Exploring the Nature of the H2 Bond. 1. Using ... - ACS Publications

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Exploring the Nature of the H2 Bond. 1. Using Spreadsheet Calculations To Examine the Valence Bond and Molecular Orbital Methods Arthur M. Halpern* and Eric D. Glendening Department of Chemistry and Physics, Indiana State University, Terre Haute, Indiana 47809, United States S Supporting Information *

ABSTRACT: A three-part project for students in physical chemistry, computational chemistry, or independent study is described in which they explore applications of valence bond (VB) and molecular orbital−configuration interaction (MO− CI) treatments of H2. Using a scientific spreadsheet, students construct potential-energy (PE) curves for several states of H2 from the kinetic and potential energies, calculated from closedform analytical expressions of the ten unique integrals arising from the Born−Oppenheimer Hamiltonian. For this project students use hydrogen 1s basis functions that include a screening parameter. From the calculated PE curves, they find the dissociation energy, De, and equilibrium internuclear distance, Re. In part I students use the Heitler−London (VB) form of the wave function to obtain the PE curves. In part II they optimize the value of the screening parameter to improve the results, and in part III they explore the treatment of H2, using both the simple MO wave function and the application of CI, with and without screening parameter optimization, to obtain the PE curves. Students compare their De and Re results with the experimental values. A set of questions, exercises, and a sample spreadsheet are provided as Supporting Information. KEYWORDS: Upper-Division Undergraduate, Physical Chemistry, Computer-Based Learning, MO Theory, Valence Bond Theory, Quantum Chemistry eginning with their first enrollment in an introductory chemistry course, students learn about chemical bonding (one of the most significant, if elusive, concepts in chemistry) through the concept of electron-pair sharing as depicted by the Lewis structure.1 Students return to chemical bonding in their physical chemistry course, but with the advantage of using the tools of quantum chemistry to explore the chemical bond.2 Starting with H2+, students set up the complete Hamiltonian operator and simplify it using the Born− Oppenheimer (BO) approximation. They encounter the approximate wave function, employ a variational treatment, and solve the secular equation to obtain the binding energy of the molecule. Students then “graduate” to the H2 molecule, where they deal with the attendant complications that arise from this twocenter, two-electron problem. Instructors usually introduce the approximate wave function based on the valence bond (VB) and molecular orbital (MO) methodologies, and illustrate the role of the Coulomb and exchange integrals that arise from the secular equation in the expression of the total interaction energy. In most cases at this point, students encounter the iconic potential-energy (PE) curves showing the attractive interaction between the two H atoms of the singlet state, as well as the repulsive interaction of the triplet state. Instructors generally feel compelled, however, to omit the mathematical

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© 2013 American Chemical Society and Division of Chemical Education, Inc.

details needed to produce these curves, and only treat the integrals as symbolic entities. Covalent bonding in H2 has been discussed in this Journal before. As early as 1936, Dushman3 published a series of articles called “Elements of the Quantum Mechanics,” of which two contained presentations of the VB treatment of H2. In 1971, Dewar and Kelemen4 presented a detailed treatment of MO theory as applied to H2, including a discussion of spin functions and configuration interaction (CI). Then in 1997, Bacskay et al.5 described covalent bonding in H2+ and H2, emphasizing the value of computer graphics as a learning tool for visualizing electron delocalization and illustrating key aspects of the covalent bond. This article describes a project in which students, armed with only a spreadsheet and basic knowledge of quantum mechanics, explore the simplest applications of the VB and MO−CI treatments of H2. Designed for students enrolled in a physical or computational chemistry course, or independent study, the project guides them through the steps needed to produce for themselves the PE curves for several states of H2. Although the difficult derivations of the internuclear distance-dependence of the kinetic and potential-energy integrals are avoided, students are given the tools needed to fill in gaps that typically separate Published: October 11, 2013 1452

dx.doi.org/10.1021/ed400234g | J. Chem. Educ. 2013, 90, 1452−1458

Journal of Chemical Education

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Table 1. Ten Integrals of the H2 Molecule Integral I1

Expression

⟨a(1)|b(2)⟩

Description

Value

Overlap integral

S

I2

1 a(1) − ∇12 a(1) 2

Kinetic energy of electron 1 centered on nucleus a

α2/2

I3

1 a(1) − ∇12 b(1) 2

Exchange kinetic energy of electron 1

−α2(K + S/2)

Coulomb attraction of electron 1 centered on nucleus a to nucleus a

−α

Coulomb attraction of electron 1 centered on nucleus a to nucleus b

αJ

Exchange Coulomb electron−nucleus attraction energy

αK

Coulomb repulsion of electron 1 centered on nucleus a and electron 2 centered on nucleus b

αJ′

Exchange Coulomb electron repulsion energy10

αK′

I4

a(1) −

1 a(1) r1a

a(1) −

1 a(1) r1b

a(1) −

1 b(1) r1a

I5

I6

I7

a(1)b(2)

1 a(1)b(2) r12

a(1)b(2)

1 b(1)a(2) r12 1 a(1)a(2) r12

Coulomb repulsion of electron 1 centered on nucleus a and electron 2 centered on nucleus a

5α/8

a(1)a(2)

1 a(1)b(2) r12

Exchange Coulomb electron repulsion energy

αL

a(1)a(2)

I8

I9

I10

complete information for performing the calculations involved in this project.

the mathematical abstraction of the secular equation from the visual interpretation of the PE curves. We believe that, by carrying out some of the basic computational steps needed to produce the PE curves, students gain a deeper understanding of both covalent bonding and the methods implemented in modern quantum chemistry programs. Students complete the project in three steps. Students begin in part I by writing the Hamiltonian for H2 and representing the VB wave function in Heitler−London (HL) form. They set up the Schrödinger equation and identify the integrals required to calculate the energy expectation value. Using closed-form expressions for the internuclear dependence of these integrals, students obtain PE curves for H2 and determine values for the binding energy, De, and equilibrium internuclear separation, Re. They also examine the relationship between the bonding interaction of H2 and electron exchange that is required by the Pauli exclusion principle. In part II, students perform variational calculations using a screening parameter of the basis functions to improve the HL wave function and PE curves. They then determine the resulting changes in the De and Re properties. In part III, students turn to MO-based approaches, examining the failure of the simplest MO wave function to describe dissociation and the application of CI to correct this deficiency. Students use a variational treatment of a simple MO−CI wave function to obtain the best possible solution of the H2 ground-state potential based on two 1s hydrogenic basis functions. All calculations of this project are carried out using an Excel (or other readily available) spreadsheet. We provide a set of questions and exercises in the Supporting Information that can be used to test students’ knowledge and to enhance their learning experience. Also included there is

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PART I: THE HEITLER−LONDON−SUGIURA (HLS) VALENCE BOND TREATMENT We assume that students have already been introduced to atomic units6 and the BO approximation7 (e.g., in studying H2+). They should readily understand the expression of the Hamiltonian operator for H2, which (in atomic units) is 1 1 1 1 1 1 1 1 Ĥ = − ∇12 − ∇22 − − − − + + 2 2 r1a r1b r2a r2b r12 R (1)

where 1 and 2 denote the electrons, a and b represent the nuclear centers, and R, the internuclear distance, is treated as a constant. Students should also have encountered the normalized 1s wave function, which will be used here to represent the basis set (the set of basis functions) from which an approximate wave function can be constructed. Thus, for electron 1 on atom a, one has a(1) =

α 3 −αr1a e π

(2)

where r1a is the distance from electron 1 to nucleus a, and α is the screening parameter that will be considered in the variational treatments of parts II and III. (Note that although the screening parameter has units of reciprocal length, it will be expressed here as a dimensionless quantity throughout.) Initially, one sets α = 1, in which case a(1) is the 1s wave function of the free H atom. Three other functions, i.e., a(2), b(1), and b(2), are denoted analogously. In 1927, in one of the earliest applications of the (then) new quantum mechanics, W. Heitler and F. London8 reasoned that at infinitely large R the H2 system, consisting of two degenerate, 1453



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Table 2. Expressions of the w-Dependence of the Fundamental Integralsa

non-interacting H atoms, can be described by the product wave functions a(1)b(2) or, equivalently, b(1)a(2). For finite values of R, they recognized that the total wave function, Ψ(1,2), of H2 can be represented (not including spin functions) as a linear combination of these two functions, or Ψ(1, 2) = C1a(1)b(2) + C2b(1)a(2)

Integral

(3)

Because the components of this approximate wave function are equivalent, the coefficients C1 and C2 must be of equal magnitude. Therefore, Ψ(1,2) must be either symmetric or antisymmetric with respect to electron permutation (i.e., C2 = ±C1), and eq 3 can be re-expressed (in unnormalized form) as Ψ± = a(1)b(2) ± b(1)a(2)

Expression

S

−w⎛ ⎜

1 ⎞ e 1 + w + w 2⎟ ⎝ 3 ⎠

S′

⎛ 1 ⎞ e w⎜1 − w + w 2⎟ ⎝ 3 ⎠

J

−

K

− e−w(1 + w)

J′

⎛1 1 11 3 1 ⎞ − e−2w⎜ + + w + w 2⎟ ⎝w w 8 4 6 ⎠

K′

⎫ ⎧ ⎛ 25 23 1 ⎞ − e−2w⎜− + w + 3w 2 + w 3⎟ ⎪ ⎪ ⎪ ⎝ 8 4 3 ⎠ 1⎪ ⎨ ⎬ 5⎪ 6 2 ⎪ 2 ⎪+ [S (γ + ln w) + S′ Ei(− 4w) − 2SS′Ei(− 2w)]⎪ ⎩ w ⎭

L

⎛ ⎛ 1 1 5 ⎞⎟ 5 ⎞⎟ e−w⎜w + + + e−3w⎜− − ⎝ ⎝ 8 8 16w ⎠ 16w ⎠

(4)

These are the HL wave functions. When treated as a variational wave function, Ψ± is associated with the expectation value of the energy as ⟨E±⟩ =

⟨Ψ±|Ĥ |Ψ±⟩ ⟨Ψ±|Ψ±⟩

(5)

a

in which Ĥ is the BO Hamiltonian of eq 1. Students can readily determine that the denominator of eq 5 is 2(1 ± S2), where S is the overlap, or orthogonality integral, ⟨a(1)|b(1)⟩. We assume that instructors will have introduced spin functions and that students will recognize that the symmetric and antisymmetric space functions (the positive and negative components of eq 4) correspond to singlet and triplet electronic states, respectively. Because the Hamiltonian contains eight terms and the wave function has two terms, the numerator of eq 5 consists, after expansion has been carried out, of 32 (2 × 8 × 2) integral terms. Eight of these express the kinetic energy of electrons 1 and 2, and the other 24 describe the pairwise Coulombic interactions of the electrons and nuclei. The integrals are presented in the Supporting Information. We recommend that students carry out the expansion of eq 5 for themselves and identify the integrals that are equivalent by symmetry. Students will find that the energy of H2 at fixed R can be fully represented by eight unique integrals, I1−I8 of Table 1,9 and the constant 1/R. I1 is the overlap (S) of the basis functions on separate atoms. The eight kinetic energy terms of eq 5 are reduced to two unique integrals, I2 and I3, and all potential energy terms are represented by the five integrals I4 −I8 and 1/R. (Integrals I9 and I10 are introduced here but will not be considered until part III.) Fortunately, each of these integrals (e.g., I1 = S, I2 = α2/2, etc.) can be expressed analytically as a function of the internuclear separation, R. These expressions are not derived here because doing so is a significant mathematical challenge (requiring the use of confocal elliptical coordinates) and, in any case, would be an inordinate distraction from the objectives of the project. Interested students can be referred to several resources.12 In their 1927 paper, Heitler and London presented closed-form expressions for integrals I1−I7 and an approximate one for I8. In another 1927 paper on H2, Sugiura13 published an exact expression for I8 (K′) that must be calculated numerically (see the discussion below).14 We present in Table 2 the closed-form expressions of the seven fundamental integrals (i.e., S, S′, J, K, J′, K′, and L)9,12 of H2. The first six of these are required to calculate the energy of the HL wave function. L will be used in part III.

⎛ 1 1⎞ + e−2w⎜1 + ⎟ ⎝ w w⎠

w = αR.

These integrals are expressed as a function of the variable w = αR. The scaling of R by the screening parameter α arises from using the confocal elliptical coordinates (ra ± rb)/R. Because ra and rb are multiplied by α in the 1s basis functions of eq 2, R must also be scaled by α. For K′, the quantity γ is Euler’s constant (which to 15 significant figures is 0.577215664901532) and Ei(−x) is the integral logarithm. In principle, Ei(−x) can be obtained by numerical interpolation of values of x (i.e., x = 2w or 4w) from published tables.15 Unfortunately, considering the range of R values (e.g., 0.6 ≤ R ≤ 6.0 in atomic units) needed for the PE calculations, the corresponding values of Ei provided in ref 15 are too widely separated for large R (large w) to obtain accurate results. We obtained Ei(−x) values for 0.6 ≤ x ≤ 24.0, in increments of 0.1, using the series expansions presented by Sugiura,13 and expressed them analytically (see the Supporting Information). Thus K′ can be calculated completely analytically. Students will have to expand the numerator of eq 5 to evaluate the energy of H2. In so doing, they may have to be reminded how to carry out (symbolically) the double integrations over the coordinates of electrons 1 and 2. It is instructive for them to separate the numerator into terms associated with ∇2 (kinetic energy) and with terms arising from Coulomb interactions (potential energy). Using the values of the integrals presented in Table 1 and recognizing the equivalence of the integrals of the same type (i.e., in electron coordinates 1 and 2, or nuclear coordinates a and b), students will obtain the following expressions for the expectation values of the kinetic and potential energies: ⟨KE±⟩ = ⟨PE±⟩ =

2(I1 ± I1I3) 1±

I12

=

α2 (1 ∓ 2KS ∓ S2) 2 1±S

2I4 + 2I5 ± 4I1I6 + I7 ± I8 2

+

(6)

1 R

1 ± I1 α 1 = ( −2 + 2J ± 4KS + J ′ ± K ′) + 2 R 1±S

(7)

Before proceeding, it is instructive for students to evaluate eqs 6 and 7 for α = 1 in the limit of infinite R, that is, for two isolated 1454



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eq 4, the HL wave functions. This term of eq 4 plays an essential role here. It ensures, together with the corresponding spin functions, that the wave functions are antisymmetric with respect to electron permutation, as required by the Pauli exclusion principle. If one discards the second term of eq 4, effectively neglecting the exclusion principle, all exchange and overlap integrals of eq 8 vanish, and only the first and third terms survive. Students should plot the sum of the first (with S = 0) and third terms of eq 8 (also shown in Figure 1). They will find a slight attraction between the H atoms at an equilibrium separation of R = 1.872 a0 with an attraction energy of 0.01961 Eh. This attraction, which arises purely from the classical electrostatic interaction of two H atoms (the electron on atom a is attracted to the nucleus of atom b, and the electron on b to nucleus a), accounts for a paltry 17% of the HLS binding energy, or 11% of the experimental De. In this context, students might ponder the result that the “covalent bond” is predominantly a consequence of the “K” integrals and, hence, the quantum mechanical “exchange energy” of eq 8. However, as Gatz18 pointed out, “While it is always a pleasant surprise to encounter certainties in quantum chemistry, this one is of dubious benefit because exchange energy is difficult to interpret intuitively.”

H atoms. They should be able to demonstrate that all integrals, including the 1/R term, approach zero in this limit. They will find ⟨KE⟩ = 1 Eh and ⟨PE⟩ = −2 Eh so that the total energy of H2 at large R is −1 Eh, or two times the energy of an isolated H atom (−1/2 Eh), as expected. This is an opportunity to introduce students to an application of the virial theorem, according to which 2 ⟨KE⟩ = n ⟨PE⟩, where the potential energy (−1/r) is a homogeneous function of degree n (in this case −1).16 They will readily see that the values of ⟨KE⟩ and ⟨PE⟩ conform in this limit to the virial theorem. At this point, students are within sight of a main objective of this project because they can now express the total energy of the H2 molecule (i.e., eq 6 + eq 7) in terms of R and α. It is helpful to rearrange the sum of eqs 6 and 7 to express the energy as follows E± =

α(α − 2 + 2J + J ′) α[(4 − α)KS + K ′ − αS2] 1 + ± 2 R 1±S 1 ± S2 (8)

Students use this expression to calculate energies of the H2 molecule by preparing a spreadsheet in which they assemble numerical values of the integrals S, J, J′, K, and K′ as a function of R (with α = 1). An appropriate range is 0.6 a0 ≤ R ≤ 6.0 a0 in steps of 0.1 a0. An example of such a spreadsheet is provided in the Supporting Information. Students plot E+ and E− as a function of R and then locate the energy minimum of the E+ PE curve to determine values for De and Re. The resulting curves are shown in Figure 1. The singlet state (E+) curve reveals an

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PART II: THE HEITLER−LONDON−SIGIURA−WANG (HLSW) VARIATIONAL TREATMENT To improve the result obtained from the HLS approach, students, recognizing the crudeness of using a pure hydrogen 1s wave function on each atom (viz., eq 2, α = 1), will readily appreciate the logic of adjusting α to minimize the energy and, thereby, improve the quality of their calculated PE curves. The parameter α is sometimes referred to as a “screening constant” that takes into account the effective nuclear charge acting on one electron under the influence of the other. Students may have encountered the use of such a parameter in the variational treatment of the He atom.19 It is preferable to describe α more generally, as a damping factor for which values deviating from α = 1 reflect the distortion of the 1s orbital on one H atom caused by the approach of a second atom. In fact, one might guide students to anticipate that α >1 (steeper damping), thus providing an increase in electron density in the internuclear region (for R ≅ Re) that is consistent with chemical bonding. The task at hand now is to find the value of α that minimizes E+ in eq 8. This can be readily accomplished using the Solver add-in feature in Excel, as described in the Supporting Information. An extra complication arises, however, because the optimal α value depends on R. At infinite separation one anticipates that α = 1. As the atoms approach each other and the covalent bond forms, one expects the electron density of the unperturbed atoms to shift somewhat into the bonding region between the two nuclei. The buildup of density in this region is enhanced by increasing α with decreasing R. Therefore, students must use Solver to optimize α at each value of R for which they have calculated E+ and construct a PE plot from eq 8 that incorporates the R-dependence of α. This variational treatment of the HLS wave function was first reported by Wang11 in 1928 and extended by Rosen20 in 1931. It should be noted that the R-dependence of α (see the Supporting Information) has some interesting interpretations, but detailed analysis is beyond the scope of this project.9,21 As expected, the resulting E+ curve (the dashed curve in Figure 1) reveals a somewhat larger binding energy (De = 0.13885 Eh) and smaller equilibrium internuclear separation

Figure 1. Potential energy curves for H2, including HLS E+ (lower solid), HLS E− (upper solid), HLS E (dotted, no exhange), and HLSW E+ (dashed).

energy minimum of −1.11577 Eh at Re = 1.642 a0. Because the value of E+ is −1 Eh in the limit of large R, the Heitler− London−Sigiura (HLS) binding energy of H2 is assigned as 0.11577 Eh, which is 66% of the experimental value of De (0.17446 Eh).17 This is the result first reported by Sigiura.13 The triplet state (E−) curve does not reveal an energy minimum but instead increases monotonically with decreasing R. To conclude part I, students should be encouraged to consider the origin of the H2 covalent bond. They may recognize that the difference between the attractive nature of the E+ curve and the repulsive character of the E− curve arises essentially entirely from the second term (the ± term) of eq 8, and that this term stems directly from exchange interactions (via the K and K′ integrals) associated with the second term of 1455



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(Re = 1.414 a0) than those of the HLS treatment. The binding energy is now 80% of the experimental value. The optimized value of α is 1.166 at Re, consistent with the augmentation of electron density associated with bond formation.

functions. Thus, students can be given the energy expectation values ⟨E1,2⟩ =

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PART III: THE MOLECULAR ORBITAL−CONFIGURATION INTERACTION TREATMENT Students may wonder at this point whether quantum mechanics can improve on the HLS and HLSW treatments of H2. The former recovers a meager 66% of the experimental binding energy; the latter does better, but still only achieves 80%. Can one do better? Indeed one can! It turns out that high-level quantum mechanical calculations, using readily available quantum chemistry applications (such as Gaussian or Spartan), can potentially yield De and Re values for H2 that are nearly in exact agreement with experiment. Such calculations are generally based on MO wave functions, not on the HL wave functions students explored in parts I and II. Thus, as a culminating experience for this project, we suggest that students complete the study of H2 using simple MO and MO−CI wave functions. In doing so, they will enhance their understanding of the relationship between MOand HL-type treatments and also learn about the wave functions and methods that are employed in contemporary electronic structure calculations. About the same time that Heitler and London examined the nature of bonding in H2 using the wave functions of eq 4, Hund and Mulliken proposed an alternative approach.22 Unlike in the HL wave function, in which an electron occupies the 1s atomic orbital (AO) of either one H atom or the other, Hund and Mulliken assumed that the electron occupies a “molecular orbital”, an orbital that spans the molecule. A simple way to represent a MO is as a linear combination of the AOs for the atoms that comprise the molecule. Thus, for H2, assuming that a single 1s AO is associated with each nucleus, two MOs can be written, an in-phase combination a(1) + b(1) and an out-ofphase combination a(1) − b(1). MO-based wave functions are expressed then (in unnormalized form) as Ψ1 = [a(1) + b(1)][a(2) + b(2)]

(9)

Ψ2 = [a(1) − b(1)][a(2) − b(2)]

(10)

α2 2α (1 ∓ 2K ∓ S) + ( −1 + J ± 2K ) 1±S 1±S ⎛5 ⎞ α 1 ⎜ + + J ′ + 2K ′ ± 4L⎟ + ⎠ R 2(1 ± S)2 ⎝ 8 (11)

for the MO wave functions without expecting that they derive these expressions. The E1 expectation value (for Ψ1) in eq 11 is calculated using the upper signs, whereas E2 (for Ψ2) uses the lower signs. Students can use eq 11, together with the R-dependent integrals of Table 2 (including the new integral L, and α = 1), to calculate and plot the PE curves for the MO wave functions. These curves are shown in Figure 2 (the dotted curves). The

Figure 2. MO and MO−CI potential-energy curves for H2, including E1 (lower dotted), E2 (upper dotted), E+ (lower solid), E− (upper solid), and variationally optimized E+ (dashed).

attractive nature of the E1 curve is similar to that which one sees for the ground-state curve in Figure 1. The E2 curve is repulsive for all R values. It is appropriate, then, to associate the in-phase combination of AOs of Ψ1 with the bonding configuration and the out-phase combination of Ψ2 with the antibonding configuration. Again, students can use Solver to locate the minimum in the E1 PE curve at Re = 1.603 a0, corresponding to De = 0.09898 Eh, which is only 57% of the experimental binding energy. Students discover that the MO method does not improve the description of H2 compared to that of the HLS treatment. Moreover, the E1 curve of Figure 2 reveals an inherent failure of the MO treatment, namely, that a single-configuration wave function, like Ψ1, cannot correctly describe bond dissociation. Dissociation of H2 should yield neutral, ground-state H atoms having a total energy of −1 Eh (−1/2 Eh per H atom). But the E1 and E2 curves asymptotically approach −11/16 Eh at infinite R. Clearly, Ψ1 describes dissociation to some state other than ground-state atoms. Multiplying out, then grouping the terms of eqs 9 and 10 reveals the problem.

Equation 9 represents a pair of electrons occupying the in-phase combination of AOs, whereas eq 10 represents the pair occupying the out-of-phase combination. Ψ1 and Ψ2 are often referred to as “configurations”. Students can now develop the energy expectation values for Ψ1 and Ψ2, using eq 5 as they did in part I for the HL wave functions. Doing so here, however, is a more daunting task than before because expansions of eqs 9 and 10 yield four terms, whereas the HL wave functions have only two terms. Thus, full expansion of the energy expectation value for Ψ1 or Ψ2 gives 128 (4 × 8 × 4) integrals, compared to 32 for HL. Furthermore, two new integral types appear in the expansion, I9 and I10 of Table 1, and, in turn, L of Table 2. It is probably too much to expect that any student, other than the most ambitious, would be interested in pursuing this expansion; moreover, students are unlikely to develop much additional insight into the manipulation of the fundamental integrals than they had already gained in part I with the HL wave

Ψ1 = [a(1)b(2) + b(1)a(2)] + [a(1)a(2) + b(1)b(2)] (12)

Ψ2 = [a(1)b(2) + b(1)a(2)] − [a(1)a(2) + b(1)b(2)] (13) 1456



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Ψ1 is equivalent to the in-phase combination of the HL wave function Ψ+ (the first two terms of eq 12, cf eq 4) and an ionic wave function (the last two terms of eq 12). Ψ2 is the out-ofphase combination of the same HL and ionic wave functions. Whereas the HL terms of the MO wave functions correctly dissociate to neutral atoms, the ionic terms impart H−H+ and H+H− character on the separated atoms, thereby raising the asymptotic energy to −11/16 Eh. Ionic contributions to the wave function should vanish as the atoms separate, but the single-configuration MO wave function will not permit this. At first glance, students may feel that all is lost, and that the MO approach is not likely to yield descriptions that improve on the HLS or HLSW treatments. But this is not the case. The key is to consider a multiconfiguration wave function constructed from a linear combination of Ψ1 and Ψ2, that is,

Ψ(CI) = c1Ψ1 + c 2 Ψ2

experimental binding energy). Importantly, students will see that the ground-state energy approaches −1 Eh as R increases, confirming that the CI wave function correctly describes the dissociation of H2 to neutral atoms. Finally, students can be encouraged to variationally optimize the CI wave function by adjusting the screening parameter α to minimize E+ at each internuclear separation R, as was done in part II for the HL wave function. The resulting, fully optimized PE curve is also shown in Figure 2 (the dashed curve). Re and De are 1.4305 a0 and 0.14777 Eh, respectively. The latter represents 85% of the experimental binding energy. The variationally optimized CI wave function yields the best possible representation of the H2 molecule of any wave function constructed from only two 1s hydrogenic basis functions. Attempts to further improve the agreement between calculated and experimental quantities require more aggressive efforts using larger numbers of basis functions (see ref 23). Table 3 contains the results of parts I−III, listing the De and Re values along with experimental data.

(14)

This is the CI expansion of the wave function. The coefficients, c1, c2, will be selected to minimize the energy. Students can show that this CI wave function can be re-expressed, using eqs 12 and 13, as Ψ

(CI)

= d1Ψ

(HL)

+ d2 Ψ

(ionic)

Table 3. Dissociation Energy and Equilibrium Internuclear Separation of H2

(15)

where the expansion coefficients, d1, d2, are related to the coefficients of eq 14 [d1 = (c1 + c2)/√2; d2 = (c1 − c2)/√2], assuming all wave functions are normalized. Equation 15 shows that the CI wave function allows the contributions of the HL and ionic terms to vary independently, unlike those of the MO wave functions (eqs 12 and 13). Students will discover that the CI wave function permits the contribution of the ionic terms to diminish as H2 dissociates. Solving the CI secular equation yields two solutions, with energy eigenvalues E± =

E1 + E2 ∓

a

(16)

where E1 and E2 are the energies of eq 11, and H12 is the coupling term that describes the interaction of configurations Ψ1 and Ψ2: H12 = ⟨Ψ|1 Ĥ |Ψ2⟩ =

⎛5 ⎞ α ⎜ − J ′⎟ 2 ⎝ ⎠ 2(1 − S ) 8

(17)

Squares of the CI coefficients, which are effectively the weights of the configurations that contribute to the CI wave function, are given by c12 =

1 1 + (E+ − E1)/(E+ − E2)

(18)

c22 =

1 1 + (E+ − E2)/(E+ − E1)

(19)

α

De/Eh

Re/a0

expt 100 × Dcalc e /De

HLS HLSW MO MO−CI MO−CI Expta

1.000 1.166 1.000 1.000 1.194

0.11566 0.13885 0.09898 0.011849 0.14777 0.17446

1.6423 1.4140 1.6030 1.6680 1.4305 1.4011

66 80 57 68 85

Reference 17.

IV. SUMMARY By exploring the nature of the HL and MO treatments of the H2 molecule in this project, we believe that students gain an enhanced understanding and appreciation of electronic structure methods. Students learn that the HL and MO approaches provide reasonable semiquantitative descriptions of molecular systems near equilibrium geometries, that variational optimization of the wave function acts to improve the reliability of calculated properties (including energy and geometry), and that the MO method fails to describe dissociation processes. Configuration interaction yields an improved description of the system and is essential for describing dissociation processes. Students can be informed that modern perturbation theory and coupled-clusters methods used in quantum chemistry calculations are simply approximate methods to treat configuration interaction.

(E1 − E2)2 + 4(H12)2 2

Method

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ASSOCIATED CONTENT

S Supporting Information *

Additional details regarding the MO−CI treatment of H2 including the R-dependence of c1, c2, d1, and d2 are provided in the Supporting Information. Armed now with eqs 16 and 17, together with the energies of the MO configurations (eq 11), students have sufficient information to plot the PE curves for the ground and excited singlet states for this simple CI calculation. These PE curves are shown in Figure 2 (the solid curves) for α = 1. The minimum in the ground-state potential is at Re = 1.668 a0 and corresponds to a De value of 0.11849 Eh (or 68% of the

Student assignments, additional instructional resources, and spreadsheets. This material is available via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

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(21) A plot of α vs R in is shown in the Supporting Information. (22) Levine, I. N. Quantum Chemistry, 5th ed.; Prentice Hall: Upper Saddle River, NJ, 2000; pp 414−420. (23) Halpern, A. M.; Glendening, E. D. Exploring the Nature of the H2 Bond. 2. Using Ab Initio Molecular Orbital Calculations To Obtain the Molecular Constants. J. Chem. Educ. 2013, DOI: 10.1021/ ed400235k.

REFERENCES

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