Electron Sharing And Chemical Bonding

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Why Does Electron Sharing Lead to Covalent Bonding?A Variational Analysis

KLAUS RUEDENBERG, MICHAEL W. SCHMIDTDepartment of Chemistry and Ames Laboratory USDOE, Iowa State University, Ames, Iowa 50011

Received 10 June 2006; Accepted 20 July 2006

DOI 10.1002/jcc.20553

Published online in Wiley InterScience (www.interscience.wiley.com).

Abstract: Ground state energy differences between related systems can be elucidated by a comparative variationalanalysis of the energy functional, in which the concepts of variational kinetic pressure and variational electrostatic

potential pull are found useful. This approach is applied to the formation of the bond in the hydrogen molecule ion. A

highly accurate wavefunction is shown to be the superposition of two quasiatomic orbitals, each of which consists to

94% of the respective atomic 1s orbital, the remaining 6% deformation being 73% spherical and 27% nonspherical in

character. The spherical deformation can be recovered to 99.9% by scaling the 1s orbital. These results quantify the

conceptual metamorphosis of the free-atom wavefunction into the molecular wavefunction by orbital sharing, orbital

contraction, and orbital polarization. Starting with the 1s orbital on one atom as the initial trial function, the value of

the energy functional of the molecule at the equilibrium distance is stepwise lowered along several sequences of wave-

function modifications, whose energies monotonically decrease to the ground state energy of H2. The contributions of

sharing, contraction and polarization to the overall lowering of the energy functional and their kinetic and potential

components exhibit a consistent pattern that can be related to the wavefunction changes on the basis of physical rea-

soning, including the virial theorem. It is found that orbital sharing lowers the variational kinetic energy pressure and

that this is the essential cause of covalent bonding in this molecule.

q 2006 Wiley Periodicals, Inc. J Comput Chem 28: 391-410, 2007

Key words: covalent bond; electron sharing; variational analysis; kinetic model; hydrogen molecule ion

Introduction

The Covalent Bond

Almost exactly 200 years ago, Volta discovered the electric battery,

Davy and Berzelius discovered electrolysis, and Dalton conceived

the atomic theory, all in the first decade of the nineteenth century!

It is therefore not surprising that Berzelius imagined all chemical

bonding as being what we would now call ionic. By the 1830s,

however, Dumas, Liebig and others had isolated and synthesized

enough nonpolar organic compounds to lead chemists to the recog-

nition of what we now call covalent bonding. This discovery paved

the way for the acceptance, between 1850 and 1860, of Avogadros(1810) and Amperes (1814) view that hydrogen and the gases in

the upper right corner of the periodic table consist of covalently

bonded diatomic molecules, an insight that proved crucial for the

definitive establishment of atomic weights and chemical stoichio-

metries and for the development of the periodic table.1

To those, however, who were searching for a physical expla-

nation of chemical bonding, this kind of short-range interatomic

attraction continued to present a puzzle for the remainder of the

19th century, as witnessed for instance by the Faraday address

Helmholtz2 gave in London at the Royal Institution in 1881. This

lacuna certainly also contributed to the long-lasting chasm be-

tween organic chemists and physicists. After Thomsons discov-

ery of the electron (1897) and Rutherfords discovery of atomic

nuclei (1911) had inspired Bohr to formulate the first model of

the structure of atoms (1913), the physical chemist G.N. Lewis3

proposed in 1916 that a covalent bond is the result of an electron

pair being shared between two atoms. This conjecture was vali-

dated in 1927 by Heitler and Londons calculation of the first

quantum mechanical wavefunction for the hydrogen molecule4

and, since then, all quantum chemical calculations have con-

firmed the connection between electron sharing and covalent

bonding. A further clarification came, however, from the calcula-

tions of Burreau,5 Pauling,6 Finkelstein and Horowitz,7 Guillemin

and Zener,8 and Hylleraas9 for the hydrogen molecule-ion, which

showed that, in fact, the sharing of a single electron establishes a

covalent bond. This finding implied that the two-electron bond is

essentially the cumulative result of the effects of each electron

Contract/grant sponsor: U.S. Department of Energy; contract/grant num-

ber: W-7405-Eng-82

Correspondence to: K. Ruedenberg; e-mail: [email protected]

q 2006 Wiley Periodicals, Inc.


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straightforward for the terms in the potential energy functional

since all of them have electrostatic forms whose energetic behav-

ior is familiar from classical theory. But that is not so for the ki-

netic energy functional. Special care must therefore be exercised

in assessing its contributions. This will prove essential for analyz-

ing the consequences of electron sharing.In the present section, we clarify certain pertinent basic ground

rules. The discussion presumes the Born-Oppenheimer separation

the unrelativistic approximation and clamped point nuclei. All

wave functions denote electronic wave functions.

The Groundstate as the Optimal Compromise

of a Variational Competition

According to quantum mechanical principles, the expectation

value of an observable with the corresponding operator L is givenby the integral h|L|i for any wave function , regardlesswhether is an eigenfunction ofL or not. Thus, h|H|i predictsthe average of a large number of energy measurements on a sys-

tem with the wave function and the Hamiltonian H. In molecu-

lar quantum mechanics, the Hamiltonian operator is the sum ofthe kinetic and the potential energy operators so that

hjHji hjT ji hjVji; (2:1)

hjT ji hjTxji hjTyji hjTzji; (2:2)

hjTwji hj@2=@w2ji (2:3)

in hartree units.

For bound states, the integral h|H|i has a further signifi-cance, namely: the minimum of this energy functional with

respect to any variations in the normalized wavefunction yieldsthe lowest eigenvalue E0 of the operator H, i.e. the exact groundstate energy, and the wave function o for which the integral isminimized is the ground state wave function. From a fundamental

theoretical perspective, the variational calculus formulation is at

least as fundamental as the associated Euler equation, i.e. here the

Schrodinger equation. The variation principle is therefore not

only the basis of most quantum chemical calculations, but it also

provides the implicit basis for much conceptual and qualitative

reasoning, as mentioned in the Introduction. A closer examination

of the components in eq. (2.1) can often lead to an understanding

of the relation between the electronic charge distribution andthe minimal value of the energy functional h|H|i.

The additive decompositions given in eqs. (2.1)(2.3) are of

fundamental relevance for the minimization of h|H|i because:

i. On the one hand, due to the attraction between electrons and

nuclei, the negative potential integral h|V|i tends towardminus infinity as concentrates closer and closer around thenuclei in a molecule.

ii. On the other hand, as described by the uncertainty relation

between position and momentum averages, the positive ki-

netic integral h|T|i tends toward plus infinity as concen-trates closer and closer around any nucleus.

Consequently, the two functionals h|V|i and h|T|i act asantagonists in the minimization of h|H|i, so that this minimiza-

tion has the character of a variational competition between what

we shall call the variational electrostatic potential pull and the

variationally resisting kinetic pressure. The ground state wave

function 0 will be shaped so as to lower the term h|V|i asmuch as possible while concomitantly increasing the term h|T|i

as little as possible. The optimal compromise between these twoantagonists determines the shape of 0 and the value of E0 h0|H|0i. Such a compromise is always possible because, nearany nucleus, h|V|i decreases proportional to the average inversedistance from the nucleus whereas h|T|i increases proportionalto the square of this inverse distance. It can thus be said that the

nuclear electronic attractions continue to pull a variational wave

function in some form together around the nuclei in a molecule

until the variational kinetic energy pressure resists further localiza-

tion and enforces a bound from below. This competition occurs

also when the potential energy contains electronic repulsions as

long as they do not destabilize the system altogether.

Variational Competition, Virial Theorem,and Shifts in the Ground State Energy

The optimal variational compromise furthermore possesses a use-

ful property. Namely, for isolated atoms as well as for molecules

at equilibria as well as transition states, the variational minimum

is characterized by the validity of the virial theorem in the form E0 h0|H|0i h0|V|0i h0|T|0i. While, in many text-books, the virial theorem is derived from the Schrodinger equa-

tion,10,11 i.e. only for eigenfunctions ofH, it is relevant in the pres-ent context that Lowdin26 has established a connection between

this theorem and the variation principle. He has shown that,

because of the above-mentioned specific dependence of the kinetic

and the potential expectation values on the average distance from

any nucleus, the virial identities are a consequence of applying the

variation principle to parameters that govern the concentration of

a wave function around the nuclei, such as notably the orbital

exponents in atom-centered LCAO expansions. Mulliken called

these orbital changes shrinking toward and swelling away

from the nuclei. Note that the theorem also holds for wave func-

tions that are optimized with respect to such parameters even if

they are not exact eigenfunctions.

The virial theorem is useful because it not only allows predict-

ing the result of the minimization with respect to orbital expo-

nents, but it also offers a conceptual understanding for this kind

of optimization. This can help in identifying differences in the

variational competition of similar systems. We shall illustrate this

approach by an example that exhibits certain features, notably

regarding the kinetic energy functional that will become relevant

in the subsequent investigation of the covalent bond.

Consider the following hydrogen atom analogue: a negative

particle with the charge of an electron in the field of a single

infinitely heavy positive particle as described, in atomic units, by

the Schrodinger equation

mr2 Z=r E; (2:4)

where m is the ratio of the negative particles mass to the mass of

the electron, and Z is the ratio of the charge of the nucleus to the

393Electron Sharing and Covalent Bonding

Journal of Computational Chemistry DOI 10.1002/jcc


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charge of the proton, e.g. m 273 for the -muon or Z 19 forthe fluorine nucleus.27 The ground state solution is given by

3=1=2 expr with min mZ; (2:5)

E mZ2; KE mZ2; PE mZ2: (2:6)

The variational competition leading to the results (2.6) is deter-

mined by the dependence of the kinetic and the potential energy

functionals on , namely

hjHji H T V; T 2=2m; V Z:

(2:7)

Figures 13 display plots of T() and [V()] as functions of for various values of m and Z. According to the virial theorem, the

optimal compromise with the lowest value of H() occurs in eachcase where the curves T() and [V()] intersect. Note that 1

is a measure of the orbital extension.Figure 1 exhibits the cases of an electron in the field of three

different nuclei, viz., m 1 with Z 1, Z< 1 and Z > 1. Thecorresponding energy functionals H() are also shown. In allcases, the positive kinetic term dominates in the region of strong

orbital contraction (large ), so that H() increases to infinity. Thenegative potential term dominates in the region of large orbital

expansion (small ), where H() eventually goes to zero. In thestandard case (m 1, Z 1), the intersection of T and []Voccurs for 1 and the total energy is 0.5 hartree. For Z> 1,i.e. a stronger nuclear attraction, the intersection moves to larger

value (min Z), the orbital contracts and the total energyH(min) []Z

2 is lowered. For a weaker nuclear attraction,

i.e. Z < 1, the opposite happens. Thus, increasing (decreasing)

the nuclear attraction binds the electron more (less) tightly, aresult that agrees with classical electrostatic intuition.

A different element enters, however, when we compare cases

having different values of m compared with the standard case

(m 1, Z 1). Figure 2 exhibits the plots for the case of (m 4,Z 1) in addition to those of the standard case. The intersectionis now shifted to a larger value ( 4), when compared withthe standard case and the energy is lowered by a factor 4. This

shift of the virial intersection is manifestly caused by the curve of

T() having been lowered for every argument value, i.e., the var-iational kinetic energy pressure, which resists localization, is

weakened by the increase in mass from m 1 to m 4 so that agreater contraction and energy lowering is variationally possible.

We thus have the superficially paradoxical situation that the over-

all lowering of the variational kinetic energy pressure for each leads to a larger kinetic energy of the optimal min.

The situation is more complex yet for cases with m > 1 and Z< 1,i.e. when both, the nuclear pull as well as the variational kinetic re-

sistance against this pull are weakened when compared with the

standard case. Consider, e.g., the cases m 9 with Z 1/9, 1/3, 2/3, which are shown in Figure 3. From eqs. (2.5) and (6), one obtains

the following optimal values forand the energies:

where the boldfaced values 1 and 1/2, respectively, are equal to the

numerical values found in the standard case (m 1, Z 1). It isapparent that, for all potentials with 1/3 < Z< 1, the orbital is morecontracted and the energy is more negative than in the standard

case, even though the potential is less attractive than in the standard

case (Z 1). This is manifestly because the resistance of the kineticpressure is sufficiently weakened. For 1/9 < Z< 1/3, the orbital ismore contracted, but the energy is less negative than in the standard

Potential parameterZ 1/9 1/3 2/3

Optimized value ofmin

1 3 6

Optimized T V/2 E 1/18 1/2 2

Figure 1. Kinetic, potential, and total energy functionals of hydrogen

atom analogues of eq. (2.7) as functions of the orbital exponent . Ki-netic functional T (red) for m 1. Potential functionals V/2 (green)and total energy functionals H (blue) for Z 0.5, 1.0, and 1.5. Themarkers indicate the minima of H and the corresponding virial inter-

sections T V/2.


atom analogues of eq. (2.7) as functions of the orbital exponent .Potential functional V/2 (green) for Z 1. Kinetic functionals T

(red) and total energy functionals H (blue) for m 1 and 4. Themarkers indicate the minima of H and the corresponding virial inter-

sections T V/2.

394 Ruedenberg and Schmidt Vol. 28, No. 1 Journal of Computational Chemistry



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case. For Z < 1/9, the orbital is less contracted and the energy isless negative than in the standard case. The reasons for these results

become readily apparent from a close examination of Figure 3,

which exhibits the plots of the kinetic and potential terms, whose

intersections yield the values in the table. Thus, when both, the

potential pull as well the variational kinetic resistance are weakened,

then various compromises are possible depending on the quantita-

tive values of Z and m. We shall encounter a similar situation later

on in our investigation.

Variational Information and Observable Information

Suppose that, for some reason, one is in a position to know the re-

sultant energy values for the problem discussed in the preceding

subsection, viz.

E K; T K; V 2K; where K> 1=2;

but that one has no information on the values of m and Z. If one is

then asked why this particle is bound more tightly to this nucleus than

in the standard case (m 1, Z 1), and if one is unfamiliar with thepreceding analysis, one might be tempted to argue naively: Since

the potential energy is more negative, the energy lowering must have

been driven by a stronger electrostatic attractive potential.

This inference would be manifestly incorrect in the cases (m > 1,

Z 1) and [m 9, (1/3 < Z < 1)] discussed above, where theweakening of the variational kinetic energy pressure is the cause

for the energy lowering. Thus even though, when comparing two

such systems, the one with the lower energy always has the lower

potential energy, by virtue of the virial theorem, this fact does not

allow the inference that the difference in the electrostatic potential

energy functionals of the two systems is always the reason for the

difference in the actual total energy. This leads to two observations.

First, the discussed cases clarify what the virial theorem can and

cannot furnish. By virtue of its validity for the eigenvalues,10,11 it

predicts the value of the kinetic and the potential energies when the

exact total energy is known. By virtue of its connection with the

variation principle,26 it offers a shortcut in identifying the varia-

tional minimum of a given energy functional with respect to trial

function variations that describe orbital shrinking toward and or-

bital swelling away from the nuclei. However, it provides no in-

sights into or information regarding the differences in detail thatdistinguish energy functionals of different systems and that are

responsible for the differences in their ground state energies. Only a

careful examination and physical analysis of the kinetic and poten-

tial functionals can reveal the specific character of the variational

kinetic pressure and the variational nuclear pull in a given system.

Consequently, the virial theorem per se cannot furnish any reasons

for shifts in energy eigenvalues upon changes in the physical

parameters of a system.

Second, the quantitative values of the energy eigenvalues of the

Schrodinger equation and of the kinetic and potential components

per se do not furnish sufficient information for identifying cogent

reasons why corresponding states in related systems have different

energies. Interpreting quantum mechanical energy changes only in

terms of measurable quantities, as is occasionally championed,excludes valuable and illuminating theoretical insights from con-

sideration. This should not come as a surprise in as much as the

Schrodinger equation itself, the source of all theoretical reasoning

pertains to the space and time dependence of a nonmeasurable

quantity, namely the wavefunction.

Bond Formation and Variational Competition

A chemical bond forms between two atoms A and B when the

ground state energy of the molecule, say E0(AB), is more nega-

tive than the sum of the ground state energies of the separate

atoms, say E0(?) [E0(A) E0B)]. Understanding the origin ofbond formation is therefore tantamount to identifying the reasons

why E0(AB) is more negative than E0(?). Since, upon separationof the two atoms to at most five times the equilibrium distance

Req, the energy of any neutral molecule AB with a predominantly

covalent bond becomes practically identical with the value at

complete separation, the question becomes: Why does the lowest

energy level drop to a lower value when the internuclear distance

decreases from 5Req to Req?

In view of the reasoning outlined in the preceding section, we

seek an answer to this question on the basis of an analysis of the

variational process in the atom and in the molecule. Since the

energy operators V and T are qualitatively quite similar at Reqand 5Re, one might conjecture that it should be possible to per-

form a comparative analysis, of the discussed variational compe-

tition for the molecule at various distances. If relevant changes in

the variational competition can be identified as the parameter Rchanges, then such an analysis may throw light on the reasons for

the dependence of the minimizing compromise E0(AB), i.e. the

groundstate energy, on the internuclear distance.

One way to accomplish this consists, e.g., of starting with the

eigenfunction 5 at 5Req. From the variation principle follows (i)that there exists no wavefunction with a lower energy at 5Req and

(ii) that, used as a trial function for the Hamiltonian Heq at the equi-librium distance Req, the function 5 yields an energy that lies abovethat of the eigenfunction eq of Heq. The task is then to identifyphysical changes in the potential and the kinetic energy functionals


atom analogues of eq. (2.7) as functions of the orbital exponent .Kinetic energy functional T (red) for m 1/9. Potential functionals

V/2 (green) and total energy functionals H (blue) for Z

1/9, 1/3,

and 2/3. The markers indicate the minima of H and the corresponding

virial intersections T V/2.




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upon morphing 5 into eq that will lower the value of the energyfunctional heq|Heq|eqi below the value ofh5|H5|5i.

The variational comparison between the equilibrium geometry

and the separated atoms is facilitated by the fact that, for both dis-

tances, the virial theorem is valid in the form 2T V, so that

this equality also holds for the binding energy.

Expectation Values as Sums of Regional Contributions

The value of any expectation value depends on the shape of ,i.e. its distribution over space. Therefore, the quantitative assess-

ment of the kinetic and potential expectation values can often be

facilitated by expressing their integrals as sums of integrals over

various regions of space, whose individual assessments are more

transparent. Working with such regional contributions for varia-

tional reasoning implies no assertions whatsoever regarding any

observable energies associated with local regions. In this context,

it is also useful to express the three expectation values of eqs.

(2.2) and (2.3) in the well known equivalent forms28

hjTwji

Zdxdydz@=@w2; (2:8)

because, here, every volume element makes a positive contribu-

tion to the expectation value, a feature that facilitates the assess-

ment of the total changes upon changes in the wave function.

This is not the case for the integrals in eqs. (2.3).

Wavefunction Analysis

The Ground State Wave Function of H21

In the present investigation, we consider the hydrogen molecule

ion at its equilibrium distance.29 All calculations are performed

with an uncontracted (14s, 6p, 3d, 2f, 1g) basis set of 26 -typespherical Gaussian AOs (atomic orbitals) on each atom, which

represents a refinement of the pc4-basis published by F. Jensen.30

The refinements were made so that the energy error in the hydro-

gen atom as well as that in the hydrogen molecule ion both did

not exceed 106 hartree.31 Our AO basis is listed in the first col-

umn of Table1. We denote these atomic orbitals by Ak and Bk,

respectively, where for the purpose of this presentation, we

assume the nonspherical Bk so oriented that they are the mirror

images of the Ak with respect to the bond-bisecting mirror plane.

We write the 1s groundstate wave function of the hydrogen

atom on A obtained in terms of these Ak as

Ax;y; z khkAkx;y; z; k 1; . . . ; 14 (3:1a)

where the A8k are the 14 spherically symmetric s-type orbitals inthe basis of Table 1. We thus have hk 0 fork 1526. Optimi-zation of this wave function yields the coefficients listed in col-

umn 2 of Table 1. The energy is 0.499,999,890 hartree, whichis 0.1 microhartree higher than the exact value of 0.5 hartree.The virial ratio is |h|V|i/h|T|i| 2.0000003, which provesthat the coefficient minimization in this basis encompasses the

scaling optimization of the orbital exponents. In the following,

we considerA as the exact wave function of the free hydrogenatom. The 1s wave function on atom B is analogously given by

Bx;y; z khkBkx;y; z; k 1; . . . ; 14; (3:1b)

To calculate the electronic wavefunction of H2, we use theunnormalized symmetry adapted molecular orbitals

Uk Ak Bk; (3:2)

Vk Ak Bk: (3:3)

to span the function space generated by the atomic basis orbitals.

While the Uj are orthogonal to the Vk, the overlap integrals

between the Us and between the Vs are given by

Ujk hUjjUki 2hAjjAki hAjjBki; (3:4)

Vjk hVjjVki 2hAjjAki hAjjBki: (3:5)

The ground state of H2 is then given by the wave function

x;y;z kckUkx;y; z kckAkBk; k 1; . . . ;26 (3:6)

with

jkUjk cj ck 1: (3:6a)

Optimization yields the theoretical equilibrium distance 1.9972

bohr, which compares well with the recently determined exact

value32 of 1.997193 bohr (the experimental value33 of 1.988 bohr

is of course not the exact minimum of the potential energy curve).

The energy at the minimum is found to be 0.602,634,066 hartree,which lies 0.55 microhartree above the exact result.32 The virial ra-

tio is |h|V|i/h|T|i| 2.0000037, showing that, here too, thecoefficient minimization in this basis encompasses the scaling opti-

mization of the orbital exponents. The coefficients ck are listed in

the third column of Table 1. As for the H atom, we consider this to

be essentially the exact H2 wavefunction.

Expression in Terms of Two Quasiatomic Orbitals

Equation (3.6) for can be written in terms of two normalizedorbitals A and B

A BN; (3:7)

where

A N1X

k

ck Ak; B N1X

k

ck Bk: (3:8)

with

N

Xjk

cjckhAjjAki

1=2

Xjk

cjckhBjjBki

1=2: (3:8a)

The expansion coefficients ofA are listed in the fifth column ofTable 1. Figure 4 displays the contours of A in a plane contain-ing the internuclear axis.




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Table 1. Various Orbitals in Terms of Mirror-Image Spherical-Harmonic Basis Sets.

Gaussian Basisa () H atom (h) H2 (c) eq. (3.10) (a and b) eq. (3.8) (N1 c)

eqs. (3.15)(3.20)

(0) (0) (@)

s: 32480.00000 0.0000012 0.0000009 0.0000014 0.0000017 0.0000011 0.0000006 0.0000000

4781.00000 0.0000095 0.0000077 0.0000116 0.0000136 0.0000092 0.0000045 0.0000000

1043.00000 0.0000516 0.0000421 0.0000637 0.0000750 0.0000501 0.0000249 0.0000000

297.20000 0.0002126 0.0001719 0.0002610 0.0003063 0.0002064 0.0001000 0.0000000

92.39000 0.0009127 0.0007443 0.0011261 0.0013268 0.0008858 0.0004410 0.0000000

29.01000 0.0037089 0.0029941 0.0045490 0.0053371 0.0035997 0.0017375 0.0000000

9.78500 0.0131954 0.0107435 0.0162649 0.0191508 0.0128068 0.0063441 0.0000000

3.52300 0.0428876 0.0341446 0.0521790 0.0608645 0.0416244 0.0192401 0.0000000

1.34900 0.1172503 0.0936915 0.1429563 0.1670097 0.1137969 0.0532129 0.0000000

0.56370 0.2428891 0.2009244 0.3021942 0.3581578 0.2357352 0.1224226 0.0000000

0.25640 0.3453130 0.2122454 0.3646410 0.3783381 0.3351422 0.0431958 0.0000000

0.13050 0.2260547 0.0598822 0.1687163 0.1067431 0.2193966 0.1126535 0.00000000.09125 0.1007768 0.0098732 0.0603221 0.0175995 0.0978085 0.0802091 0.00000000.05152 0.0517047 0.0004384 0.0268527 0.0007815 0.0501818 0.0494004 0.0000000

p: 8.62500 0.0005475 0.0004847 0.0009760 0.0009760

2.14000 0.0041845 0.0037044 0.0074590 0.00745900.90650 0.0160468 0.0142058 0.0286041 0.0286041

0.47930 0.0243458 0.0215527 0.0433975 0.0433975

0.27360 0.0235936 0.0208869 0.0420568 0.0420568

0.14460 0.0097485 0.0086301 0.0173771 0.0173771

d: 1.96700 0.0013186 0.0011673 0.0023504 0.0023504

0.77270 0.0049697 0.0043995 0.0088586 0.0088586

0.32580 0.0039810 0.0035243 0.0070963 0.0070963

f: 2.24500 0.0002766 0.0002448 0.0004930 0.0004930

0.96580 0.0006154 0.0005448 0.0010969 0.0010969

g: 3.10800 0.0000485 0.0000429 0.0000865 0.0000865

s: 32480.00000 0.0000009 0.0000014

4781.00000 0.0000077 0.0000116

1043.00000 0.0000421 0.0000637

297.20000 0.0001719 0.0002610

92.39000 0.0007443 0.001126129.01000 0.0029941 0.0045490

9.78500 0.0107435 0.0162649

3.52300 0.0341446 0.0521790

1.34900 0.0936915 0.1429563

0.56370 0.2009244 0.3021942

0.25640 0.2122454 0.3646410

0.13050 0.0598822 0.1687163

0.09125 0.0098732 0.0603221

0.05152 0.0004384 0.0268527

p: 8.62500 0.0005475 0.0004847

2.14000 0.0041845 0.0037044

0.90650 0.0160468 0.0142058

0.47930 0.0243458 0.0215527

0.27360 0.0235936 0.0208869

0.14460 0.0097485 0.0086301

d: 1.96700 0.0013186 0.0011673

0.77270 0.0049697 0.0043995

0.32580 0.0039810 0.0035243

f: 2.24500 0.0002766 0.0002448

0.96580 0.0006154 0.0005448

g: 3.10800 0.0000485 0.0000429

aJensens pc4 basis of 11s, 6p, 3d, 2f, 1g basis was uncontracted and expanded to 13s, before optimization

of all exponents specifically for the ground state of H2 at 1.9972 Bohr. Then, a 14th s exponent was added

to these H2 primitives, and optimized for the H atom. While this basis is accurate to better than 1 micro-

hartree (the energies of H and H2 are given in Section 3.1), it has no overcompleteness problems, the

smallest eigenvalue of the matrix U of eq. (3.4) being 3.6 104.




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To assess whether the orbitals A and B might be basis-set-dependent, we solved the following problem: To find the resolu-

tion of of the form

A B=21 hAjBi1=2 (3:9)

(with A and B being each others mirror images), where Aand B are those normalized orbitals that are closest to the

ground state 1s orbital of the free atoms A and B, respectively,

say by having maximal overlap with them. Since they are defined

by a basis set independent definition, they are intrinsic to the mo-

lecular wave function.34,35 As shown in the appendix, this prob-

lem has the unique solution

A X

k

ak AkX

k

bk Bk; B X

k

bk Ak X

k

ak Bk;

(3:10)

where

ak C ck hk=D; (3:11a)

bk C ck hk=D; (3:11b)

with

C X

jk

Ujk cj hk (3:12)

D2 X

jk

Ujk cj hk

!2X

jk

Vjk hj hk (3:13)

where the quantities hk, Ujk, Vjk are defined in eqs. (3.1), (3.2),

and (3.3). The expansion coefficients ak, bk are listed in the fourth

column of Table 1.

The overlap integrals between the free atom orbitals A of eq.(3.1) and the two types of orbitals, viz. of eqs. (3.8) and (3.10),

respectively, are found to be

hAjAi 0:9705463; hAjAi 0:9768673;

hAjAi 0:9935293: 3:14

Manifestly:

iii. Both kinds of orbitals represent only slight deformations of

the free-atom ground state orbital and we therefore call them

quasi-atomic orbitals.

iii. The one-center quasi-atomic orbitals of eq. (3.8) are nearly

as close to the free-atom orbitals as is possible under the con-

straint of eq. (3.7).

iii. Since the one-center quasi-atomic orbitals of eq. (3.8) are ex-

tremely close to the basis-set-invariant orbitals of eq. (3.10), we

expect the one-center orbitals to be near-basis-set independent.

In the following analysis, we shall use the simpler quasi-

atomic orbitals of eq. (3.8).

Analysis of the Quasiatomic Deformation

Let us examine the character of the deformation of the quasi-atomic

orbital A of eq. (3.8) with respect to the free-atom orbital A on Agiven by eq. (3.1a). To this end, we divide the orbitals on atom A in

two groups: The 14 spherically symmetric s-type orbitals A8i and the12 other orbitals A@k. The expansion (3.8) can then be rewritten as

A sA

00A; (3:15)

where

sA N1i

Ai ci; i 1 to 14; (3:16)

00A N1k Ak ck; k 15 to 26: (3:17)

We further decompose the spherical part sA as the sum of its pro-jection onto the (spherical) free-atom orbital A and its (spheri-cal) component perpendicular to it:

sA hsAjAi A

0A; h

0AjAi 0; (3:18)

so that the entire quasi-atomic orbital is now decomposed as follows

A A 0A;00A: (3:19)

where

A hsAjAi A: (3:20)

The expansion coefficients of 8A, 0A, @A are listed in the lastthree columns of Table 1. By virtue of the mutual orthogonality

of the three terms, one has

1 hAjAi h

0Aj

0Ai h

00Aj

00Ai (3:21)

Figure 4. Contour plots exhibiting the decomposition of the quasi-

atomic orbital A in terms of its projection 8A on the 1sA orbital, itscontraction deformation 0A and its polarization deformation @A,according to eq. (3.19). The contour increment forA and 8A is 0.05bohr3/2, while it is 0.01 bohr3/2 for 0A and @A. Positive contours:Solid. Nodes and negative contours: Dotted-dashed. The solid straight

lines connect the nuclei.




9/22

and also

hAjAi hsAjAi h

AjAi h

Aj

Ai

1=2; (3:22)

whose value was already given in eq. (3.14). The calculation of

the remaining quantities yields for the breakdown of eq. (3.21),

the quantitative values

1 0:9419601 0:0422308 0:0158091: (3:23)

Thus, the quasi-atomic orbital A consists to 94% of the free-atom 1s orbital A, and the 6% deformation is 73% spherical and27% nonspherical in character.

Figure 4 displays contours in a plane containing the internu-

clear axis illustrating the decomposition of eq. (3.19). The four

panels of Figure 4 exhibit the contours of the functions A, 8A,0A, @A, respectively. Figure 5 exhibits plots of the functions A(green), A (red),

0A (blue), @A (purple) along the internuclear

axis. Note that 8A 0.9705463A.It is apparent from these figures that the nonspherical deforma-

tion @A has largely p-character. This deformation is therefore essen-tially a polarization of the spherical part. That the addition of a single

scaled 2p AO to a scaled 1s AO on each atom substantially improvesthe H2

wave function was already shown in 1933 by Dickinson.36

Regarding the spherical deformation, we recall that Finkelstein

and Horowitz7,13 found a good approximation to the H2 wavefunc-

tion by choosing the quasi-atomic orbitals simply as optimally

scaled hydrogen 1s functions, the optimized orbital exponent being

1.239. It seems, therefore, likely that the spherical part sA ofour quasi-atomic orbital is similar to such an orbital. We therefore

constructed from eq. (3.1a) the corresponding normalized scaled

free-atom orbital A() and then determined the orbital exponent by maximizing the overlap with the spherical part sA of our quasi-

atomic orbital, renormalized to unity i.e. [sA/hsA|

sAi

1/2]. This

yields the contractively scaled 1s orbital *A A(*) with

1:265; hAjsAi=j

sAj

sAi

1=2 0:999562: (3:24)

Thus, 99.96% of the spherical deformation of A represents a

contraction.37

The quasi-atomic orbital B can manifestly be decomposedanalogously in terms of orthogonal components:

B sB

00B

B

0B

00B;

B h

sBjBi B: (3:25)

Variational Analysis

In the following discussions, the hamiltonian operators of H and

H2 are denoted as

HA T rA1 (4:1)

H T V R1

; V rA1

rB1

; (4:2)

respectively. Unless otherwise stated, all energies will be quoted

in millihartree (mh) units in this section.

The Atomic Ground State xA as Initial

Trial Function for H21

In the spirit of the discussion in Section 2.4, we begin by choos-

ing the ground state wavefunction A of the hydrogen atom at Aas the initial trial function for the H2

system. The corresponding

energy functional hA|H|Ai is the energy expectation value ofan electron occupying the orbital A on A with respect to the H2

hamiltonian H of eq. (4.2), with the two protons A and B at thetheoretical equilibrium distance R 1.9972 bohr.

The value of this energy functional differs from the hydrogen

atom energy by

hAjHjAi EH hAjHjAi hAjT rA1jAi; (4:3)

and this difference can also be viewed as the change in the energy

functional of a hydrogen atom that results when a second proton

is brought from infinity to the position it occupies in the H 2 mol-

ecule without changing the electronic wavefunction of the atom.

The second term on the right hand side of eq. (4.3) cancels

identical integrals in the first term, including the kinetic energy

terms, so that

hAjHjAi EH R1 Zdxdydz A2=rB: (4:4)

The right hand side of this equation is simply the coulombic elec-

trostatic energy between a proton B and the atom A, i.e. nucleus

plus ground state electron density at A. We call it the zeroth-

order quasi-classical energy and denote it by EQC.

Since A is spherically symmetric around A, Newtons theo-rem of potential theory applies to the integral on the right hand

side, which therefore becomesZdxdydz A

2=rB R1

Zdxdydz A

2; (4:5)

Figure 5. Orbital plots along the internuclear axis exhibiting the

decomposition of the quasi-atomic orbital A (green) in term of its

projection 8A 0.970546 times the drawn free Hydrogen atom A(red), its contraction deformation 0A (blue), and its polarization de-formation @A (purple), according to eq. (3.19). All orbital amplitudesare in bohr3/2.




10/22

where the integral $*dxdydz covers only the inside of the spherewith radius R 1.9972 bohr around nucleus A. Since this spherecontains 94.48% of the charge of the hydrogen (1s) orbital, inser-

tion of eq. (4.5) into eq. (4.4) yields

EQC hAjHjAi EH 1 0:9448=1:9972 27:64 mh:

(4:6)

Note that, by virtue of Newtons theorem, adding a second nu-

cleus B in any direction at a finite distance from any spherical or-

bital charge plus a proton at A will increase the energy. It is of

course true that the presence of the nucleus B will very greatly

lower the electrostatic potential energy of the electron in the orbital

at A [by close to 0.5 hartree in the present case, see eq. (4.6)]. But

this increase in electronnuclear attraction is overcompensated by

the concomitant increase in the repulsion between the two nuclei.

Morphing the Wave Function From xAto the Molecular Ground State c

By virtue of the variation principle, the energy of the eigenfunction

of the molecular hamiltonian H will lie below the expectationvalue hA|H|Ai. In fact, since we saw in Section (3.1) that theground state energy of H2

is 602.634066 mh, it follows that re-placing A by will lower the value of the H2

energy functional by

hAjHjAi hjHji 602:634 500 27:641 130:275 mh:

(4:7)

To explain this decrease is our task. We shall accomplish it by

generating the wavefunction change A ? via a sequence ofsteps whose individual kinetic and potential energy changes can

be understood and assessed on physical grounds.

Since we have seen in Sections 3.2 and 3.3 that is the super-position of two quasiatomic orbitals A and B that differ fromthe free-atom orbitals A and B by contractive and polarizingdeformations, we shall construct intermediate wave functions in

terms of the corresponding components of , which were identi-fied in Section 3.3. To this end, we consider the following eight

intermediate wave functions, where the factors Nk denote appro-

priate normalization constants.

1. The initial function, i.e. the free-atom ground state orbital at A:

1 A (4:8)

2. The atomic ground state orbital on atom A deformed by the polar-

izing component @A of the quasiatomic orbital [see eqs. (3.17)and (3.19)]:

2 pA A 00A N2 (4:9)

3. The atomic ground state orbital on atom A deformed by the

contractive component 0A of the quasiatomic orbital [see eqs.(3.16), (3.18)(3.20)]:

3 sA

A

0A N3 (4:10)

4. The quasiatomic orbital A, which contains contractive andpolarizing deformation ofA [see eqs. (3.8) and (3.19)]:

5. A molecular orbital that is shared between the free-atom or-

bital on A and B, i.e. A and B:

5 A B N5 (4:12)

6. A molecular orbital that is shared between the polarized atomic

ground state orbitals on A and B, i.e. p

A and p

A of eq. (4.9):

6 pA

pB N6 (4:13)

7. A molecular orbital that is shared between the contracted atomic

ground state orbitals on A and B, i.e. sA and sA of eq. (4.10):

7 sA

sB N7 (4:14)

8. The molecular orbital that is shared between the atomic

ground state orbitals on A and B, each contracted as well

polarized, i.e. the exact wavefunction of eqs. (3.7):

8 A B N8 (4:15)

It is helpful to place the symbols for the eight orbitals (4.8)

(4.15) at the corners of a cube as exhibited in Figure 6. The

changes in the wave function leading from 1 A in the upperleft back corner to 8 in the lower right front corner can thenbe achieved along six different paths by way of various edges. As

can be seen from the figure, the orbitals have been arranged in

such a fashion that the three Cartesian coordinate directions are

associated with the three kinds of wavefunction changes that

were found in the wavefunction analysis of Section 3.3, viz.

polarization, contraction, and sharing between atoms. Analogous

wave function changes are therefore associated with parallel

edges as indicated in the figure: Left ? right edges correspond

to contraction, back ? front edges correspond to polarization,

Figure 6. Polarization, contraction and sharing pathways represent-

ing the stepwise morphing of the hydrogen atom wavefunction 1 A into the H2

molecular wavefunction 8 , via the intermediatewavefunctions 2 to 7 defined by eqs. (4.9) to (4.14).4 A

A

0A

00A 4:11




11/22

up? down edges correspond to sharing between atoms. Different

pathways from A to correspond to making these three kinds ofwave function changes in a different order.

Changing the Energy Functional From H(xA) to H(c)

The values of the energy functionals for the wave functions 1,2, 3, . . . ,8 defined in the previous section, as well as their ki-netic and potential components, are listed in Table 2. Relevant to

us are the changes of these quantities that are associated with

each edge of the cube in Figure 6. The quantitative values of

these changes are entered on the edges of the cube in Figure 7.

For any path starting with 1 A and ending with 8 , theedge values of the total energy add up to 130.275 mh, the value

given in eq. (4.7). Furthermore, at each corner of the cube the vir-

ial ratio |2T/V| of the corresponding wave function is listed.

Reasoning in the spirit of a variational analysis is manifestly

cleanest when one has a sequence of wave function changes that

lower the energy functional at each step and thus approach the

exact energy monotonically. Of the six possible paths from corner

1 to corner 8 in Figure 7, the three paths including corner 4 all

involve however an increase of the energy functional upon con-traction. This implies that contraction lowers the energy func-

tional only if it is applied after sharing. The three monotonic

sequences are thus

P-S-C: Polarization?sharing?contraction (path 1?2?6?8) S-P-C: Sharing?polarization?contraction (path 1?5?6?8) S-C-P: Sharing?contraction?polarization (path 1?5?7?8).

The various changes of the energy functional for these three

variational sequences are collected in Table 3. The changes in the

Table 2. Energy Expectation Values (in Millihartree) of the Functions 1,2, . . . , 8, Defined by eqs. (4.84.15), corresponding to the Corners ofthe Cubes in Figures 6 and 7.

hEi hVi hTi |2hTi/hVi|

1 472.3595 972.3593 499.9997 1.02842582 508.2970 1018.7474 510.4504 1.00211383 461.6155 1220.0998 758.4842 1.24331504 494.2487 1258.6559 764.4072 1.21464055 553.6202 939.9338 386.3136 0.82200176 571.5597 985.0910 413.5313 0.83957997 590.2039 1178.4054 588.2015 0.99830088 602.6341 1205.2659 602.6318 0.9999981

Figure 7. Quantitative changes (in millihartree) of the kinetic (red),

potential (green) and total (blue) energy functionals of H2 for the

twelve possible intermediate steps in morphing the hydrogen atom

wavefunction 1 A into the H2 molecular wavefunction 8 ,

as defined in Figure 6. The corner circles also contain the intermedi-

ate virial ratios |2T/V|.

Table 3. Energy Functional Changes (in Millihartree) for SuccessiveWavefunction Changes from 1 A to 8 .

Character of

wavefunction changes

Sequence of wavefunction changes

P-S-Ca

(1268)bS-P-Ca

(1568)bS-C-Pa

(1578)b

Total energy functional changes DEPolarization 35.9 17.9 12.4Sharing 63.3 81.3 81.3Contraction 31.1 31.1 36.61 A? 8 130.3 130.3 130.3

Potential energy functional changes DVPolarization 46.4 45.2 26.9Sharing 33.7 32.4 32.4

Contraction 220.2 220.2 238.51 A? 8 232.9 233.0 233.0

Kinetic energy functional changes DTPolarization 10.5 27.2 14.4

Sharing 96.9 113.7 113.7Contraction 189.1 189.1 201.9

1 A? 8 102.7 102.6 102.6

Bond-parallel kinetic changes Dh|Tz|iPolarization 0.2 13.1 8.5


1 A? 8 25.9 26.0 25.8

Bond-perpendicular kinetic changes Dh|Tx|i Dh|Ty|iPolarization 5.1 7.1 3.0


1 A? 8 64.3 64.4 64.4

Virial ratio changes D|2T/V|Polarization 0.028 0.02 0.002Sharing 0.162 0.21 0.21Contraction 0.16 0.16 0.178

1 A? 8 0.03 0.03 0.03

aP Polarization, S Sharing, C Contraction.bSequence of comers traversed on path from A to in Figures 6 and 7.




12/22

virial ratio are also listed. For the sake of clarity, all quantities

are rounded to 0.1 millihartree.

Although the individual values for the three paths differ some-

what, they clearly exhibit a consistent pattern, which leads to the

following conclusions.

The polarization of the quasiatomic orbitals always lowers theenergy because of a lowering of the potential energy, which man-

ifestly results from skewing each quasiatomic orbital toward the

other proton. The kinetic energy increases because the addition

of the polarizing orthogonal scaled p-type orbital of eq. (3.17)

has a node while approximately maintaining the spatial extension

of the 1s orbital (see Figures 4 and 5). This is obvious in the case

P-S-C where the polarization occurs before sharing (yielding the

one-center wavefunction 2) but the analogous energetic effectsare obviously also operative when polarization occurs after shar-

ing. The kinetic and potential energy changes are always such

that the virial ratio |2T/V| remains nearly the same. We also note

that the polarization yields at best a very weak binding, viz. for

the one-center orbital 2 [27.635.9 8.4&8% of the actual

binding energy, see eq. (4.6) or Table 2]. Its virial ratio of about1 illustrates the fact that the virial ratio of 1 is a necessary but not

a sufficient minimum condition, as was mentioned in the first

paragraph of Section 2.2.

The sharing delocalization from a one-center to a two-center

orbital always yields the largest energy lowering and this lower-

ing is always due to a large lowering of the kinetic energy in spite

of a modest increase in the potential energy. The latter is in fact

so modest that the virial ratio |2T/V| decreases substantially in this

step. The origin of these changes will be discussed in Section 4.4.

The contraction of the quasiatomic orbitals always lowers the

total energy by a moderate amount. This is however the result of

a very large lowering in the potential energy and a very large

increase in the kinetic energy, but the latter being not quite as

large as the former. The substantial increase of the virial ratio inthis step compensates its decrease in the sharing step so that the

final |2T/V| change with respect to the initial variational function

1 A value is 0.3, which is what is needed to establish thevirial ratio |2T/V| 1 for the exact wavefunction 8 . The ori-gin of these changes will be discussed in Section 4.5.

It is apparent that the earlier a contribution type occurs in the

sequence, the larger its energy functional change is. Thus, the

quantitative value for polarization decreases from P-S-C to S-P-C

to S-C-P, and sharing and contraction behave similarly. This

shows that the wave function changes accomplished by the three

types of adjustments are not entirely independent but can, to

some degree, substitute for each other. However, the marked con-

sistency of the discussed patterns shows that this substitution abil-

ity is limited and that they do in fact describe three fairly distinctand independent physical adjustments.

Another relevant observation is the importance of the bond-

parallel component of the kinetic energy. Not only does it con-

tribute the largest lowering in the sharing step, but it also experi-

ences the smallest increase in the contraction and polarization

steps so that even its contribution to the total binding energy is

still negative.

It is apparent from Table 3 that the essential element for the

covalent energy lowering in H2 is the sharing feature of the wave

function. No surprise here. The table also shows, however, that,

with or without polarization, it is the kinetic energy change that

lowers the energy functional in the sharing step. The potential

energy change in that step is always positive and nearly the same

with or without polarization. As a result, this crucial energy low-

ering decreases the virial ratio significantly below the value of

unity, the value for the optimized wavefunction 8. This wave-function must therefore embody another adjustment that readjusts

the virial ratio to unity while preserving the sharing stabilization.

This is accomplished by the contraction step.

The order of importance of the energy lowering contributions,

namely sharing, contraction, polarization agrees with the order of

importance of these contributions to the wavefunction, as shown

by eq. (3.23). It would therefore seem physically most sensible to

consider the sequence S-C-P, listed in the last column of Table 3,

as the preferred path formodeling the deformation A? .While the physical origin of the kinetic and potential energy

changes of polarization step is transparent and has been discussed

above, the changes due to sharing and contraction deserve a

closer examination.

Energy Functional Lowering Through Electron Sharing

Why does electron sharing between the quasiatomic orbitals

lower the energy expectation value of the molecular electronic

wavefunction? In the present context, we have defined orbital

sharing as the change from a one center orbital, say A, to thetwo center molecular orbital N(A B) consisting of thesame kind of one-center orbitals. The difference in the energy

functional due to sharing is thus

ESH hjHji hAjHjAi: (4:16)

Figure 8. Contours of interference densities [2 (A2 B

2)/2] for

the sharing steps (1 ? 5), (2? 6), (3? 7), and (4? 8) in Figure 6.

The contour increment is 0.002 bohr3 in all cases. Positive contours:

Solid. Nodes and negative contours: Dotted-dashed.




13/22


14/22

It is thus clear why, for all three wavefunction sequences, we

find negative values for all three components in the kinetic energy

integral change (4.19), which results from electron sharing, the

effect being strongest for the bond parallel component. Thisattenuation of the kinetic energy functional by electron sharing in

a bonding orbital is basically the same as the lowering of the ki-

netic energy of a free particle in a box when the length of the box

is increased. It is the result of delocalization and related to the

uncertainty principle between position and momentum.10,24,25

Thus, the orbital interference associated with orbital sharing

attenuates the variational kinetic energy pressure as well as the

variational potential electrostatic energy pull. The kinetic interfer-

ence term (4.18) is, however, considerably larger than the poten-

tial interference term (4.19) so that the addition of the two yields

for the sharing energy (4.17) the negative value 63.3 on the P-S-C path and the value 81.3 on the S-P-C and S-C-P paths aslisted in Table 3.

Energy Functional Lowering Through Contraction

of Quasiatomic Orbitals

Why does orbital contraction of the quasiatomic orbitals lower the

energy expectation value of the molecular wavefunction? By defini-

tion, the contraction of the quasiatomic orbitals (3.15) pertains to

their spherical deformation part defined by eqs. (3.16) and (3.18).

To analyze this deformation, we recall that the discussion in connec-

tion with eq. (3.24) showed this spherical deformation to be 99.96%

identical with a contractive scaling of the hydrogen 1s orbital. This

similarity allows us to approximate the wavefunction of eq. (3.7)very closely through replacing, in the quasiatomic orbitals A andB of eq. (3.15), their spherical parts

sA and

sB by the slightly

modified spherical parts that result from the substitutions

sA ! hsAj

sAi

1=2 A; sB ! h

sBj

sBi

1=2 B; (4:20)

where A(*) is the scaled 1sA orbital of eq. (3.24) with

* 1.2654.

On the path S-C-P, the contraction occurs before polarization

and corresponds to the wavefunction change from 5 *(A B)to 7 *(

sA

sB), as defined by eqs. (4.12) and (4.14). By virtue

of the approximation (4.20), the progress of this contraction can be

monitored by changing * from 1 to 1.2654 in the orbit.

NA B: (4:21)

On the paths P-S-C and S-P-C, contraction occurs after polariza-

tion and corresponds to the wavefunction change from 6 *(RA

RB) to 8 *(A B), as defined by eqs. (4.13) and(4.15). By virtue of the approximation (4.20), the progress of this

contraction can be expressed as

NfhsAjsAi

1=2A 00A

hsBjsBi

1=2B 00Bg: 4:22

Figure 13 exhibits plots of the changes of the kinetic, potential,

and total energy functionals during contraction, using these

approximations. Panel 13a displays the contraction 5 ? 7 asfunction of, according to eq. (4.21). Panel 13b displays the con-traction 6 ? 8 as function of , according to eq. (4.22). Thepatterns exhibited in the two panels are very similar. According

to eq. (4.20), the contraction covers the range of from 1 to * 1.2654. It is apparent from Panels 13a and 13b that this end value

* is very close to the minima of the total (blue) energy curves,

which occur at about # 1.24 on both panels and are marked bydiamonds (The deviation * # 0.025 is about the width ofthe diamond marker).

The accuracy of the used approximations is seen from the fol-

lowing errors. For the contraction 5? 7, one finds

E7; approx; E7; approx;

# 3:7 mh

E7; approx; # E7; exact 0:4 mh:

Figure 11. Contours of the contributions of (a) the z-component and

(b) the sum of the x and y components to the kinetic interference den-

sity contours displayed in Figure 10. All closed contours are negative;

the open contours are zeros. Contour increment: 0.005 hartree/bohr3.

Figure 12. Plots along (a) the internuclear axis and (b) an axis per-

pendicular to the bond axis and passing through one of the nuclei for:

The kinetic interference density (blue) of Figure 10 and its two parts,

viz. the molecular squared gradient (!5)2 (red) and the average of

the squared gradients of the atoms [(!A)2 (!B)

2]/2 (green).

Kinetic densities in hartree/bohr3.




15/22

For the contraction 6? 8, one finds

E8; approx; E8; approx;

# 1:5 mh

E8; approx; # E8; exact 0:3 mh:

These deviations are small enough compared with the total energy

changes due to contraction (about 30 mh, see Table 3) to permit

the following conclusions.

The patterns of Panels 13a and 13b are manifestly very similar

to the patterns of the energy functionals of the hydrogen atom ana-

logues we have discussed in Section 2.2 (see Figs. 13). This is

demonstrated by Panel 13c where we show the plots of the hydro-gen atom analogue defined by eq. (2.7) with (m 1.27573, Z 0.97199). The graphs for this atomic analogue manifestly mimic

those in Panel 13b extremely closely. This is because m and Z

were determined from eqs. (2.5) and (2.6) using the minimum val-

ues of Panel 13b for (# 1.24) and the energy (0.6023).Using the energy minimum value of Panel 13a, one would obtain

an atom analogue mimicking Panel 13a equally closely.

Panels 13a, 13b differ from Panel 13c in the following

respects:

i. In Panel 13c, increasing describes a contraction toward onenucleus; in Panels 13a and 13b, it describes a simultaneous

contraction toward both nuclei.

iii. In Panel 13c, the potential electrostatic pull is weakened,compared with the hydrogen atom situation, because of the

charge Z being 1. In Panels 13a and 13b, it is weakened as a conse-quence of electron sharing as discussed in Section 4.4.

Nonetheless, the variations of the kinetic and the potential

functionals with exhibited in Panels 13a and 13b are very similar

to those in Panel 13c. The similarity suggests that the potential and

kinetic energy changes associated with contraction have their ori-

gin predominantly in the interaction of the spherical component

sA with nucleus A and of the spherical component sB with nu-

cleus B, an inference that is supported by the similarity to the

energy changes listed in Figure 7 for the one-center contraction1?3. This conclusion has been confirmed by more detailed cal-culations.38 The nature of the variational competition between the

kinetic pressure and the potential pull is thus essentially the same

in the molecular and in the atomic case, namely: The kinetic pres-

sure and the potential pull are weaker than in the hydrogen atom,

but the kinetic pressure reduction is sufficiently stronger so that or-

bital contraction (toward one or two nuclei, respectively) is

induced to reach the virial intersections 2T V where the totalenergy functional has its minimum.

Thus, the simultaneous contraction toward both nuclei occurs

because, in the context of the variational competition, the resist-

ance of the kinetic energy pressure against the simultaneous elec-

trostatic potential pull from both nuclei has been weakened by

electron sharing. The contraction proceeds until the virial ratio ofunity has been reached. The contraction is, therefore, a conse-

quence of the strong attenuation of the variational kinetic energy

functional that results from orbital sharing.

Conclusions

Summary

The preceding analysis has shown that the exact wavefunction of

H2 at the equilibrium distance can be obtained from the wavefunc-

tion of the hydrogen atom (on nucleus A say) by orbital sharing,

orbital contraction and orbital polarization. Each of these three or-

bital modifications is associated with a characteristic lowering ofthe energy functional of H2

in the context of variational calculus.

The largest contribution comes from the establishment of a

shared orbital between the two nuclei. This energy lowering is due

to a decrease of the kinetic energy functional, which is akin to the

kinetic energy lowering associated with the increasing wavelength

of an electron in a box upon delocalization when the box is length-

ened. It is stronger than the concomitant increase in the potential

energy functional that also occurs with orbital sharing.

In the context of the variational competition, the effect of or-

bital sharing on the energy functional represents a weakening of

the kinetic energy pressure and lowers the virial ratio |2 T/V|. It

therefore induces an orbital contraction that further lowers the total

energy functional until the virial ratio |2T/V| 1 is re-established.

Thus, electron sharing lowers the energy functional in twoways: First, directly by orbital sharing, as expressed by the

sharing energy of eq. (4.17) and, second, indirectly by inducing

the orbital contraction, which lowers the energy functional fur-

ther. Both are in response to the change in the kinetic energy

functional caused by orbital sharing.

By comparison, the sum total of the quasiclassical electro-

static changes is small. Although orbital polarization yields some

energy lowering, this is approximately compensated by the elec-

trostatic repulsion energy between an unpolarized hydrogen atom

and a proton.

Figure 13. Variation of the kinetic (red), potential (green) and total

(blue) energy functionals in two contraction steps of Figure 7, as

functions of the contraction parameter of the model formulated inthe text. Panel (a): Contraction step (5? 7). Panel (b): Contraction

step (6 ? 8). Panel (c): Atomic analogue with (m 1.27537, Z 0.97199).




16/22

Essence

What is the upshot of the variational analysis?

The first observation is that the variational competition in the

hydrogen molecule ion is in fact quite similar to that in the atom.

This is seen, e.g., by comparing the energy functional of the

hydrogen atom, viz.

hAjHAjAi with HA given by eq. 4:1;

and A 3

=1=2

exprA; 4:23

with two energy functionals of the hydrogen molecular ion, viz.

hjHji with H given by eq. 4:2;

and NA B; 4:24

where we consider the following two choices forA(). The first is

A A; (4:25)

which takes into account only the contraction of the shared wave-

function and contains the step 5?7 of Figures 6 and 7. Thesecond is

A f1 h0Aj

0Ai h

00Aj

00Aif

2g1=2A1

0A 00Af; 4:26

where

f 1= 1 with 1:265 of eq: 3:24;

(4:27)

and 0A and @A are the contraction and polarization componentsof the exact wavefunction, respectively, given in eq. (3.19). Mani-

festly, eq. (4.26) is an interpolative function that leads from the

shared wavefunction 5 in Figures 6 and 7 by simultaneous con-traction and polarization to the exact wavefunction 8 in Figures

6 and 7. By virtue of eq. (3.24), one has very closely

A 1 h00Aj

00Ai

1=2A 00A; (4:28)

i.e. the spherical part of A(*) is a straight atomic contraction,

which makes a comparison with the atomic case possible. The pa-

rameterhas of course a somewhat different meaning for the twosystems: For the atom, it describes a contraction towards one

atom (A), for the molecule it describes a simultaneous shrinking

toward both atoms.

Figure 14 exhibits the variational competition of the atomic

energy functional (4.23) together with that of the molecular func-

tional (4.24), Panel 14a using the molecular function (4.25), Panel

14b using the molecular function (4.26). The kinetic, potential, and

total energy functionals are indicated by red, green, and blue

curves, respectively. The plots are similar to those discussed in

Section 2.2. The molecule is indicated by solid curves, the atom by

curves marked (). The two sets of curves are remarkablysimilar and, in both cases, the energy functional reaches its mini-

mum at the virial intersection of the kinetic and potential func-

tionals. That is to say, in both systems, the nuclear electrostaticattraction variationally pulls the electronic charge towards the nu-

clear center(s) until the resistance of the variational kinetic pressure

puts a stop to it, namely when 2 h|T|i has reached h|V|i.The second observation is that the optimized energy of H2

is

lower than that of H because of the shifts of the molecular kinetic

and potential functionals relative to their atomic counterparts.

Although both are weakened (i.e. the molecular curves lie below

the corresponding atomic curves in Fig. 14), the attenuation of the

kinetic energy functional is much stronger and, hence, determin-

ing. That is to say, for any given value of the potential energy asso-

ciated with a certain charge concentration around the nuclear cen-

ter(s), the molecular kinetic energy functional h|T|i lagsbehind the value of the corresponding atomic functional. Because

of this relative weakening of the molecular kinetic energy pressure,a lower value ofh|V|i can be reached in the molecule by shrink-ing toward the nuclei before the value of2 h|T|i catches upwith it. At that point h|H|i h|V|i is then also lower.

The essential question as regards binding is therefore: What is

the physical origin of these shifts? Determining the answer to this

question has been the main subject of the preceding analysis and

it has substantially shown the following. Because, in the mole-

cule, the electrostatic potential energy functional can be lowered

by simultaneously approaching two nuclei, this potential lowering

can be achieved while maintaining the delocalization over two

centers and this persisting delocalization has an attenuating

effect on h|T|i. It is because of this inherent delocalization,that a given amount of negative electrostatic potential energy can

be acquired with a lesser increase in the molecular kinetic energy

functional than is possible in the atom.

Model

In view of these results, the simplest valid model for the origin of

binding in this molecule must focus on the reason why the curve

of the molecular kinetic energy functional is shifted downward

from the corresponding atomic curve. The simple model therefore

has to be that covalent bonding is a consequence of the lowering

of the kinetic energy functional that is caused by the delocaliza-

tion inherent in electron sharing.

Figure 14. Kinetic (red), potential (green) and total (blue) energy

functionals for the H atom ( curves) and the H2 molecule

(solid curves) as functions of the orbital exponent , according to eqs.

(4.23) and (4.24) respectively. Panel (a): Molecular functionals calcu-lated with eq. (4.25), with the minimum at 1.239, E 0.586505hartree. Panel (b): Molecular functionals calculated with eq. (4.26),

with the minimum at * 1.265, E 0.602634 hartree.




17/22

One can give a semiquantitative model estimate for this lower-

ing of the kinetic energy functional by comparing it with the ki-

netic energy lowering of a particle in a linear box parallel to the

z-axis when the box length is doubled, corresponding to the z-

dimension of H2 being about twice that of H. Consider a particle

in a box of length L and let L be such that its kinetic energy(*const/L2) is equal to the kinetic energy contribution of the z-component in the H atom, i.e. [1/3]TH. Increasing L by a factor 2will decrease this part of the kinetic energy by a factor 4. Taking

this factor as the modification, we have to apply to the z-compo-

nent in the kinetic energy functional of the hydrogen atom, TH() ([]2), we obtain for the kinetic energy functional of H2

(including the unchanged x-, y-, and z- components) the estimate

TH2

TH1=3 1=3 1=3 1=4 0:75 TH

0:751=22: 4:29

The actual ratio (TH2/TH) in the range between the optimal values

for H and H2, is in fact about 0.77 in Figure 14a and 0.76 in Fig-

ure 14b, an agreement that exhibits the physical reasonableness

of the simple model.

It would be helpful if the basic relationship between concep-

tual models and the variational interpretation of quantum chemi-

cal energies would be more widely appreciated. If that were the

case, then it would no longer seem puzzling to some how the low-

ering of the kinetic energy functional can lead to a minimized

energy that, by virtue of the virial theorem, has in fact a higher

kinetic component than in the atom.

Such counter-active relaxation phenomena are not uncom-

mon in physics, as Kutzelnigg has discussed in detail.1719 He has

also given a delightful analogy17,18 from the business world by

comparing total, kinetic, and potential energy with net income,

expenses, and gross income: Consider two businesses that reduce

their overhead expenses by fusion, which eliminates certain dupli-

cations. As a consequence, they are more successful and generate

more income than the sum of their previous incomes. Because of

greatly increasing business, the overhead expenses too eventually

exceed the sum of their previous expenses. Thus, although they end

up with a higher overhead, it is nevertheless the overhead reduc-

tion by fusion that is the reason for their success. Examples from

various other fields have been quantitatively discussed by bitter.39a

Since the use of the variation principle for obtaining quantitative

results is ubiquitous in contemporary computational chemistry codes,

it seems only appropriate that taking advantage of these numerical

capabilities should be complemented by accurate variational think-

ing regarding the conceptual interpretation of these calculations.

The kinetic model was first introduced by Hellmann10 in 1933.

He was aware that it conflicted with the virial theorem, but could

not resolve the conflict since the variational connection between

the virial theorem and orbital contractions had not yet been

attracted attention26 (although orbital contraction in H2 had been

numerically found7 in 1928). Subsequently, Peierls24 as well as

Platt25 have followed Hellmanns lead.

Role of the Virial Theorem

The virial theorem is characteristic of the coulombic nature of the

potential energy functional,10,11,26 and, therefore, associated with

the similarity between the contractive variational competition in the

molecule and that in the atom, as shown by the overall shapes of

the curves in Figure 14. The virial theorem has however no connec-

tion with those physical features that are responsible for the differ-

ences between the atomic and the molecular variational competi-

tion, notably the shift of the kinetic energy functional in Figure 14.

Thus, for all its usefulness, the virial theorem is limited to identify-ing and elucidating the connection between variational minimiza-

tion and orbital shrinking toward or swelling away from nuclei,

once the detailed behavior of the kinetic energy functional and the

potential energy functional with respect to relevant Hamiltonian

and orbital changes are known. The virial theorem by itself cannot

furnish any information regarding the relevant changes in the

behavior of these functionals from one system to another. This

knowledge has to come from an independent physical analysis. But

it is just this specific behavior of the kinetic and potential energy

functionals in specific systems that is the basis of the physical origin

of the energy lowering that establishes specific bonds. These con-

clusions are in agreement with the observations made earlier in

Section 2.3. In this context, it should also be noted that the virial

theorem holds not only at bonded energy minima but also at transi-tion states where a bond is broken.

Unfortunately, an unfounded overestimate of the information

content of the virial theorem as regards the origin of chemical bind-

ing has lingered in the chemistry literature for over half a century.

This bias has furthermore led to the conjecture that electron sharing

causes chemical binding through the increased electrostatic attrac-

tion experienced by the charge that is accumulated in the bond. The

detailed variational analysis given here as well as in earlier

papers1220 invalidates this hypothesis for the case of H2 and H2.

Beyond H21 at the Equilibrium Distance

Other Aspects of the H21

Ion, the H2 Molecule

Since the relation between variational calculus and the eigenvalues

of the Schrodinger equation is completely general, it can be applied

to understand any molecular energy. It can thus also be used to ex-

amine the energies along the entire potential energy curve of H2.

In this context, the virial theorem plays a different role because

its general form11 is in fact

2TR VR R dER=dR 0: (5:1)

which can also be obtained by applying the variation principle to

an orbital exponent scale parameter.26 By combining eq. (5.1)

with the relation E T V, one obtains

TR ER R dER=dR; VR 2ER R dER=dR;

(5:2)

so that T(R) and V(R) are determined when E(R) is known. Appli-

cation to typical covalent diatomic dissociation curves (e.g. the

Morse potential) yields the well-known shapes of E(R), T(R), V(R).

On the other hand, since the simple form 2T V, is lost alongthe potential energy curve it can no longer be used as a short cut in

the variational reasoning and more detailed analyses are required.

Such analyses have been carried through for H2.13,14,1619

The curves of H2 are displayed in Figure 15 and they exhibit

the typical reversals in the signs of the kinetic and potential com-




18/22

ponents to the energy difference [E(H2) E(H)] with increasing

internuclear distance. From about twice the equilibrium distance to

infinity, the binding energy is due to a lowering of its kinetic com-

ponent while the potential component opposes binding. The analy-

sis shows that, in this range, there exists no quasiatomic orbital

contraction so that, here, the kinetic energy lowering caused by

electron sharing, which was discussed in Section 4.4, is directly

reflected in the minimized total energy. The turnover of the kineticand potential curves from about 2Req inwards is clearly related to

the steadily increasing contraction. On the other side, it is seen that

the increase in the total energy just short of the equilibrium dis-

tance is due to a steep rise in kinetic energy as a consequence of

the increasing localization of the entire system. The potential

energy starts to increase only at about []Req. It is thus apparent

that the kinetic energy plays a crucial role in shaping the entire dis-

sociation curve except at very short distances. The repulsive char-

acter of the antibonding state, too, is essentially related to the ki-

netic energy; its analysis has been discussed in ref. 13.

An analysis of the electron pair bond in H2 has shown12,18,19

that, here, the bonding effect is the result of each electron contrib-

uting individually to the energy lowering in essentially the same

way as has been described for the one-electron bond of H2

. Thereare two differences: On the one hand, the quasiclassical electro-

static terms contribute more strongly to the binding energy lower-

ing; on the other hand, the electron repulsions interfere to some

degree with the two electrons being independently shared

between the two centers and force an electron pair correlation

that creates an exchange-like pattern. The lowering of the kinetic

energy functional by electron sharing is again the crucial factor

for establishing this covalent bond.

Bonding Involving Other Atoms

The hydrogen molecule ion and the hydrogen molecule mani-

festly have the simplest energy functional of all molecules. Many

additional interactions are contained in the energy functionals

that are relevant for bond formation in other systems. The varia-

tional analysis will therefore have to take into account more com-

plex relationships, such as for instance:

Interactions involving primarily p-, d- or f-orbitals. Interactions involving many electrons and many orbitals. Increases in interelectronic repulsions due to electron sharing. Changes in interelectronic correlations due to bond formation. Facilitation of polarization due to near-degeneracies between

orbitals.

Electrostatic attractions betweenneutral atoms with many electrons. Electrostatic attractions generated by electronic charge transfer

between atoms.

Nonbonded repulsions caused by exclusion effects due to elec-tron antisymmetry.

Differences between cores and valence shells in bonding activ-ities.

Kutzelnigg has given an early overview over many of these

aspects.18,19 Bitter and Schwarz have elucidated the problems of

inner shells using the pseudo-potential approach.39 Kovacs, Ester-

huysen, and Frenking have shown the important strength of the

quasiclassical coulombic attractions in many electron systems.40

Bickelhaupt and Baerends have recently given a thorough discus-

sion of many of these problems from the point of view of density

functional theory.41 Bitter et al. examine further aspects in this

issue.39c These publications contain further references to the large

literature.

At present, it seems likely that the lowering of the kinetic

energy pressure through electron sharing will remain a valid basic

ingredient for nonionic covalent bonds in general, subject to the

mentioned additional influences. In general, too, there must exist

wave function relaxations that yield the further energy lowering

required for the establishment of the virial theorem. They can

however be different and more complex than the simple contrac-

tion found in H2. It is thus likely that the kinetic model will

remain a valid part for nonionic covalent bonds in general, but

that models for the additional influences will complement this

specific aspect of electron sharing.

An interesting and promising fortunate fact is that variational

analyses of electronic energies at transition states can also make

use of the virial theorem in the simple form 2T V.

Acknowledgments

K.R. wishes to express his warmest gratitude to the late John R.

Platt for his friendship, inspiration, and stimulation half a century

ago. He also acknowledges repeated instructive interactions with

W.H.E. Schwarz and W. Kutzelnigg. The authors also thank them

as well as E. J. Baerends for a critical reading of the manuscript.

Figure 15. Kinetic, potential and total energies of the H2 ground-

state dissociation curve as functions of the internuclear distance. Dis-

tance in bohr. Energies in hartree.




19/22

Appendix

Basis-Set-Independent Definition of Quasiatomic Orbitals

In the discussion of eqs. (3.9)(3.14), we broached the subject of

defining basis-set-independent quasiatomic orbitals in H2. We

shall now obtain such orbitals by expressing the molecular wave-

function in terms of two unique normalized orbitals that are assimilar as possible to the free-atom groundstate orbitals on A and

B, respectively.

We introduce a normalized function f that is antisymmetric

with respect to the molecular symmetry plane, but otherwise as

yet arbitrary. It can be expanded as

fx;y; z kfkVkx;y; z; k 1; . . . ; 26 (A:1)

where the Vk are defined in eq. (3.3) and the coefficients fk satisfy

jkVjk fjk 1: (A:2)

With the help of f, we now define the two functions

A cos f sin ; B cos f sin : (A:3)

Since andfmanifestly form an orthonormal pair, the functions AandB arenormalized but notorthogonal. Their overlap integral is

hAjBi cos2 sin2 cos2: (A:4)

They are each others mirror image with respect to the molecular

symmetry plane and the molecular wavefunction can mani-festly be expressed in terms of them as

A B=2 cos A B=21 hAjBi1=2:

(A:5)

We shall now determine the function fand the angle in such away that the normalized function A differs as little as possible from

the groundstate orbital A of the free hydrogen atom A, as given byeq. (3.1), and, similarly, B differs as little as possible from the

groundstate orbital B of the free hydrogen atom B. We accomplishthis by maximizing the overlap integral betweenA andA

hAjAi cos hjAi sin k fkhVkjAi (A:6)

with respect to the angle and the coefficients fk under the con-straint that f remains normalized. An alternative would be to

determine and the fk by minimizing the energy hA|HA|Ai,where HA is the hamiltonian of the free hydrogen atom at A. Theresult would presumably be very similar to maximizing the over-

lap hA|Ai, in as much as A is in fact the function that yieldsthe variational atomic energy minimum for an unrestricted trial

function. This implies maximization of the Lagrange functionL cos hjAi sin kfkhVkjAi ljkVjk fj fk (A:7)

with the side condition of eq. (A.2). However, we find from eqs.

(3.2)(3.5) that

hUjjAki hUjjBki hAjjAki hAjjBki Ujk; (A:8)

hVjjAki hVjjBki hAjjAki hAjjBki Vjk; (A:9)

so that

hUkjAi hUkjBi jhjUjk; (A:10)

hVkjAi hVkjBi jhjVjk; (A:11)

Hence, the Lagrange function can be written

L cos jkckhjUjk=2 sin jkfkhjVjk=2 ljkfjfkVjk;

(A:12)

where eq. (3.6) has also been used. Differentiation with respect to

the fk now yields

sin jhjVjk=2 2ljfjVjk;

whence

hj sin 2lfj;

and, by renormalization according to the side condition (A.2),

fj hjikVikhihk1=2: (A:13)

The overlap of eq. (A.6) becomes therefore

hAjAi cos hjAi sin hfjAi; (A:14)

where now

hjAi jkUjkcjhk (A:15)

hfjAi jkVjkhjhk1=2: (A:16)

We note for later use that, because of eq. (A.16), one can write

eq. (A.13) as

fj hj=2hf=Ai: (A:17)

Maximization of (A.14) with respect to then yields

cos hjAi=hjAi2 hfjAi

21=2 (A:18a)

sin hfjAi=hjAi2 hfjAi

21=2: (A:18b)

Insertion of these results into eq. (A.14) yields for the overlap

between A and A the expression

hAjAi2 hjAi

2 hfjAi2 jkUjkcjhk

2

1=4jkVjkhjhk: A.19

It is readily seen that the values found for fk and also maxi-mize the overlap integral hB|Bi. Insertion of eqs. (A.17) and(A.18) into eq. (A.3) therefore yields for these quasiatomic orbi-

tals the result

A hjAikckUk khkVk=hAjAi (A:20)

B hjAikckUk khkVk=hAjAi; (A:21)




20/22

where h|Ai and hA|Ai are given by (A.15) and (A.19). Inser-t

Documents

Electron Sharing And Chemical Bonding