Orbital Exponent Optimization in Elementary VB Calculations of the

Research: Science and Education

Orbital Exponent Optimization in Elementary VB Calculations of the Chemical Bond in the Ground State of Simple Molecular Systems Valerio Magnasco Dipartimento di Chimica e Chimica Industriale dell’Università, Via Dodecaneso 31, 16146 Genova, Italy; [email protected]

The nature of the chemical bond has attracted the attention of the chemical community since the early days of the application of quantum mechanics to chemistry (1, 2) up to the recent Discussion of the Faraday Society (3). Ruedenberg (4) was the first person who stressed the importance of what he called “interference” effects on the electron density in the formation of the chemical bond, while more recently Kutzelnigg (5) outlined the importance of the kinetic energy terms. The nature of the “exchange-overlap” terms arising necessarily from any wavefunction satisfying the antisymmetry principle was analyzed theoretically by the author and McWeeny (6) for the weak interactions occurring in short range between closed-shell molecules and by the author and Costa (7, 8) in the case of the chemical bond. The “exchange-overlap” term has a meaning strictly correlated to the “interference” term of Ruedenberg (4). Recently, Corongiu (9, 10) presented accurate numerical calculations on homonuclear diatomics and diatomic hydrides of the first-row atoms using a generalized valence bond (VB) approach that mixes Hartree– Fock with Heitler–London theory. In a recent article on this Journal (11), I proposed a simple model of the chemical bond for homonuclear diatomics based on solution of a 2 × 2 Hückel-type secular equation including overlap. I found that bonding and antibonding in H2+, H2 , He2+, and He2 could be explained in terms of occupation of molecular orbitals (MOs), the resulting bond energies strictly following the filling of such levels by electrons, Pauli repulsion (He2) occurring when both levels are full. The bond energies of the series were expressed in terms of the single-electron bondenergy parameter [(β − αS)/(1 + S)] < 0, which represents the one-electron part of the exchange-overlap component of the interaction (6–8). It is of some pedagogical interest to further examine this matter on the basis of elementary ab initio VB calculations on the same series of molecules, with the aim of (i) analyzing the relative importance of physically (even if not observable) recognizable terms that account for the origin of the chemical bond, and (ii) providing in a homogeneous way quantitative numerical results that are not easily found in most current textbooks for the simplest models of chemical bonds and of Pauli repulsion between closed shells. The Heitler–London theory of the chemical bond in H2 can be considered as the first approximation including exchange in a Rayleigh–Schroedinger perturbation expansion of the interatomic interaction energy (6, 12). In a VB calculation, variational optimization of the orbital exponent c0 of a simple Slater-type (STO) basis is tantamount to include in first order most of the spherical distortion, so expecting a fair description of the quantum exchange-overlap component of the interaction that is dominant in the bond region (12). It will be shown that,

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in this case, correct bond lengths and about 85% of the experimental bond energies are obtained from optimized ab initio VB calculations for 1-electron (H2+) (13, 14), 2-electron (H2) (14, 15), and 3-electron (He2+) (14, 16, 17) bonds, with an adequate description of the corresponding Pauli repulsion in He2 (18, 19). Atomic units1 (12) are used throughout in this article. The VB Wavefunctions for the Ground States The simplest description of our systems is achieved in terms of a STO basis of two normalized 1s atomic orbitals (AOs) centered at A and B, respectively, 1/ 2

a

c03 Q

1 /2

b

c 03 Q

exp c 0 rA

(1) exp c 0 r B

where c0 is the orbital exponent that can be optimized variationally. In terms of the conventionally defined Slater determinants (12) for 1-, 2-, 3-, and 4-electrons, the normalized VB wavefunctions for the ground states are H2 : :

2

4 g

S a b

H2 : : 1 4 g

a b ab b a 2 2S 2

the Heitler–London (HL) wavenfunction, He 2 : :

2

4 u

S a ab

(2)

2 2S

(3)

a ab bba 2 2S bba

(4)

He 2 : : 1 4 g

(5) a abb where ∙...∙ denotes a normalized Slater determinant of order N, a = aα, a = aβ, b = bα, b = bβ are the usual short notations for the normalized spin-orbitals, and S is the overlap integral.

Journal of Chemical Education • Vol. 85 No. 12 December 2008 • www.JCE.DivCHED.org • © Division of Chemical Education


For H2, the performance of the semilocalized Coulson– Fischer (CF) AOs (20) was also examined ab

bb

for the ground state E

a Mb 1 M 2 2 MS

2

4 g

(6)

b Ma

c1b c 2b

1 M 2 2 MS 2M 1 M 2 2 MS

2 2S

2

(7)

2 2S b 2 2 2S 2

1 /2

(8)

2 2S b 2

S b ab bb

SM

2M S

1 M 2 2 MS

VB

a

2

(11)

rB1

where EA

c 02 c 0 EH 2

c 02 1 c0 2 2

(12)

is the variational energy for H(2S), with EH = ‒½Eh for c0 = 1;

1 R

%E cb a 2 r B1

(9)

(13)

c 02 1 c0 2 2

E pol ∆E pol

rB1 ± dr rB1 a2 r

a r b r

a b a b ± ± dr1dr2 2 r 2 a r1 b r1

12

is the semiclassical Coulombic interaction (the electrostatic energy) between the H atom at A and the proton at B;

In the following, we make use of the charge density notation (12) for either 1-electron or 2-electron integrals 2

1 R

1 S

1 S

The Bond Energy and Its Components

a

r B1

E H %E cb E pol %E exch-ov 2 4 g

2

2

1 a b rA1 S c 02 c 0

ab S a

1/ 2

S a b

a

2

c1b : covalent c 2b : ionic

1 M

c 0

1 M 2 2 MS

2

c 02 c0 2

which for λ = 0 reduce to the original AOs. The normalized HL wavefunction built from CF AOs is seen to be equivalent to a normalized covalent plus ionic VB wavefunction : 1 4 g

(10)

The first equation is the attraction of the spherical electron density a2(r) by the nucleus at B, and the second equation is the 2-center 2-electron exchange integral describing repulsion between the two 2-center electron densities {a(r1)b(r1)} and {a(r2)b(r2)}. The Born–Oppenheimer molecular energies E for the ground states of the molecules are easily calculated from the wavefunctions given in the previous section. We define the interaction (or bond) energies as the difference E − E∞, where E∞ is the value of the variational molecular energy at infinite separation of the constituent atoms. Rather than giving lengthy formulas not particularly relevant in the present context, for didactical reasons we concentrate on the energy expression for the simplest molecule of the series, the hydrogen molecular ion H2+ giving the 1-electron chemical bond. Little calculation gives

c 0 1 a b rA11 S c 0 2 c 0

(14)

1 S

is the energy contribution due to the spherical polarization of the H atom by the proton, a term that is zero for c0 = 1 (hydrogenic AOs);

a b S a

2

%E

exch-ov 2

4 g

rB1

1 S

(15)

is the quantum mechanical “exchange-overlap” term that depends on the electronic state of H2+ (attractive for the 2Σg+ ground state, repulsive for the 2Σu+ excited state). This term is the main factor determining the existence of the 1-electron chemical bond in H2+. Equation 15 contains the exchange-overlap density (see the Appendix in the online material) a r b r S a 2 r

(16)

having the property

± dr ©«a r b r

S a2 r ¸º 0

(17) Accurate representation of such kind of densities, which are essentially those given in ref 11, is expected to give a reasonably

© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 12 December 2008 • Journal of Chemical Education

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rors that become increasingly larger at great distances, obtaining the polarized exchange-overlap component as difference from the interaction energy. Direct calculation of ΔE cb for H2+, H2, He2+, and He2 gives (ρ = c0 R) –2

H2 : % E cb c 0 exp 2S 1 S

–1

Disp

lace

1

men

–2

2

t/a

–1

0

1

2

3

Displacement / a0

3

0

H2 : % E cb c 0

Figure 1. Coulson–Fischer optimized AO for ground state H2 at R = 1.4a0 (left) and its section in the zx plane (right).

exp 2S

5 3 1 1 S S 2 S3 8 4 6 S

quantitative account of the chemical bond in the ground state. It is interesting to note that eq 15 exactly parallels the bond energy parameter of our previous model (11), if we put

C ab r

B a 2 r B1

1 B

(18)

To simplify the analysis, it is convenient to calculate the bond energy as (19) %E E Ee %E cb %E exch-ov + cb where Δ E is given explicitly by eqs 24–27 for H2 , H2, He2+, and He2, and Δ E exch-ov is the exchange-overlap between spherically polarized charge distributions absorbing the polarization term. The expressions from which Δ E was calculated for H2+, H2, He2+, and He2 are H2

H2

2

4 g : % E

1

4 g

2

4 g

E

2

4 g Ee

1 Ee EH E h 2 : %E 1 4 g E 1 4 g Ee

(20)

(21)

Ee 2 EH 1E h He2

2

4 u : %E

2

4 u

E

2

4 u Ee

3 2 5 c 0 3Z c0 (22) 2 8 5 43 Z for Z 2 24 24

Ee c 0e He2

1

4 g : %E

4 E 4 E 1

g

1

g

e

5 c0 (23) 16 5 27 c 0e Z for Z 2 16 16 In fact, for variationally optimized c0, it is found simpler and more accurate to directly evaluate at any internuclear distance R the Coulombic (electrostatic) component Δ E cb of the interaction through its analytic expressions, avoiding the round-off erEe c 0 2 2 Z

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(25)

He 2 : % E cb 4c 0 exp 2S 1 13 S 3 S2 1 S3 (26) S

(24)

S

0

He 2 : % E cb 4c 0

16

8

12

exp 2S

1 5 3 1 S S2 S3 6 8 4 S

(27)

For all systems studied, including the molecular ions, the Coulombic interaction has the nature of a charge-overlap term that decreases exponentially with the internuclear distance R. It is seen that for H2+ the Coulombic interaction is repulsive at any distance, so that classically it is not possible to form any 1-electron bond, while for the remaining systems it is repulsive for small values and attractive for large values of ρ. At large values of R, all 2-center integrals go to zero, and the variationally optimized molecular energy correctly gives the energy EA + EB of the separate atoms for H2+ (E∞ = ‒0.5Eh for c0 = 1) and H2 (E∞ = ‒1Eh for c0 = 1), and the corresponding variational results for He2+ [He(1S) + He+(2S), E∞ = ‒(3/2)(43/24)2 for c0 = 43/24] and He2 [2He(1S), E∞ = ‒2(27/16)2 for c0 = 27/16]. So, even if the absolute values of the calculated molecular energies may be rather poor compared to the accurate values, bond energies (differences between variationally estimated quantities) can nonetheless be obtained in a sufficiently accurate way. Results and Discussion The results obtained are given in Tables 1–6. The values at the minima in the potential energy calculations for H2+, H2, and He2+ are given as the last row of Tables 2, 3, and 5. These quantitative data, which clearly show the origin of the chemical bond, are not available in a homogeneous way in quantum chemistry textbooks. Table 1 shows the results of the calculations for the hydrogenic values of c0 (c0 = 1 for H2+ and H2, c0 = 2 for He2+ and He2). These results are unsatisfactory either for the bond distances or the bond energies. The bond distances taken at the minimum of the potential energy curves are 25% too long for H2+ (about 2.5a0) (13, 14) and 18% for H2 (about 1.65a0) (14, 15), 8% too short for He2+ (about 1.88a0) (14, 17), with corresponding weaker (H2+ and H2) or stronger (He2+) bond energies. The Pauli repulsion between He atoms is severely underestimated in the medium range region (no more than 55% at R = 3a0 and 27% at R = 4a0) (18, 19). The c0 optimization (Tables 2–5) contracts the AOs shortening the bond to the experimental value for H2+ (from 2.5a0 to 2a0 for c0 = 1.24) and nearly so for H2 (from 1.65a0 to 1.42a0 for c0 = 1.17), while it expands the AOs for He2+ lengthening the bond (from 1.88a0 to 2.06a0, nearly the exact bond length, for c0 = 1.83). The c0-optimized bond energies are now within 84–85% for H2+ and H2, and over 99% for He2+, a remarkably good and rather unexpected result. The c0–optimized Pauli repulsion energy (Table 6) for the interaction between ground


Research: Science and Education Table 1. Hydrogenic Interaction Energies (c0 = 1 for H2+ and H2; c0 = 2 for He2+ and He2)

R/a0

H2+/(Eh x10–2)

H2/(Eh x10–2)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.5 2.6 2.8 3 3.5 4 4.5 5 5.5 6

21.16337 9.758522 2.865430 –1.351176 –3.900570 –5.377150 –6.149239 –6.454169 –6.482939 –6.449620 –6.242878 –5.908260 –4.817392 –3.686612 –2.702921 –1.920316 –1.331936 –0.9067662

0.3576199 –7.198681 –10.54739 –11.57096 –11.30491 –10.35513 –9.088880 –7.730608 — –6.414417 –5.214909 –4.167404 –2.240237 –1.128546 –0.5428369 –0.2524731 — –0.05091801

He2+/(Eh x10–2)

He2/(Eh x10–2)

50.75862 16.53329 –0.4442915 –8.248667 –11.12231 –11.40897 –10.43490 –8.948979 — –7.361077 –5.879420 –4.594600 –2.322478 –1.104469 — –0.2256733 — –0.04265488

— — — — — 11.65662 — — 3.051962 — — 0.7435668 0.1695968 0.03657354 0.007532202 0.001492938 — —

Table 2. c0-Optimized Bond Energies and Their Components for Ground State H2+(2∑g+) R/a0 0.5 1 1.2 1.4 1.6 1.8 2 2.5 3 4 5 6 7 8 2.0002

c0/a0–1 1.779224 1.538102 1.459847 1.391916 1.333143 1.282345 1.238623 1.153585 1.094669 1.028295 1.002295 0.995495 0.995667 0.995247 1.238581

ΔE cb/(Eh x10–2) 63.78162 11.70929 6.899700 4.274606 2.748650 1.817299 1.226018 0.4856869 0.2005753 0.03419533 0.005334579 0.0007537196 0.0001005924 0.00001319804 1.225552

ΔE exch–ov/(Eh x10–2) 13.04247 –5.809126 –8.491863 –9.808583 –10.28057 –10.23494 –9.876617 –8.361437 –6.645396 –3.767643 –1.925876 –0.9087941 –0.4047131 –0.1734292 –9.876152

ΔE(2∑g+)/(Eh x10–2) Accurate ∆E/(Eh x10–2) 76.824080 5.900164 –1.592163 –5.533977 –7.531918 –8.417645 –8.650599 –7.875751 –6.444821 –3.733448 –1.920542 –0.9080403 –0.4046125 –0.1734160 –8.650600

76.50120 4.821369 –2.897452 –6.998353 –9.093722 –10.02536 –10.26342 –9.382351 –7.756286 –4.608488 –2.442029 –1.196905 –0.5594004 –0.2570388 —

Note: The accurate values are from ref 13.

Table 3. c0-Optimized Bond Energies and Their Components for Ground State H2(1∑g+) R/a0

c0/a0–1

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 3 4 5 6 7 8 1.419

1.286578 1.253185 1.222744 1.194869 1.169492 1.146511 1.125751 1.107005 1.090165 1.075101 1.056844 0.995375 0.991778 0.996987 0.999255 0.999856 0.999979 1.165041

ΔE cb/(Eh x10–2)

ΔE exch–ov/(Eh x10–2)

1.584685 –0.001308811 –0.9928711 –1.595334 –1.942307 –2.119554 –2.183107 –2.170527 –2.107627 –2.012083 –1.899395 –0.7016216 –0.1677576 –0.03290781 –0.006006299 –0.001051293 –0.0001792907 –1.987593

–10.44321 –11.40208 –11.90274 –12.06085 –11.96264 –11.67602 –11.25415 –10.73820 –10.15971 –9.543187 –8.901615 –3.468312 –0.9680654 –0.2205829 –0.04498264 –0.008577527 –0.001560399 –11.92031

ΔE(1∑g+)/(Eh x10–2) Accurate ∆E/(Eh x10–2) –8.858521 –11.40339 –12.89561 –13.65618 –13.90495 –13.79557 –13.43726 –12.90873 –12.26734 –11.55527 –10.80101 –4.169934 –1.135823 –0.2534907 –0.05098894 –0.009628820 –0.001739690 –13.90790

–12.45396 –15.0057 –16.49352 –17.23471 –17.44476 –17.28550 –16.85833 –16.24587 –15.50687 — –13.81329 –5.732623 –1.639023 –0.3785643 –0.0835702 –0.0197910 –0.0055603 —



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Research: Science and Education Table 4. c0- and λ- Optimized Bond Energies for the Single Coulson–Fischer Covalent VB Structure for Ground State H2(1∑g+) R/a0

c0/a0–1

λ

ΔE(1∑g+)/(Eh x10–2)

Accurate ∆E/(Eh x10–2)

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 3 4 5 6 7 8

1.277953 1.249469 1.224354 1.200973 1.179314 1.159912 1.141592 1.124769 1.109405 1.078934 1.014706 0.997786 0.998278 0.999464 0.999876 0.999977

0.129301 0.132650 0.134559 0.134915 0.134059 0.132428 0.129870 0.126694 0.123035 0.118521 0.071289 0.031703 0.012105 0.004373 0.001563 0.000560

–12.02490 –13.61720 –14.46170 –14.77770 –14.71900 –14.39560 –13.88760 –13.25350 –12.53700 –11.74300 –4.695300 –1.278060 –0.2779240 –0.0544762 –0.0100996 –0.0018025

–15.00573 –16.49352 –17.23471 –17.44476 –17.28550 –16.85833 –16.24587 –15.50687 — –13.81329 –5.732623 –1.639023 –0.3785643 –0.0835702 –0.0197910 –0.0055603


Table 5. c0-Optimized Bond Energies and Their Components for Ground State He2+(2∑u+) R/a0

c0/a0–1

1 1.2 1.4 1.6 1.8 1.9 1.95 2 2.05 2.1 2.2 2.4 3 4 5 6 10 2.0604

1.962093 1.933367 1.904229 1.877931 1.855573 1.845911 1.841430 1.836915 1.832951 1.829454 1.822631 1.811482 1.793294 1.788060 1.789807 1.791003 1.791445 1.832438

ΔE cb/(Eh x10–2) 4.117996 –0.5603882 –1.489278 –1.353544 –0.9975039 –0.8246597 –0.7450050 –0.6709627 –0.6019660 –0.5381917 –0.4272901 –0.2623586 –0.05226690 –0.002628478 –0.0001099342 –0.000004233514 –0.00000000000642838 –0.5879553

ΔE exch–ov/(Eh x10–2)

ΔE(2∑u+)/(Eh x10–2) Accurate ∆E/(Eh x10–2)

52.92881 22.94101 6.188459 –2.632875 –6.807837 –7.810852 –8.121559 –8.328238 –8.446735 –8.490096 –8.394771 –7.722307 –4.641788 –1.379302 –0.3420699 –0.07833218 –0.0001511549 –8.461885

57.04681 22.38062 4.699181 –3.986419 –7.805341 –8.635512 –8.866564 –8.999201 –9.048701 –9.028288 –8.822061 –7.984666 –4.694055 –1.381930 –0.3421798 –0.07833641 –0.0001511549 –9.049840

— — 0.3879050 –5.563789 –8.251056 –8.831154 — –9.068892 –9.088497 –9.054864 –8.859313 –8.128068 –5.206322 –1.905905 –0.624148 –0.203952 –0.007699 —

Note:The accurate values are from ref 17.

Table 6. c0-Optimized Pauli Repulsion Energies and Their Components for the He(1S)–He(1S) Interaction in Medium Range R/a0 2 2.5 3 3.5 4 4.5 5

c0/a0–1 1.694575 1.693473 1.690886 1.689065 1.688130 1.687729 1.687577

ΔE cb/(Eh x10–2) –2.728214 –0.7549173 –-0.1930696 –0.04687464 –0.01095241 –-0.002486387 –0.0005521989

ΔE exch–ov/(Eh x10–2) 16.38988 5.021854 1.489472 0.4268538 0.1182692 0.03176747 0.008300243

ΔE(1∑g+)/(Eh x10–2) 13.66167 4.266937 1.296402 0.3799792 0.1073168 0.02928108 0.007748044

Accurate ∆E/(Eh x10–2)

1.352 0.1355 0.01250


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state He atoms is now remarkably improved (96% of the accurate result at R = 3a0 and about 80% at R = 4a0). The results of the simple Heitler–London–Coulson– Fischer wavefunction are also remarkably good. At R = 1.4a0, optimization of both c0 and λ gives c0 = 1.200973, λ = 0.134915, E = ‒1.147777Eh, within about 98% of the accurate Wolniewicz (15) result for the molecular energy and about 85% of the bond energy. The optimized 2-center semilocalized CF AO,

(28) a b 0. 91281 a 0. 12311 b shows a little cusp at nucleus B, which improves interorbital overlap by over 18% (S′ ≈ 0.797 instead of 0.675) and is sketched in Figure 1. At the optimized value of λ the ionic structure disappears. Examining the components of the bond energies, Tables 2–6 show that the quantum exchange-overlap component Δ E exch-ov is by far the most important contribution to either the bond energy or the Pauli repulsion in He2, confirming the validity of the recently proposed model of the chemical bond (11). Conclusions From this work it is apparent that c0 optimization (i) gives the correct variational approximation at any value of R, (ii) includes most of the spherical distortion of the AOs, so important in the bond region, and (iii) yields correct bond lengths and about 85% of the bond energies. This is a remarkable result for this simple approach when compared to the more elaborate calculations based on large basis sets.2 Furthermore, it allows for wavefunctions satisfying the virial theorem, ensuring a correct partition between kinetic and potential energy as required by Kutzelnigg (5). The change in the shape of the AOs occurring in the optimization further suggests that, in the elementary VB theory of the chemical bond, the guiding principle should be the maximum of the exchange-overlap energy (7, 8) and not the maximum overlap (1, 2), since the latter behaves in the opposite way (overlap decreases, contracting the AOs; increases, expanding the AOs ). These conclusions are of great pedagogical value, since, besides the origin of chemical bond and Pauli repulsion, they show that fair quantitative results can be obtained whenever the most important physical effects are accounted for in an elementary way (11). Acknowledgments Support by the Italian Ministry of Education University and Research (MIUR) under Grant No. 2006 03 0944 003 and by the University of Genoa is gratefully acknowledged. Thanks are due to Roberto Peverati for help in the calculations.

2. The larger basis sets are (i) for H2, a 279–term expansion in spheroidals for the two electrons including powers of the interelectronic distance (15); for He2+, a Full–CI calculation from a Hartree–Fock 4s3p2d1f STO basis onto each center (16) and a 320–term expansion of fully optimized Single Exponentially Correlated Spherical Gaussians (SECSGs) with 2240 non–linear parameters (17); and for He2, a SCF Hartree–Fock calculation using the previous 4s3p2d1f STO basis onto each center (18). The latter data were confirmed by a recent accurate calculation by the Jeziorski group (21).

Literature Cited 1. Pauling, L. J. Am. Chem. Soc. 1931, 53, 1367–1400. Pauling, L. J. Am. Chem. Soc. 1931, 53, 3225–3237. 2. Coulson, C. A. Proc. Cambridge Phil. Soc. 1937, 33, 111– 114. 3. Faraday Discussions 2007, 135, 1–515. 4. Ruedenberg, K. Rev. Mod. Phys. 1962, 34, 326–376. 5. Kutzelnigg, W. The Physical Origin of the Chemical Bond. In Theoretical Models of Chemical Bonding, Part 2; Maksic, Z. B., Ed.; Springer: Berlin, 1990; pp 1–43. 6. Magnasco, V.; McWeeny, R. Weak Interactions Between Molecules and Their Physical Interpretation. In Theoretical Models of Chemical Bonding, Part 4; Maksic, Z. B., Ed.; Springer: Berlin, 1991; p 133–169. 7. Magnasco, V.; Costa, C. Chem. Phys. Lett. 2005, 403, 303– 307. 8. Magnasco, V. Chem. Phys. Lett. 2005, 407, 213–216. 9. Corongiu, G. Int. J. Quantum Chem. 2005, 105, 831–835. 10. Corongiu, G. J. Phys. Chem. 2006, A110, 11584–11598. 11. Magnasco, V. J. Chem. Educ. 2004, 81, 427–435. 12. Magnasco, V. Elementary Methods of Molecular Quantum Mechanics; Elsevier: Amsterdam, 2007. 13. Peek, J. M. J. Chem. Phys. 1965, 43, 3004–3006. 14. Huber, K. P.; Herzberg, G. Molecular Spectra and Molecular Structure: IV. Constants of Diatomic Molecules; Van Nostrand Reinhold: New York, 1979. 15. Wolniewicz, L. J. Chem. Phys. 1993, 99, 1851–1868. 16. Liu, B. Phys. Rev. Lett. 1971, 27, 1251–1253. 17. Cencek, W.; Rychlewski, J. J. Chem. Phys. 1995, 102, 2533–2538. 18. Liu, B.; McLean, A. D. J. Chem. Phys. 1973, 59, 4557–4558. 19. Feltgen, R.; Kirst, H.; Koehler, K. A.; Pauly, H.; Torello, F. J. Chem. Phys. 1982, 76, 2360–2378. 20. Coulson, C. A.; Fischer, I. Phil. Mag. 1949, 40, 386–393. 21. Patkowski, K.; Cencek, W.; Jeziorska, M.;Jeziorski, B.; Szalewicz, K. J. Phys. Chem. A 2007, 111, 7611–7623.

Supporting JCE Online Material

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Notes

Abstract and keywords

1. Atomic units (au) are obtained by posing e = m = h = 4πε0 =1 (12). SI equivalents of the quantities that are of interest to us here are: 2 au of length (bohr), a0 = 4πε0h (me2)–1 = 5.291772 × 10–11 m; au of 2 energy (hartree), Eh = e (4πε0 a0)–1 = 4.359744 × 10–18 J.

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Origin of the quantum mechanical exchange-overlap density


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Orbital Exponent Optimization in Elementary VB Calculations of the

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