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Article Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX

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Orbital Configuration: Terms, States, and Configuration State Functions Joseph M. Brom* Department of Chemistry, University of St. Thomas, 2115 Summit Avenue, St. Paul, Minnesota 55105, United States

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S Supporting Information *

ABSTRACT: Molecular orbital configurations give rise to electronic terms comprising corresponding electronic states. Electronic terms possess symmetry classifications, which are denoted by a term symbol, based on the symmetry properties of the electronic wave functions associated with the states. Using the methylene radical, and both planar and twisted ethylene molecules as examples, the symmetry classifications of the wave functions connected to orbital configurations are determined. The wave functions associated with the open-shell ···(2e)2 ground configuration of twisted, D2d ethylene present an informative example of how to use group theory to determine the configuration state functions associated with the six states arising from this configuration. KEYWORDS: Upper-Division Undergraduate, Graduate Education/Research, Physical Chemistry, Problem Solving/Decision Making, Group Theory/Symmetry, Quantum Chemistry

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INTRODUCTION Previous articles1−3 in this Journal have addressed the question of which molecular electronic terms arise from a given molecular orbital configuration. After a brief review of the orbital approximation in quantum chemistry, this article will address the question of which approximate electronic wave functions correspond to the molecular electronic states composing the spectroscopic terms. What is the connection between the symmetry properties of a molecular state and the wave function representing that state? The answer will involve a rudimentary application of group theory that is appropriate for an upper-division course in physical chemistry. In particular, the answer will show how to arrive at wave functions known as configuration state functions (CSFs) that are built from one or more determinants possessing proper spin and spatial symmetry properties. Using the direct product rules from group theory will be an important consideration.

challenges. The problem arises when students at introductory levels ask the following question: What is an orbital? The correct answer is that an orbital is a one-electron wave function, i.e., a mathematical construct. The only problem with this answer is that then students must ask this question: Ok, what is a wave function? Attempts to answer this difficult question at the introductory level often only serve to present students with misconceptions concerning orbitals.4 While the Schrödinger equation might be mentioned briefly in lower-division collegiate chemistry courses, methods for solving the equation almost necessarily must wait until an upper-division course in physical chemistry. Thus, while students in the lower-division organic chemistry course can and do make good use of introductory molecular orbital theory, a better understanding of an orbital really must wait until the student is ready to tackle the Schrödinger equation head on. At some point in the rigorous upper-division physical chemistry course students will learn to solve the Schrödinger equation for atomic hydrogen and, indeed, for other oneelectron systems such as the helium cation, He+. These solutions to the Schrödinger equation for one-electron species are the mathematical functions otherwise known as AOs. Soon enough, students are taught the truth that the Schrödinger equation cannot be solved exactly for many-electron systems, not even for two-electron systems such as atomic helium, He.

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ORBITAL APPROXIMATION Chemists tend to make pervasive use of the orbital approximation because it offers an explanation and understanding of phenomena occurring at the atomic/molecular level. For this reason, students enrolled in an introductory chemistry course at the secondary level likely learn about atomic orbitals (AOs), AO quantum numbers, AO shapes, AO electron configurations of the atomic elements and stable atomic ions, and AO occupation diagrams. Certainly, this is the case for students enrolled in a general chemistry course at the collegiate level. However, there are potential pitfalls to teaching orbital concepts at lower-division levels. There certainly are © XXXX American Chemical Society and Division of Chemical Education, Inc.

Received: March 13, 2018 Revised: June 14, 2018

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DOI: 10.1021/acs.jchemed.8b00187 J. Chem. Educ. XXXX, XXX, XXX−XXX

Journal of Chemical Education

Article

Thus, it is the case that one resorts to approximate solutions to the Schrödinger equation for many-electron systems, and thus it is that one is led to the very useful orbital approximation. The orbital approximation has everything to do with using one-electron wave functions to construct reasonably good approximate solutions to the Schrödinger equation for manyelectron systems.

Due to the Pauli Principle, there is only one allowable orbital occupation diagram for this configuration. In other words, were one to express the orbital occupation diagram with parallel spins, then one would deal with a Slater determinant with two identical rows, and mathematics tells us that this determinant will vanish.

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SPIN QUANTUM NUMBERS Now, provided one can reasonably neglect any spin−spin or spin−orbit interactions in one’s atomic/molecular system, the exact eigenfunctions of the spin-free Hamiltonian operator are also eigenfunctions of the square total spin, Ŝ 2, and the zcomponent total spin, Ŝ z, operators:

SLATER DETERMINANTS How does one go about this process of using the orbital approximation for many-electron systems? Part of the answer lies in what we know from quantum theory about the unknowable exact solutions to the Schrödinger equation. Quantum theory and the Pauli Principle inform us that, because electrons are Fermions, if we interchange the spacespin coordinates q of any two electrons in the true wave function ψ then the wave function must change sign, i.e., the wave function must be antisymmetric with respect to the electron permutation operator: P̂ 12ψ(q1, q2) = ψ(q2, q1) = −ψ(q1, q2).5 If we also expect our approximate wave functions to be antisymmetric with respect to the electron permutation operator, then we can build approximate wave functions using orbitals if we express the approximate wave functions as determinants, i.e., Slater determinants.6 A Slater determinant is a determinant of orbitals. Consider, for example, the ground-state wave function for atomic He. As one learns early on, the ground AO electron configuration for He is 1s2. This is really shorthand for saying that the groundstate wave function for this two-electron system is, in the orbital approximation, the following Slater determinant, or CSF: ψ (q1 , q2) ≅ ψ (q1 , q2)orbital =

2

S ̂ ψ = S(S + 1)ψ and Sẑ ψ = MSψ

where S and MS are “good” quantum numbers, and where the MS quantum numbers range in value −S, −S + 1, ···, S. It is not difficult to show8 that the 1s2 ground orbital configuration, Slater determinant, approximate wave function for He is an eigenfunction of the two spin operators with S = 0 and MS = 0. The 2S + 1 spin multiplicity is 1. The ground electronic state for He is nondegenerate, and the (Russell−Saunders) term symbol for this state is 1S0. Again, the closed-shell orbital configuration gives rise to a single 1S0 term comprising a single electronic state represented, in the orbital approximation, by the Slater determinant, or CSF, of eq 1. Similarly, one can express approximate wave functions for excited states of He using the orbital approximation for openshell configurations such as 1s12s1 or 1s12p1. From an introductory point-of-view that avoids issues of orbital angular momentum coupling, it is more straightforward here to consider the ground and excited states of molecules. The two-electron wave function for the ground electronic state of molecular hydrogen may also be expressed in the orbital approximation with configuration (1σ+g )2

1 ϕ1sα(1) ϕ1sβ(1) 2 ! ϕ1sα(2) ϕ1sβ(2) (1)

Here, ϕ1sα(1) designates a space-spin orbital for electron 1 with the space function being a 1s AO and spin function that is α (“spin-up” or ms = +1/2) spin, with similar designations for an electron possessing the β (“spin-down” or ms = −1/2) spin function. The 1/ 2! factor in eq 1 is a normalization constant for the two-electron wave function. (The normalization factor for an N-electron Slater determinant is 1/ N! .) The best 1s AO to use in this orbital approximation is determined by a Hartree−Fock atomic electronic structure calculation.7 It is easy to see that if the Slater determinant is operated upon by the P̂ 12 electron permutation operator, this will interchange the two rows of the determinant and thereby mathematically change the sign of the determinant, as required by the Pauli principle. It is customary to express the Slater determinant in a shorthand notation ψ (1, 2)orbital = |1s(1) 1s(2)|

(3)

ψ (1, 2) ≅ ψ (1, 2)orbital =

ϕ1σ +α(1) ϕ1σ +β(1) g g 1 2 ! ϕ1σ +α(2) ϕ1σ +β(2) g

g

(4)

ϕ1σg+α(1)

only now where designates a molecular spin−orbital for electron 1 with the orbital function that is a 1σ+g bonding molecular orbital9 (MO) and spin function that is α spin, with similar designations for an electron possessing the β spin function. It is well-known that MOs are usually expressed as linear combinations of atom-centered basis functions, e.g., 631G(d), and the best MOs are determined by the Hartree− Fock approach. Again, it is customary to express the Slater determinant CSF in a shorthand notation, as described above: ψ (1, 2)orbital = |1σg+(1) 1σg+(2)|

(5)

The diagrammatical approach to denote the determinant is the MO occupation diagram:

(2)

where the normalization constant is not displayed, the overstrike is used to designate β spin, and only the diagonal elements of the determinant are given. The often used diagrammatical approach to denote the 1s2 Slater determinant is the orbital occupation diagram:

(1σ+g )2

Slater

Just as for the ground configuration of He, there is only one allowable orbital occupation diagram for this configuration, and this approximate two-electron wave function for groundstate H2 is also an eigenfunction of the Ŝ 2 and Ŝ z spin operators with spin quantum numbers S = 0 and MS = 0. The spin B



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multiplicity is 1 and the (Mulliken) term symbol for the ground electronic state of H2 is 1∑+g .

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GROUP THEORY AND CONFIGURATION STATE FUNCTIONS Because the two-electron examples considered thus far are ground orbital configurations with filled-shell, doubly occupied orbitals, the configurations give rise to a single electronic term consisting of one electronic state. The next case to consider is a molecule where there exist several orbital configurations, terms, and electronic states of importance. Before moving on to specific details, consider that there is an important relationship between quantum theory and group theory. Because the Hamiltonian operator in the Schrödinger equation commutes with any symmetry operation R̂ belonging to the molecular point group, for wave functions with nondegenerate eigenvalues Ei of the Hamiltonian operator it must be that R̂ ψi = c ψi

Figure 1. Minimal basis valence MOs of methylene. The geometry is optimized at the RHF level, and the orbital contours correspond to 0.1 bohr −3/2. By convention, the molecule is placed in the yz-plane. The MO symmetry classifications and orbital energies (hartrees) are given.

(6)

and if the wave function is to be normalized then c = ±1.10 For this reason, the wave function for nondegenerate electronic eigenstates11 must form a basis for an irreducible representation of the molecular point group. Wave functions in the orbital approximation are also devised to form a basis for an irreducible representation of the molecular point group. The question becomes the following: Which wave functions form a basis for which irreducible representation? As shown below, often the requirement that an orbital wave function form a basis for an irreducible representation of the molecular point group, as well as being an eigenfunction of the spin operators, means that linear combinations of Slater determinants must be used. As mentioned in the Introduction, spatial and spin symmetry-adapted linear combinations of one or more Slater determinants are termed CSFs. Several examples below show how one determines the CSFs for electronic states, of particular spatial and spin symmetry, that arise from a given orbital configuration.

Ψ(1, 2, ···, 8)orbital = |1a1(1)1a1(2)2a1(3)2a1(4)1b2(5)1b2(6)3a1(7)3a1(8)| ≈ |··· 3a13a1|

(7)

As another example of a filled-shell, doubly occupied orbital configuration, this Slater determinant is an eigenfunction of the spin operators with S = 0 and MS = 0. This Slater determinant also forms a basis for the totally symmetric A1 irreducible representation of the C2v point group. For these reasons the term symbol for this singlet electronic state is 1A1. This orbital configuration gives rise to a single term comprising a single electronic state. However, as is known from both theory and experiment, this 1A1 term is not the ground electronic term of methylene.12,14 This singlet term is actually the lowest-energy, first excited term of methylene, and the conventional symbol for this term is ã1A1. As seen in Figure 1, the highest-occupied MO (HOMO) for the closed-shell configuration is the 3a1 MO, and the lowestunoccupied MO (LUMO) is the 1b1 MO. As shown by the Walsh orbital correlation diagram for AH2 molecules,13 the 3a1 and the 1b1 MOs approach each other in orbital energy as the AH2 bond angle opens toward 180°. In fact, for linear AH2 molecules these two MOs together form a basis for the doubly degenerate πu irreducible representation of the centrosymmetric D∞h point group. Indeed, early spectroscopic experiments on methylene were interpreted as evidence that the molecule was linear. Later, new spectroscopic and computational experiments showed the molecule to be paramagnetic and bent, with a bond angle of 134° for ground term CH2.12,14 For this reason, the ground orbital configuration of CH2 which is consistent with all experimental and computational results is not the closed-shell configuration predicted by the Walsh diagram but actually is the open-shell (1a 1 ) 2 (2a 1 ) 2 (1b2)2(3a1)1(1b1)1 configuration. Unlike the case for filledshell orbital configurations, there are several terms and states arising from this open-shell configuration.

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METHYLENE CSFS The methylene molecule played an important role in the historical development and acceptance of computational quantum chemistry.12 Methylene, CH2, possesses C2v point group symmetry, and the molecular orbitals used to make up the molecular wave function in the orbital approximation are also chosen to form a basis for irreducible representations in this point group. Consider only the six [i.e., 4(C) + 2(H′s) = 6(CH2) ] minimal basis valence set of MOs, shown in Figure 1 for the depicted closed-shell configuration of CH2. The MOs shown are the canonical restricted Hartree−Fock (RHF) MOs with C2v symmetry classifications. [See Supporting Information for computational details.] The orbital occupation diagram shown in Figure 1 corresponds to the orbital configuration (1a1)2(2a1)2(1b2)2(3a1)2. This configuration would be predicted as the ground orbital configuration for a bent CH2 molecule according to the popularized Walsh orbital correlation diagram.13 Again, this configuration is shorthand for the eight-electron Slater determinant wave function, expressed in shorthand notation as C



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For reasons explained below, we need only consider the open shells to determine the terms, states, and CSFs arising from an open-shell configuration. In this (1a1)2(2a1)2(1b2)2(3a1)1(1b1)1 example there are four possible open-shell orbital occupation diagrams, shown in Figure 2.

σv̂ (yz)|··· 3a11b1| = |σv̂ (yz)(3a1)σv̂ (yz)(1b1)| = |( +1)(3a1)( −1)(1b1)| = ( −1)|··· 3a11b1|

(10)

Checking with the character table for the C2v point group shows that this Slater determinant wave function transforms with B1 symmetry. Indeed, the direct product rule in this case gives a1 ⊗ b1 = b1. Since each of the four Slater determinants belonging to the ···(3a1)1(1b1)1 configuration possess the same spatial MOs, all four determinants belong to the B1 symmetry classification. However, not every determinant shown in Figure 2 is an eigenfunction of both spin operators. We have seen that the |···3a11b1| is an eigenfunction with S = 1 and MS = 1. The determinant |···3a11b1| (bottom right in Figure 2) similarly is an eigenfunction with S = 1 and MS = −1. Neither of the two remaining determinants in Figure 2, |···3a11b1| and |···3a11b1|, however, are eigenfunctions of the Ŝ 2 spin operator because operating on either determinant gives8 the same linear combination of both determinants: 2

2

S ̂ |··· 3a11b1| = S ̂ |··· 3a11b1| = |··· 3a11b1| + |··· 3a11b1| (11)

In other words, neither of these determinants alone can serve as CSFs for electronic states arising from the ···(3a1)1(1b1)1 orbital configuration. Now, we know there must be four electronic states and that the wave functions representing these states must be eigenfunctions of both spin operators, as well as eigenfunctions of the symmetry operations in the C2v point group. We can resolve our conundrum here when we realize that we can take normalized linear combinations of determinants in order to produce CSFs that are eigenfunctions of the spin operators; i.e., we can form CSFs using more than one determinant. Since we have two determinants that neither alone are eigenfunctions of the Ŝ 2 spin operator, eq 11 suggests we can form two orthogonal linear combinations that are eigenfunctions8 of Ŝ 2 and Ŝ z:

Figure 2. Four allowable determinants arising from the (3a1)1(1b1)1 MO configuration.

This means there are four Slater determinants to work with and there are four electronic states to consider. Consider first the |···3a11b1| determinant (bottom left in Figure 2). This determinant is an eigenfunction of the spin operators with S = 1 and MS = 1. To determine the spatial symmetry classification of this wave function, one operates on it with each of the symmetry operations belonging to C2v. To operate on the determinant is to form the direct product of operating on each MO in the N-electron determinant since R̂|ϕ1ϕ2 ··· ϕN | = |R̂ϕ1R̂ϕ2 ··· R̂ϕN |

(8)

Since the direct product15,16 of a nondegenerate canonical MO with itself belongs to the totally symmetric irreducible representation, only the singly occupied MOs in the configuration will determine the ultimate symmetry classification of the determinant. In other words, in this case we need only consider the 3a1 and 1b1 MOs: R̂|··· 3a11b1| = |··· R̂(3a1)R̂(1b1)|

ψ[S = 1, MS = 0] =

1 (|··· 3a11b1| + |··· 3a11b1|) 2

(12)

ψ[S = 0, MS = 0] =

1 (|··· 3a11b1| − |··· 3a11b1|) 2

(13)

(Note that the factor 1/√2 in these equations is a normalization constant for the orthogonal linear combinations, over and above the normalization constants assumed in each Slater determinant.) We now have four electronic states and four CSFs. Two of the CSFs are single determinant functions, and two are spin-adapted linear combinations of two determinants. Three of the states [S = 1, MS = 1, 0, −1] have the same energy and compose the 3B1 term that is a spin triplet, while the remaining state [S = 0, MS = 0] composes the 1 B1 term that is an open-shell spin singlet. It is important to note that the open-shell spin singlet CSF cannot be expressed as a single Slater determinant. All four states and CSFs arise from the same orbital conf iguration. In accord with Hund’s rule, the 3B1 triplet term is lower in energy than the 1B1 singlet term and, in fact, is the ground term of the methylene molecule.12,14 [Conventional symbols for the ground triplet term and the now second excited singlet term are X̃ 3B1 and b̃1B1.]

(9)

For the symmetry operations in C2v, other than the trivial identity operation, we use the character table to determine: Ĉ2|··· 3a11b1| = |Ĉ2(3a1)Ĉ2(1b1)| = |( +1)(3a1)( −1)(1b1)| = ( −1)|··· 3a11b1| σv̂ (xz)|··· 3a11b1| = |σv̂ (xz)(3a1)σv̂ (xz)(1b1)| = |( +1)(3a1)( +1)(1b1)| = ( +1)|··· 3a11b1| D



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with the HOMO and LUMO for methylene, there are six ways one can distribute two electrons within these two MOs. Also, as with methylene, we know there are six electronic states arising from these six determinants: one state belonging to the closed-shell (···1b23u)1A1g ground term, one state belonging to the closed-shell (···1b22g)1A1g excited term, one state belonging to the open-shell (···1b13u1b12g)1B1u excited term, and three states belonging to the open-shell (···1b13u1b12g)3B1u excited term. Spectroscopists17 have commonly labeled these valence terms as belonging to the N (normal or ground singlet), Z (zwitterionic singlet), V (valence lowest excited singlet), and T (triplet) terms, respectively. Note that a shortcut has been applied to determine the symmetry classification of the two 1 1 terms arising from the open-shell (···1b3u 1b2g ) ∼ π1π*1 configuration. One can use group theory tables of direct products18 that show that in D2h symmetry one has b3u ⊗ b2g = b1u. Note that, as with the methylene example, the CSFs for the open-shell state composing the (···1b13u1b12g)1B1u term and for the S = 1, MS = 0 state belonging to the (···1b13u1b12g)3B1u term require spin-adapted linear combinations of two Slater determinants. When one allows for configuration mixing between the two 1A1g terms and optimizes the geometry for the ground term with the TCSCF wave function, the computation gives mixing coefficients c1 = 0.9772 and c2 = 0.2122, showing that the N(1A1g) ground term has significant multiconfigurational character. The terms, states, and CSFs for ethylene become more challenging to determine when one twists the molecule so as to break the horizontal plane of symmetry. As one begins to twist the two ···CH2 planes against one another, the molecular symmetry descends to the D2 point group. When the two planes are twisted by a full 90°, the molecule possesses D2d point group symmetry. One can imagine that the full π-bond present in D2h symmetry completely breaks when the molecule twists to D2d symmetry. What about the orbital configurations as the molecule twists? An interesting thing happens because, as the molecule twists, the orbital energy of the π-MO increases and the orbital energy of the π*-MO decreases until, at the 90°-twist with D2d symmetry, these two MOs form a doubly degenerate pair. The lowest-energy orbital configuration in the twisted D 2d symmetry is (1a 1 ) 2 (1b 2 ) 2 (2a 1 ) 2 (2b2)2(3a1)2(1e)4(2e)2. The highest-occupied 2e MOs are shown in Figure 4. The degenerate MOs in the figure are displayed to show that they are nonsuperimposable mirror images of one another. Once again there are six allowable ways to distribute two electrons into the open-shell doubly degenerate 2e MOs so we know there are six electronic states that arise from this openshell ···(2e)2 MO configuration. We now determine the terms and CSFs for the states composing the terms. We can use the D2d direct product table18 to determine the terms: e ⊗ e = a1 + a2 + b1 + b2. However, because of the orbital degeneracy in this case, this approach alone will not give us the CSFs. We must return to determining the symmetry classifications of the six Slater determinants. There are eight symmetry operations composing the D2d point group, but there is a shortcut as we only need to use consistently one operation from each of the five symmetry classes within the D2d point group. Figures 5 and 6 show the results of operating on the ea and eb MOs, respectively, with the necessary symmetry operations of D2d.

If we consider the total number of allowable ways that two electrons can occupy the 3a1 HOMO and 1b1 LUMO of Figure 1, there are six Slater determinants to consider. We have discussed five of the six, which give rise to the ã1A1, b̃1B1, and X̃ 3B1 terms, but we should include the closed-shell ···1b12 configuration that gives rise to yet another 1A1 term. The states arising from the ···3a12 and ···1b12 configurations are of same 1 A1 symmetry and close enough in energy that ···1b12 and ··· 3a12 configuration mixing is important. In other words, better wave functions for both 1A1 terms would express this mixing as linear combinations of two CSFs. This proper consideration of configuration interaction gives rise to multiconfiguration, selfconsistent-field (MCSCF) wave functions for these states. The MCSCF wave functions are then linear combinations of determinant wave functions, but now the determinants arise from different configurations. The two-configuration, selfconsistent-field (TCSCF) wave functions for the two 1A1 terms of methylene, with conventional symbols ã1A1 and c̃1A1, are ψ [a1̃ A1] = c1|··· 3a13a1| − c 2|··· 1b11b1| and ψ [c1̃ A1] = c 2|··· 3a13a1| + c1|··· 1b11b1|

(14)

Optimizing the geometry of methylene for the lower-energy ã1A1 term at the TCSCF level of theory gives the coefficient values of c1 = 0.9793 and c2 = 0.2022. This relatively large value of the c2 coefficient shows that the electronic state of methylene belonging to the lowest-excited ã1A1 term has significant multiconfigurational character. In fact, qualitatively correct computational predictions of the singlet−triplet energy separation between the ã1A1 and X̃ 3B1 terms require a TCSCF treatment of the low-lying singlet term.12

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ETHYLENE CSFS The ethylene molecule, C2H4, in its ground electronic state has planar D2h symmetry. The HOMO and LUMO for the 16electron RHF ground-state wave function are shown in Figure 3. The HOMO is the bonding π-MO with b3u symmetry while the LUMO is the antibonding π*-MO with b2g symmetry. As

Figure 3. HOMO and LUMO for ground-state ethylene. The geometry is optimized at the RHF level, and the orbital contours correspond to 0.1 bohr−3/2. By convention, the molecule is placed in the yz-plane. The symmetry classifications and orbital energies (hartrees) are presented. E



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Figure 4. Shown are the computed (top) highest occupied doubly degenerate MOs for D2d ethylene. The orbital contours correspond to 0.1 bohr−3/2. Also shown are the corresponding schematic MOs (bottom) along with the axis system and a projection diagram showing several D2d symmetry operations (principal axis z is perpendicular to plane).

Figure 6. Shown are the results of performing symmetry operations on the schematic eb MO of ethylene in the D2d point group. Proper and improper rotations are taken to be clockwise.

Ĉ2′|··· eaeb| = |···(− ea)(eb)| = − 1|··· eaeb| σ̂d|··· eaeb| = |···(eb)(ea)| = − 1|··· eaeb|

This determinant is an eigenfunction of the symmetry operations, and a check of the eigenvalues with the character table for D2d, given in Table 1, shows that this determinant belongs to the A2 symmetry classification in D2d. Table 1. Character Table for D2d Point Group D2d

Ê

2Ŝ 4

Ĉ 2

2Ĉ ′2

2σ̂ d

A1 A2 B1 B2 E

1 1 1 1 2

1 1 −1 −1 0

1 1 1 1 −2

1 −1 1 −1 0

1 −1 −1 1 0

It is straightforward to see that the |···ea̅ eb̅ | determinant will also have A2 symmetry only with S = 1 and MS = −1. Next, consider the |···eaea̅ | determinant. This determinant is an eigenfunction of the two spin operators with S = 0 and MS = 0, but what about the symmetry operations? We apply eq 8 using the results shown in Figures 5 and 6:

Figure 5. Shown are the results of performing symmetry operations on the schematic ea MO of ethylene in the D2d point group. Proper and improper rotations are taken to be clockwise.

S4̂ |··· ea ea̅ | = |···( −eb)( − eb̅ )| = +1|··· eb eb̅ |

These results allow for the determination of the symmetry classifications of the six Slater determinants arising from the ··· (2e)2 MO configuration. Consider first the |···eaeb| determinant. This determinant is an eigenfunction of the two spin operators with S = 1 and MS = 1. To determine if it forms a basis for an irreducible representation in the D2d point group we apply eq 8 using the results shown in Figures 5 and 6:

Ĉ2|··· ea ea̅ | = |···( −ea)( − ea̅ )| = +1|··· ea ea̅ | Ĉ2′|··· ea ea̅ | = |···(− ea)(− ea̅ )| = +1|··· ea ea̅ |

σ̂d|··· ea ea̅ | = |···(eb)( eb̅ )| = +1|··· eb eb̅ |

This determinant is not an eigenfunction of the symmetry operations. The results suggest, however, that orthogonal linear combinations of the two determinants |···eaea̅ | and |···ebeb̅ | would be eigenfunctions of the symmetry operations. That is indeed the case.

S4̂ |··· eaeb| = |···(− eb)(ea)| = −|···(eb)(ea)| = + 1|··· eaeb| Ĉ2|··· eaeb| = |···(− ea)(− eb)| = + 1|··· eaeb| F



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1 (σd̂ |··· ea ea̅ | − σd̂ |··· eb eb̅ |) 2 1 = (|··· eb eb̅ | − |··· ea ea̅ |) 2

First consider the linear combination, which remains a spinsinglet CSF: ψ[S = 0, MS = 0] =

σd̂ ψ =

1 (|··· ea ea̅ | + |··· eb eb̅ |) 2

= −1ψ

1 (S4̂ |··· ea ea̅ | + S4̂ |··· eb eb̅ |) S4̂ ψ = 2 1 (|··· eb eb̅ | + |··· ea ea̅ |) = 2

(16)

The D2d character table shows this wave function has B1 symmetry. The term symbol for this spin-singlet state is 1B1. Having considered four of the six Slater determinants arising from the (···2e2) MO configuration, attention turns to the remaining two: |···eaeb̅ | and |···ea̅ eb|. Application of eq 8 shows that neither of these determinants are eigenfunctions of either the spin operators or the symmetry operations. We again must use orthogonal linear combinations of the two. One determines the spin quantum numbers as before,8 but the spatial symmetry properties also need to be addressed. First consider the spin-triplet CSF:

= +1ψ 1 (Ĉ2|··· ea ea̅ | + Ĉ2|··· eb eb̅ |) 2 1 = (|··· ea ea̅ | + |··· eb eb̅ |) 2

Ĉ2ψ =

= +1ψ ψ[S = 1, MS = 0] = 1 (Ĉ2′|··· ea ea̅ | + Ĉ2′|··· eb eb̅ |) 2 1 = (|··· ea ea̅ | + |··· eb eb̅ |) 2

Ĉ2′ψ =

1 (|··· ea eb̅ | + |··· ea̅ eb|) 2

1 (S4̂ |··· ea eb̅ | + S4̂ |··· ea̅ eb|) 2 1 = (|···( −eb)( ea̅ )| + |···(− eb̅ )(ea)|) 2

S4̂ ψ =

= +1ψ

= +1ψ

1 (σd̂ |··· ea ea̅ | + σd̂ |··· eb eb̅ |) 2 1 = (|··· eb eb̅ | + |··· ea ea̅ |) 2

σd̂ ψ =

= +1ψ

1 (Ĉ2|··· ea eb̅ | + Ĉ2|··· ea̅ eb|) 2 1 = (|···( −ea)( − eb̅ )| + |···(− ea̅ )( −eb)|) 2

Ĉ2ψ = (15)

= +1ψ

This CSF is a basis for the A1 irreducible representation in D2d. Since the wave function is a spin-singlet the term symbol for this state is 1A1. Next consider the orthogonal linear combination, which also remains a spin-singlet CSF: ψ[S = 0, MS = 0] =

1 (Ĉ2′|··· ea eb̅ | + Ĉ2′|··· ea̅ eb|) 2 1 = (|···( − ea)( eb̅ )| + |···(− ea̅ )(eb)|) 2

Ĉ2′ψ =

1 (|··· ea ea̅ | − |··· eb eb̅ |) 2

= −1ψ

1 (S4̂ |··· ea ea̅ | − S4̂ |··· eb eb̅ |) 2 1 = (|··· eb eb̅ | − |··· ea ea̅ |) 2

S4̂ ψ =

1 (σd̂ |··· ea eb̅ | + σd̂ |··· ea̅ eb|) 2 1 = (|···(eb)( ea̅ )| + |···( eb̅ )(ea)|) 2

σd̂ ψ =

= −1ψ

= −1ψ

1 (Ĉ2|··· ea ea̅ | − Ĉ2|··· eb eb̅ |) 2 1 = (|··· ea ea̅ | − |··· eb eb̅ |) 2

(17)

This CSF thus has A2 symmetry, and it clearly is the MS = 0 component of the spin triplet, 3A2 term. Finally, consider the orthogonal spin-singlet CSF:

Ĉ2ψ =

ψ[S = 0, MS = 0] =

= +1ψ 1 (Ĉ2′|··· ea ea̅ | − Ĉ2′|··· eb eb̅ |) 2 1 = (|··· ea ea̅ | − |··· eb eb̅ |) 2

1 (|··· ea eb̅ | − |··· ea̅ eb|) 2

1 (S4̂ |··· ea eb̅ | − S4̂ |··· ea̅ eb|) 2 1 = (|···( −eb)( ea̅ )| − |···(− eb̅ )(ea)|) 2

Ĉ2′ψ =

S4̂ ψ =

= +1ψ

= −1ψ G



■

1 (Ĉ2|··· ea eb̅ | − Ĉ2|··· ea̅ eb|) 2 1 = (|···( −ea)( − eb̅ )| − |···( − ea̅ )( −eb)|) 2

Article

CONCLUSION Computations of the electronic states of molecules using MO theory require that one knows how to build wave functions in the orbital approximation. This article has given examples of how to build CSFs that correctly describe the spatial and spin symmetry properties of any electronic state of interest.

Ĉ2ψ =

= +1ψ

■

1 (Ĉ2′|··· ea eb̅ | − Ĉ2′|··· ea̅ eb|) 2 1 = (|···( − ea)( eb̅ )| − |···( − ea̅ )(eb)|) 2

Ĉ2′ψ =

ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available on the ACS Publications website at DOI: 10.1021/acs.jchemed.8b00187.

= −1ψ 1 (σd̂ |··· ea eb̅ | − σd̂ |··· ea̅ eb|) 2 1 = (|···(eb)( ea̅ )| − |···( eb̅ )(ea)|) 2

σd̂ ψ =

= +1ψ

■

Details of the RHF and MCSCF computations done here to provide the orbitals displayed in the figures of this article (PDF, DOCX)

AUTHOR INFORMATION

Corresponding Author

(18)

*E-mail: [email protected].

This CSF thus has B2 symmetry, and the term symbol for this spin singlet state is 1B2. Summarizing, the ···(2e)2 MO configuration of twisted C2H4 with D2d symmetry gives rise to four terms comprising six states. Three of the states belong to the triplet term 3A2, and the remaining states are nondegenerate belonging to singlet terms 1A1, 1B1, and 1B2. The orbital CSFs for the six states have been determined. Having determined the CSFs for the states of both the planar D2h and twisted D2d conformations of ethylene, one could ask if there is a connection, or correlation, between them. This is an important question if one is interested in the potential energy surfaces (PESs) of ethylene. For example, the lowest-energy, ground electronic state of ethylene belongs to the planar N(1A1g) ground term. This is a stationary state of minimum energy on the ground term PES. As the molecule twists and begins to break the π-bond, the molecule must climb the PES until a maximum energy state on the surface is reached at the 90° twist angle. At this point the molecule has reached a transition state (TS) on the PES, and this stationary state must belong to the 1B1 term of D2d: N(1A1g) → N(1B1). This is known because, along the surface, the planar π(b3u), π*(b2g) MOs evolve into the twisted, degenerate ea, eb pair of MOs, and one can compare the CSFs for the two stationary states. The TCSCF wave function for the ground-state 1A1g (ψ[N1A1g] = 0.9772|···1b3u1b3u | − 0.2122|···1b2g1b2g |) and the CSF for TS 1B1 (ψ[N1B1] = 0.7071|···ea ea | − 0.7071|···eb eb|) are both linear combinations of two determinants comprising correlating MOs. Only the mixing coefficients and MO symmetry labels change with rotation. The ground term TCSCF wave function has some significant configuration mixing with mixing coefficients, noted above, of c1 = 0.9772 and c2 = 0.2122, but the 1B1 TS shows “complete mixing” with 1 coefficient values of c1 = c 2 = 2 = 0.7071. While a mixing of two MO configurations is required for a better description of the ground-state minimum, only the single MO configuration is used to describe the TS maximum. MSCSF computations19,20 confirm the TS-character of the 1B1 stationary state. They show the TS maximum is 65 kcal/mol higher in energy than the ground-state minimum, and this is thus a measure of the π-bond strength in ethylene.

ORCID

Joseph M. Brom: 0000-0003-3818-9040 Notes

The author declares no competing financial interest.

■

ACKNOWLEDGMENTS The author thanks Mike Schmidt, Frank Rioux, and Josh Layfield for feedback and critical comments on the manuscript.

■

REFERENCES

(1) Ford, D. I. Molecular Term Symbols by Group Theory. J. Chem. Educ. 1972, 49, 336−340. (2) Mazo, R. M. Molecular Electronic Terms and Molecular Orbital Configurations. J. Chem. Educ. 1990, 67, 135−138. (3) Novak, I. Electronic States and Configurations: Visualizing the Difference. J. Chem. Educ. 1999, 76, 135−137. (4) Autschbach, J. Orbitals: Some Fiction and Some Facts. J. Chem. Educ. 2012, 89, 1032−1040. (5) Levine, I. N. Quantum Chemistry, 4th ed.; Prentice-Hall: Englewood Cliffs, NJ, 1991; pp 262−264. (6) Levine, I. N. Quantum Chemistry, 4th ed.; Prentice-Hall: Englewood Cliffs, NJ, 1991; pp 271−272. (7) Pilar, F. L. Elementary Quantum Chemistry; Dover: Mineola, NY, 2001; pp 269−277. (8) Pilar, F. L. Elementary Quantum Chemistry; Dover: Mineola, NY, 2001; pp 203−209. (9) The customary notation for this bonding MO is abbreviated 1σg. The complete 1σ+g symbol is used here to emphasize the MO forms a basis for the totally symmetric irreducible representation in the linear, centrosymmetric D∞h point group. (10) Eyring, H.; Walter, J.; Kimball, G. E. Quantum Chemistry; John Wiley & Sons: New York, 1944; pp 184−187. (11) Similar treatment for spatially degenerate electronic states is beyond the scope of this article. (12) Schaefer, H. F., III Methylene: A Paradigm for Computational Quantum Chemistry. Science (Washington, DC, U. S.) 1986, 231, 1100−1107. (13) Levine, I. N. Quantum Chemistry, 4th ed.; Prentice-Hall: Englewood Cliffs, NJ, 1991; pp 503−506. (14) Kalemos, A.; Dunning, T. H., Jr.; Mavridis, A.; Harrison, J. F. CH2 Revisited. Can. J. Chem. 2004, 82, 684−693. (15) Strommen, D. P. A Pedagogical Approach to the Direct Product. J. Chem. Educ. 1979, 56, 248. H



Article

(16) Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy. An Introduction to Vibrational and Electronic Spectroscopy; Dover: New York, 1989; p 57. (17) Merer, A. J.; Mulliken, R. S. Ultraviolet Spectra and Excited States of Ethylene and Its Alkyl Derivatives. Chem. Rev. 1969, 69, 639−656. (18) Harris, D. C.; Bertolucci, M. D. Symmetry and Spectroscopy. An Introduction to Vibrational and Electronic Spectroscopy; Dover: New York, 1989; pp 479−491. (19) Schmidt, M. W.; Truong, P. N.; Gordon, M. S. π−Bond Strengths in the Second and Third Periods. J. Am. Chem. Soc. 1987, 109, 5217−5227. (20) Yamaguchi, Y.; Osamura, Y.; Schaefer, H. F., III Analytic Energy Second Derivatives for Two-configuration Self-consistent-field Wave Functions. Application to Twisted Ethylene and to the Trimethylene Diradical. J. Am. Chem. Soc. 1983, 105, 7506−7511.

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