic ground states) and lowest RHF eigenvalues (molecules), each calculated at 6-31G and 6-4G levels.
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row hydrides, the behaviour is similar, but only HCl shows a negative AC,,_for the innermost orbital. For orbitals outside the core, the eigenvalue differences between the flexible and constrained level calculations are as follows. For BHJ, CH, and SiH4, Atflex is positive for all MOs including the core orbital, but across the periodic table the other compounds exhibit negative Aeflexvalues for progressively more and more MOs until HF and HCl are reached, when A+,,, becomes negative for all MOs. (There is a single exception for HCl, the HOMO.) The magnitude of the eigenvalue differences is generally much smaller for the outer orbitals than for the core orbitals. Attention to the individual MO energies shows how important the redistribution of charge around atoms can be, when the split valence basis makes it possible. Normally, when results of calculations at a single basis-set level on different molecules are compared, an increase in the core eigenvalue is taken to indicate raised core/valence interelectronic repulsion and a lowering in the effective charge of the atom [ 111. By analogy, a positive AC,,_for the core eigenvalue may also be interpreted as due to a lowered effective charge, this time as a result of the changes in valence shell radial distribution accompanying electronegativity equalization. Analysis of this behaviour corroborates the hypothesis above that orbital deformation is responsible for an enhanced one-electron energy term in compounds like CH4 and BHs and a reduced interelectronic repulsion term in molecules like HF and H20. The core orbital energy of CH, is useful as a first example. The density of the C,, function, the principal component of this MO, is very high. It changes from 0.991 to 0.996 when the 6-4G set is replaced by the flexible 6-3 1G set. By itself, this change should lower the MO energy by about 0.06 hartree, this tigure being based on the presumed relationship between C,, density and orbital energy. Instead, Acflex= + 0.19 hartree, showing that electron repulsion between the carbon core electrons and valence shell electrons must have risen to offset this stabilization by about 0.25 hartree, a rather large quantity. Altogether, repulsion rises by 1.05 hartree as the AOs contributing to the valence shell become more compact. However, the overall stabilization (AEfleX = -0.20 hartree) shows that contraction has increased the attractive term even more. 228
7 November 1986
CHEMICAL PHYSICS LETTERS
With
HF,
the
core
eigenvalue
is reduced
( Aeaex = - 0.19 hartree), and again this movement must be attributed to a change in repulsion between core and valence electrons, since the density of the F,, A0 in the core MO is reduced by changing from the 6-4G to the 6-3 1G basis; by itself this reduction should cause the MO energy to rise. In this case, the overall stabilization (A&,, = - 0.09 hartree) is due to a large reduction in total repulsion (0.72 hartree) offsetting the reduction in the attractive term (0.63 hartree)
.
Analysis of the one- and two-electron components of the molecular energies calculated at the 6-31G* basis-set level reveals the effects of supplementation of the s,p basis. In CH4, supplementation increases both the attractive and the repulsive energy terms, but the former more than the latter, leading to a total energy increment of - 0.0 13 hartree. Contrariwise, in HCl the overall energy response to supplementation (0.026 hartree) occurs, because the attractive energy rises by only 0.17 1 hartree while the electron repulsion term drops by 0.197 hartree. The effect of the higher-order functions is simply to enhance the two kinds of effects referred to above - increased attractive terms in the energies of CH4 and similar molecules, and reduced repulsive terms in the energies of HF, HCl, etc. Acknowledgement I thank Dr. Ross Nobes for computational assistance and the referee for suggesting the extension of the analysis to the supplemented basis set results. References [1] L. Pauling, J. Chem. Sot. (1948) 1461. [2] S. Wang, Phys. Rev. 31 (1928) 579. [ 31 R.G. Parr, R.A. Donnelly, M. Levy and W.E. Palke, J. Chem. Phys. 68 (1978) 3801; R.G. Parr, Ann. Rev. Phys.Chem. (1983) 631; N.H. March, Roy. Sot. Chem. Spec. Per. Rept. (Theoret. Chem.) 4 (1984) 92; R.G. ParrandR.G.Pearson, J.Am. Chem. Sot. 105 (1983) 7512; R.T. Sanderson, Science 114 (195 1) 670; Chemical bonds and bond energy, 2nd ed. (Academic Press, New York, 1976); P. Politzer and H. Weinstein, J. Chem. Phys. 71 (1979) 4218.
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[4] P. Politzer, R.G. Parr and D. Murphy, J. Chem. Phys. 79 (1983) 3859. [ 51 E. Magnusson, to be published. [6] W.J. Hehre, R. Ditchtield and J.A. Pople, J. Chem. Phys. 56 (1972) 2257; J.D. Dill and J.A. Pople, J. Chem. Phys. 62 (1975) 2921; M.M. Francl, W.J. Pietro, W.J. Hehre, J.S. Binkley, D.J. DeFrees and J.A. Pople, J. Chem. Phys. 77 (1983) 3654; J.S. Binkley, J.A. Pople and W.J. Hehre, J. Am. Chem. Sot. 102 (1980) 939; M.S. Gordon, J.S. Binkley, J.A. Pople, W.J. Pietroand W.J. Hehre, J. Am. Chem. Sot. 104 (1982) 2797; W.J. Pietro, M.M. Francl, W.J. Hehre, D.J. DeFrees, J.A. Pople and J.S. Binkley, J. Am. Chem. Sot. 104 (1982) 5039; R. Ditchfield, W.J. Hehre and J.A. Pople, J. Chem. Phys. 54 (1971) 724.
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[ 71 E. Magnusson, to be published. [8] R.A. Whiteside, M.J. Frisch and J.A. Pople, eds., Camegie-Mellon Quantum Chemistry Archive, 3rd ed., Department of Chemistry, Carnegie-Mellon University, Pittsburgh (1983). [ 91 J.S. Binkley, R.A. Whiteside, R. Krishnan, R. Seeger, D.J. DeFrees, H.B. Schlegel, S. Topiol, L.R. Kahn and J.A. Pople, GAUSSIAN 80, QCPE 12 (1980) 406. [lo] E.A. Magnusson, J. Am. Chem. Sot. 106 (1984) 1177. [ 1 I] W.L. Jolly and W.B. Perry, J. Am. Chem. Sot. 95 (1973) 5442; P. Politzer and A. Politzer, J. Am. Chem. Sot. 95 (1973) 5450.
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