Cramer C.J. Essentials of Computational Chemistry Theories and Models

4.3 CONSTRUCTION OF TRIAL WAVE FUNCTIONS

111

In general, the Born–Oppenheimer assumption is an extremely mild one, and it is entirely justified in most cases. It is worth emphasizing that this approximation has very profound consequences from a conceptual standpoint – so profound that they are rarely thought about but simply accepted as dogma. Without the Born–Oppenheimer approximation we would lack the concept of a potential energy surface: The PES is the surface defined by Eel over all possible nuclear coordinates. We would further lack the concepts of equilibrium and transition state geometries, since these are defined as critical points on the PES; instead we would be reduced to discussing high-probability regions of the nuclear wave functions. Of course, for some problems in chemistry, we do need to consider the quantum mechanical character of the nuclei, but the advantages afforded by the Born–Oppenheimer approximation should be manifest.

4.3 Construction of Trial Wave Functions

Equation (4.16) is simpler than Eq. (4.2) because electron–nuclear correlation has been removed. The remaining correlation, that between the individual electrons, is considerably more troubling. For the moment we will take the simplest possible approach and ignore it; we do this by considering systems with only a single electron. The electronic wave function has thus been reduced to depending only on the fixed nuclear coordinates and the three Cartesian coordinates of the single electron. The eigenfunctions of Eq. (4.16) for a molecular system may now be properly called molecular orbitals (MOs; rather unusual ones in general, since they are for a molecule having only one electron, but MOs nonetheless). To distinguish a one-electron wave function from a many-electron wave function, we will designate the former as ψel and the latter as el. We will hereafter drop the subscript ‘el’ where not required for clarity; unless otherwise specified, all wave functions are electronic wave functions.

The pure electronic energy eigenvalue associated with each molecular orbital is the energy of the electron in that orbital. Experimentally, one might determine this energy by measuring the ionization potential of the electron when it occupies the orbital (fairly easy for the hydrogen atom, considerably more difficult for polynuclear molecules). To measure Eel, which includes the nuclear repulsion energy, one would need to determine the ‘atomization’ energy, that is, the energy required to ionize the electron and to remove all of the nuclei to infinite separation. In practice, atomization energies are not measured, but instead we have compilations of such thermodynamic variables as heats of formation. The relationship between these computed and thermodynamic quantities is discussed in Chapter 10.

4.3.1The LCAO Basis Set Approach

As noted earlier, we may imagine constructing wave functions in any fashion we deem reasonable, and we may judge the quality of our wave functions (in comparison to one another) by evaluation of the energy eigenvalues associated with each. The one with the lowest energy will be the most accurate and presumably the best one to use for computing other properties by the application of other operators. So, how might one go about choosing

112	4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY

mathematical functions with which to construct a trial wave function? This is a typical question in mathematics – how can an arbitrary function be represented by a combination of more convenient functions? The convenient functions are called a ‘basis set’. Indeed, we have already encountered this formalism – Eq. (2.10) of Chapter 2 illustrates the use of a basis set of cosine functions to approximate torsional energy functions.

In our QM systems, we have temporarily restricted ourselves to systems of one electron. If, in addition, our system were to have only one nucleus, then we would not need to guess wave functions, but instead we could solve Eq. (4.16) exactly. The eigenfunctions that are determined in that instance are the familiar hydrogenic atomic orbitals, 1s, 2s, 2p, 3s, 3p, 3d, etc., whose properties and derivation are discussed in detail in standard texts on quantum mechanics. For the moment, we will not investigate the mathematical representation of these hydrogenic atomic orbitals in any detail, but we will simply posit that, as functions, they may be useful in the construction of more complicated molecular orbitals. In particular, just as in Eq. (4.10) we constructed a guess wave function as a linear combination of exact wave functions, so here we will construct a guess wave function φ as a linear combination of atomic wave functions ϕ, i.e.,

N
φ = ai ϕi	(4.17)

i=1

where the set of N functions ϕi is called the ‘basis set’ and each has associated with it some coefficient ai . This construction is known as the linear combination of atomic orbitals (LCAO) approach.

Note that Eq. (4.17) does not specify the locations of the basis functions. Our intuition suggests that they should be centered on the atoms of the molecule, but this is certainly not a requirement. If this comment seems odd, it is worth emphasizing at this point that we should not let our chemical intuition limit our mathematical flexibility. As chemists, we choose to use atomic orbitals (AOs) because we anticipate that they will be efficient functions for the representation of MOs. However, as mathematicians, we should immediately stop thinking about our choices as orbitals, and instead consider them only to be functions, so that we avoid being conceptually influenced about how and where to use them.

Recall that the wave function squared has units of probability density. In essence, the electronic wave function is a road map of where the electrons are more or less likely to be found. Thus, we want our basis functions to provide us with the flexibility to allow electrons to ‘go’ where their presence at higher density lowers the energy. For instance, to describe the bonding of a hydrogen atom to a carbon, it is clearly desirable to use a p function on hydrogen, oriented along the axis of the bond, to permit electron density to be localized in the bonding region more efficiently than is possible with only a spherically symmetric s function. Does this imply that the hydrogen atom is somehow sp-hybridized? Not necessarily – the p function is simply serving the purpose of increasing the flexibility with which the molecular orbital may be described. If we took away the hydrogen p function and instead placed an s function in between the C and H atoms, we could also build up electron density in the bonding region (see Figure 4.1). Thus, the chemical interpretation of the coefficients in Eq. (4.17) should only be undertaken with caution, as further described in Chapter 9.

4.3 CONSTRUCTION OF TRIAL WAVE FUNCTIONS

113

C

H

Figure 4.1 Two different basis sets for representing a C– H σ bonding orbital with the size of the basis functions roughly illustrating their weight in the hybrid MO. The set on the right is the more chemically intuitive since all basis functions are centered on the atoms. Note, however, that the use of a p function to polarize the hydrogen density goes beyond a purely minimalist approach. The set on the left is composed entirely of s functions distributed along the bond. Such a basis set may seem odd in concept, but is quite capable of accurately representing the electron density in space. Indeed, the basis set on the left would have certain computational advantages, chief among them the greater simplicity of working with s functions than with p functions

One should also note that the summation in Eq. (4.17) has an upper limit N ; we cannot work with an infinite basis in any convenient way (at least not when the basis is AOs). However, the more atomic orbitals we allow into our basis, the closer our basis will come to ‘spanning’ the true molecular orbital space. Thus, the chemical idea that we would limit ourselves to, say, at most one 1s function on each hydrogen atom is needlessly confining from a mathematical standpoint. Indeed, there may be very many ‘true’ one-electron MOs that are very high in energy. Accurately describing these MOs may require some unusual basis functions, e.g., very diffuse functions to describe weakly bound electrons, like those found in Rydberg states. We will discuss these issues in much more detail in Section 6.2, but it is worth emphasizing here, at the beginning, that the distinction between orbitals and functions is a critical one in computational molecular orbital theory.

4.3.2The Secular Equation

All that being said, let us now turn to evaluating the energy of our guess wave function. From Eqs. (4.15) and (4.17) we have

E	=		i	ai ϕi H j	aj ϕj dr

ai ϕi

aj ϕj dr

ij

114	4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY
		ij	ai aj			ϕi H ϕj dr
	=
		ij	ai aj				ϕi ϕj dr
			ai aj Hij
	=	ij					(4.18)

ai aj Sij

ij

where we have introduced the shorthand notation Hij and Sij for the integrals in the numerator and denominator, respectively. These so-called ‘matrix elements’ are no longer as simple as they were in prior discussion, since the atomic orbital basis set, while likely to be efficient, is no longer orthonormal. These matrix elements have more common names, Hij being called a ‘resonance integral’, and Sij being called an ‘overlap integral’. The latter has a very clear physical meaning, namely the extent to which any two basis functions overlap in a phasematched fashion in space. The former integral is not so easily made intuitive, but it is worth pointing out that orbitals which give rise to large overlap integrals will similarly give rise to large resonance integrals. One resonance integral which is intuitive is Hii , which corresponds to the energy of a single electron occupying basis function i, i.e., it is essentially equivalent to the ionization potential of the AO in the environment of the surrounding molecule.

Now, it is useful to keep in mind our objective. The variational principle instructs us that as we get closer and closer to the ‘true’ one-electron ground-state wave function, we will obtain lower and lower energies from our guess. Thus, once we have selected a basis set, we would like to choose the coefficients ai so as to minimize the energy for all possible linear combinations of our basis functions. From calculus, we know that a necessary condition for a function (i.e., the energy) to be at its minimum is that its derivatives with respect to all of its free variables (i.e., the coefficients ai ) are zero. Notationally, that is

∂E	= 0	k	(4.19)
∂ak

(where we make use of the mathematical abbreviation meaning ‘for all’). Performing this fairly tedious partial differentiation on Eq. (4.18) for each of the N variables ak gives rise to N equations which must be satisfied in order for Eq. (4.19) to hold true, namely

N

ai (Hki − ESki ) = 0

k

(4.20)

i=1

This set of N equations (running over k) involves N unknowns (the individual ai ). From linear algebra, we know that a set of N equations in N unknowns has a non-trivial solution if and only if the determinant formed from the coefficients of the unknowns (in this case the ‘coefficients’ are the various quantities Hki − ESki ) is equal to zero. Notationally again,

¨	115
4.4 HUCKEL THEORY

that is

H11	− ES11	H12	− ES12	· · ·		H1N − ES1N
H21	− ES21	H22	− ES22	· · ·		H2N − ES2N	=	0	(4.21)
	.		.	. .		.
	.		.		.	.
	.		.			.
HN 1 − ESN 1		HN 2 − ESN 2		· · · HN N − ESN N

Equation (4.21) is called a secular equation. In general, there will be N roots E which permit the secular equation to be true. That is, there will be N energies Ej (some of which may be equal to one another, in which case we say the roots are ‘degenerate’) where each value of Ej will give rise to a different set of coefficients, aij , which can be found by solving the set of linear Eqs. (4.20) using Ej , and these coefficients will define an optimal wave function φj within the given basis set, i.e.,

N
φj = aij ϕi	(4.22)
i=1

In a one-electron system, the lowest energy molecular orbital would thus define the ‘ground state’ of the system, and the higher energy orbitals would be ‘excited states’. Obviously, as these are different MOs, they have different basis function coefficients. Although we have not formally proven it, it is worth noting that the variational principle holds for the excited states as well: the calculated energy of a guess wave function for an excited state will be bounded from below by the true excited state energy (MacDonald 1933).

So, in a nutshell, to find the optimal one-electron wave functions for a molecular system, we:

1.Select a set of N basis functions.

2.For that set of basis functions, determine all N 2 values of both Hij and Sij .

3.Form the secular determinant, and determine the N roots Ej of the secular equation.

4.For each of the N values of Ej , solve the set of linear Eqs. (4.20) in order to determine the basis set coefficients aij for that MO.

All of the MOs determined by this process are mutually orthogonal. For degenerate MOs, some minor complications arise, but those are not discussed here.

4.4 Huckel¨ Theory

4.4.1Fundamental Principles

To further illuminate the LCAO variational process, we will carry out the steps outlined above for a specific example. To keep things simple (and conceptual), we consider a flavor of molecular orbital theory developed in the 1930s by Erich Huckel¨ to explain some of the unique properties of unsaturated and aromatic hydrocarbons (Huckel¨ 1931; for historical

116	4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY

insights, see also, Berson 1996; Frenking 2000). In order to accomplish steps 1–4 of the last section, Huckel¨ theory adopts the following conventions:

(a)The basis set is formed entirely from parallel carbon 2p orbitals, one per atom. [Huckel¨ theory was originally designed to treat only planar hydrocarbon π systems, and thus the 2p orbitals used are those that are associated with the π system.]

(b)The overlap matrix is defined by

Sij = δij

(4.23)

Thus, the overlap of any carbon 2p orbital with itself is unity (i.e., the p functions are normalized), and that between any two p orbitals is zero.

(c)Matrix elements Hii are set equal to the negative of the ionization potential of the methyl radical, i.e., the orbital energy of the singly occupied 2p orbital in the prototypical system defining sp2 carbon hybridization. This choice is consistent with our earlier discussion of the relationship between this matrix element and an ionization potential. This energy value, which is defined so as to be negative, is rarely actually written as a numerical value, but is instead represented by the symbol α.

(d)Matrix elements Hij between neighbors are also derived from experimental information. A 90◦ rotation about the π bond in ethylene removes all of the bonding interaction between the two carbon 2p orbitals. That is, the (positive) cost of the following process,

E = Ep

E = 2Ep

is E = 2Ep − Eπ . The (negative) stabilization energy for the pi bond is distributed equally to the two p orbitals involved (i.e., divided in half) and this quantity, termed β, is used for Hij between neighbors. (Note, based on our definitions so far, then, that Ep = α and Eπ = 2α + 2β.)

(e)Matrix elements Hij between carbon 2p orbitals more distant than nearest neighbors are set equal to zero.

4.4.2 Application to the Allyl System

Let us now apply Huckel¨ MO theory to the particular case of the allyl system, C3H3, as illustrated in Figure 4.2. Because we have three carbon atoms, our basis set is determined from convention (a) and will consist of 3 carbon 2p orbitals, one centered on each atom. We will arbitrarily number them 1, 2, 3, from left to right for bookkeeping purposes.

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			4.4 HUCKEL THEORY
	H
H 1 C 3 H f3 =					a − √2b
C	2		C

HH

f2 =

a

E

pC

f1 =

a + √2b

Figure 4.2 Huckel¨ MOs for the allyl system. One pC orbital per atom defines the basis set. Combinations of these 3 AOs create the 3 MOs shown. The electron occupation illustrated corresponds to the allyl cation. One additional electron in φ2 would correspond to the allyl radical, and a second (spin-paired) electron in φ2 would correspond to the allyl anion

The basis set size of three implies that we will need to solve a 3 × 3 secular equation. Huckel¨ conventions (b)–(e) tell us the value of each element in the secular equation, so that Eq. (4.21) is rendered as

α − E	β	0	= 0	(4.24)
β	α − E	β
0	β	α − E

The use of the Kronecker¨ delta to define the overlap matrix ensures that E appears only in the diagonal elements of the determinant. Since this is a 3 × 3 determinant, it may be expanded using Cramer’s rule as

(α − E)3 + (β2 · 0) + (0 · β2) − [0 · (α − E) · 0] − β2(α − E) − (α − E)β2 = 0					(4.25)
which is a fairly simple cubic equation in E that has three solutions, namely
E = α + √		β, α, α − √		β	(4.26)
	2		2

Since α and β are negative by definition, the lowest energy solution is α + √2β. To find the MO associated with this energy, we employ it in the set of linear Eqs. (4.20), together

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4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY

with the various necessary H and S values to give

a1[α − (α + √

β) · 1] + a2[β − (α + √

β) · 0] + a3[0 − (α + √

β) · 0] = 0

2

2

2

a1[β − (α + √

β) · 0] + a2[α − (α + √

β) · 1] + a3[β − (α + √

β) · 0] = 0

2

2

2

a1[0 − (α + √

β) · 0] + a2[β − (α + √

β) · 0] + a3[α − (α + √

β) · 1] = 0 (4.27)

2

2

2

Some fairly trivial algebra reduces these equations to

= √

a1

a2

2

a3

= a1

(4.28)

While there are infinitely many values of a1, a2, and a3 which satisfy Eq. (4.28), the requirement that the wave function be normalized provides a final constraint in the form of Eq. (4.11). The unique values satisfying both equations are

	1			√				1
a11 =		, a21	=		2	, a31	=		(4.29)
	2			2				2

where we have now emphasized that these coefficients are specific to the lowest energy molecular orbital by adding the second subscript ‘1’. Since we now know both the coefficients and the basis functions, we may construct the lowest energy molecular orbital, i.e.,

	1			√				1
ϕ1 =		p1	+		2	p2	+		p3	(4.30)
	2			2				2

which is illustrated in Figure 4.2.

By choosing the higher energy roots of Eq. (4.24), we may solve the sets of linear equations analogous to Eq. (4.27) in order to arrive at the coefficients required to construct φ2 (from E = α) and φ3 (from E = α − √2β). Although the algebra is left for the reader,

the results are
	√											√
a12 =	2			, a22		=	0, a32 = −					2
	2											2
	1						√					1
a13 =		, a23			= −		2			, a33	=					(4.31)
	2						2					2

and these orbitals are also illustrated in Figure 4.2. The three orbitals we have derived are the bonding, non-bonding, and antibonding allyl molecular orbitals with which all organic chemists are familiar.

Importantly, Huckel¨ theory affords us certain insights into the allyl system, one in particular being an analysis of the so-called ‘resonance’ energy arising from electronic delocalization in the π system. By delocalization we refer to the participation of more than two atoms in a

4.5 MANY-ELECTRON WAVE FUNCTIONS

119

given MO. Consider for example the allyl cation, which has a total of two electrons in the π system. If we adopt a molecular aufbau principle of filling lowest energy MOs first and further make the assumption that each electron has the energy of the one-electron MO that it occupies (φ1 in this case) then the total energy of the allyl cation π system is 2(α + √2β). Consider the alternative ‘fully localized’ structure for the allyl system, in which there is a full (doubly-occupied) π bond between two of the carbons, and an empty, non-interacting p orbital on the remaining carbon atom. (This could be achieved by rotating the cationic methylene group 90◦ so that the p orbital becomes orthogonal to the remaining π bond, but that could no longer be described by simple Huckel¨ theory since the system would be non-planar – the non-interaction we are considering here is purely a thought-experiment). The π energy of such a system would simply be that of a double bond, which by our definition of terms above is 2(α + β). Thus, the Huckel¨ resonance energy, which is equal to Hπ − Hlocalized , is 0.83β (remember β is negative by definition, so resonance is a favorable phenomenon). Recalling the definition of β, the resonance energy in the allyl cation is predicted to be about 40% of the rotation barrier in ethylene.

We may perform the same analysis for the allyl radical and the allyl anion, respectively,

by adding the energy of φ to the cation with each successive addition of an electron,

2 √ √

i.e., Hπ (allyl radical) = 2(α + 2β) + α and Hπ (allyl anion) = 2(α + 2β) + 2α. In the hypothetical fully π -localized non-interacting system, each new electron would go into the non-interacting p orbital, also contributing each time a factor of α to the energy (by definition of α). Thus, the Huckel¨ resonance energies of the allyl radical and the allyl anion are the same as for the allyl cation, namely, 0.83β.

Unfortunately, while it is clear that the allyl cation, radical, and anion all enjoy some degree of resonance stabilization, neither experiment, in the form of measured rotational barriers, nor higher levels of theory support the notion that in all three cases the magnitude is the same (see, for instance, Gobbi and Frenking 1994; Mo et al. 1996). So, what aspects of Huckel¨ theory render it incapable of accurately distinguishing between these three allyl systems?

4.5 Many-electron Wave Functions

In our Huckel¨ theory example, we derived molecular orbitals and molecular orbital energies using a one-electron formalism, and we then assumed that the energy of a many-electron system could be determined simply as the sum of the energies of the occupied one-electron orbitals (we used our chemical intuition to limit ourselves to two electrons per orbital). We further assumed that the orbitals themselves are invariant to the number of electrons in the π system. One might be tempted to say that Huckel¨ theory thus ignores electron–electron repulsion. This is a bit unfair, however. By deriving our Hamiltonian matrix elements from experimental quantities (ionization potentials and rotational barriers) we have implicitly accounted for electron–electron repulsion in some sort of average way, but such an approach, known as an ‘effective Hamiltonian’ method, is necessarily rather crude. Thus, while Huckel¨ theory continues to find use even today in qualitative studies of conjugated systems, it is

120	4 FOUNDATIONS OF MOLECULAR ORBITAL THEORY

rarely sufficiently accurate for quantitative assessments. To improve our models, we need to take a more sophisticated accounting of many-electron effects.

4.5.1 Hartree-product Wave Functions

Let us examine the Schrodinger¨ equation in the context of a one-electron Hamiltonian a little more carefully. When the only terms in the Hamiltonian are the one-electron kinetic energy and nuclear attraction terms, the operator is ‘separable’ and may be expressed as

N
H = hi	(4.32)

i=1

where N is the total number of electrons and hi is the one-electron Hamiltonian defined by

1		i2 −	M	Zk
hi = −					(4.33)
	2		k=1	rik

where M is the total number of nuclei (note that Eq. (4.33) is written in atomic units). Eigenfunctions of the one-electron Hamiltonian defined by Eq. (4.33) must satisfy the

corresponding one-electron Schrodinger¨ equation

hi ψi = εi ψi

(4.34)

Because the Hamiltonian operator defined by Eq. (4.32) is separable, its many-electron eigenfunctions can be constructed as products of one-electron eigenfunctions. That is

HP = ψ1ψ2 · · · ψN

(4.35)

A wave function of the form of Eq. (4.35) is called a ‘Hartree-product’ wave function. The eigenvalue of is readily found from proving the validity of Eq. (4.35), viz.,

H HP = H ψ1ψ2 · · · ψN

N

=hi ψ1ψ2 · · · ψN

i=1

=(h1ψ1)ψ2 · · · ψN + ψ1(h2ψ2) · · · ψN + . . . + ψ1ψ2 · · · (hN ψN )

=(ε1ψ1)ψ2 · · · ψN + ψ1(ε2ψ2) · · · ψN + . . . + ψ1ψ2 · · · (εN ψN )

N

=εi ψ1ψ2 · · · ψN

i=1

=	N	εi HP	(4.36)

i=1