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

on the r.h.s. of the last equality in Eq. (C.14) differs only by assignment of the electron labels 1 and 2, it also must have a value of 1. By the same symmetry argument, the second and third integrals must be equal to one another. Evaluating the second using Eq. (C.16) gives

α(1)β(2)|S2 |α(2)β(1) = α(1)β(2) 12 ( 12 + 1)α(2)β(1)dω(1)dω(2)

+α(1)β(2) 12 ( 12 + 1)α(2)β(1)dω(1)dω(2)

+α(1)β(2) 12 β(2)α(1)dω(1)dω(2)

+α(1)β(2) 12 β(2)α(1)dω(1)dω(2)

−α(1)β(2) 12 α(2)β(1)dω(1)dω(2)

=0 + 0 + 12 + 12 − 0

= 1 (C.18)

Thus, the expectation value of S2 from Eq. (C.14) for the closed-shell state is simply 12 (1 − 1 − 1 + 1) = 0.

Another wave function of interest is the one formed from two α-spin electrons in two

In this case, integration over the normalized non-spin-dependent spatial portion of the wave function leaves only a fairly simple integral to evaluate for the expectation value of S2, namely

α(1)α(2)|S2 |α(1)α(2) = α(1)α(2) 12 ( 12 + 1)α(1)α(2)dω(1)dω(2)

+α(1)α(2) 12 ( 12 + 1)α(1)α(2)dω(1)dω(2)

+α(1)α(2) 12 β(1)β(2)dω(1)dω(2)

−α(1)α(2) 12 β(1)β(2)dω(1)dω(2)

+α(1)α(2) 12 α(1)α(2)dω(1)dω(2)

=12 ( 12 + 1) + 12 ( 12 + 1) + 0 − 0 + 12

In the case of the Sz = −1 state (i.e., two β electrons instead of α), it is straightforward to show that the expectation value may be evaluated by the analog of Eq. (C.20) with all spin functions permuted α to β and vice versa. The resulting expectation value is still 2.

Thus far, we have described wave functions for which

of the integrals in the expectation values of Eqs. (C.14) and (C.20) makes this a simple exercise.) This situation defines what is meant by a singlet (s = 0) or triplet (s = 1) wave function.

Let us now consider the wave function with an α electron in spatial orbital a and a β

which has been superscripted 50:50 and subscripted ± for reasons that will be apparent later. Evaluation of the expectation value for S2 involves

50:50 |S2|50:50 = 12 a(1)α(1)b(2)β(2)|S2 |a(1)α(1)b(2)β(2)

−a(1)α(1)b(2)β(2)|S2 |a(2)α(2)b(1)β(1)

−a(2)α(2)b(1)β(1)|S2 |a(1)α(1)b(2)β(2)

+ a(2)α(2)b(1)β(1)|S2 |a(2)α(2)b(1)β(1) (C.23)

Note that the second and third integrals on the r.h.s. are zero because of the orthonormality of the spatial orbitals a and b, whose products appear over the same electronic coordinate in those integrals. The spatial functions integrate to one in the first and fourth integrals, and the remaining spin expectation values are just those of Eq. (C.17). Thus, the expectation value of Eq. (C.23) is 12 (1 − 0 − 0+1) = 1. With additional work, it can be shown that 50:50 is not an eigenfunction of S2.

570 APPENDIX C

By permutational symmetry, it is easy to show that the expectation value of S2 for the

is also 1, and it too fails to be an eigenfunction of S2. In order to construct proper eigenfunctions, we must take linear combinations of the two 50:50 functions, thereby creating two-determinantal wave functions. In particular, we can construct

where the spin function is identical to that appearing in the closed-shell singlet wave function of Eq. (C.13). Since S2 operates only on the spin part of the wave function, OSS must be an eigenfunction of S2 with eigenvalue s = 0 just as is true for CSS . Thus, OSS is a singlet, and in particular it is an open-shell singlet (hence the ‘OSS’ superscript), i.e., at least two electrons are in singly occupied orbitals.

The other linear combination of the 50:50 wave functions is

After integration over the spatial coordinates, we may evaluate the expectation value of S2 as

where the integrals on the r.h.s. were simplified using Eqs. (C.17) and (C.18). Thus, the wave function of Eq. (C.26) is a triplet wave function, and it is the so-called Sz = 0 triplet.

Equation (C.25) and (C.26) make it apparent that we may also write

if a and b are spatially identical in the singlet and triplet wave functions. Equation (C.28) is the foundation of the sum method, described in Section 14.4.

572 APPENDIX C

by removing the undesirable higher spin states through a process known as projection or annihilation. Consider the case where the UHF wave function for the desired state s is contaminated by the next higher possible spin state (s+1) , i.e.,

where each pure spin wave function is normalized and the sum of the squares of the coefficients c is 1 for normalization of UHF . When the annihilation operator of Eq. (14.19) is applied to the spin-contaminated wave function we have

Thus, the annihilation operator completely removes the next higher spin state and delivers a wave function that is a pure s spin state. Note, however, that it is not a normalized wave function, since cs < 1 (otherwise the original wave function would not have been spin contaminated). Normalization is simple in this case, since we have

UHF |As+1|UHF = cs s + c(s+1)(s+1) |cs s

=cs s |cs s + c(s+1)(s+1) |cs s

=cs2 s |s + cs c(s+1) (s+1) |s

With the annihilated wave function in hand, any property may be computed in the usual fashion as an expectation value of the appropriate operator. The Hamiltonian operator is a particularly simple operator to work with because we can make good use of the original UHF wave function in evaluating the expectation value. Thus

UHF |H As+1|UHF

UHF |As+1|UHF

cs s + c(s+1)(s+1) |H |cs sUHF |As+1|UHF

where the final line is the desired result. To solve for the projected UHF (PUHF) energy, one employs the so-called ‘resolution of the identity’ technique in the first line of Eq. (C.33), giving

which is the result expressed in Eq. (14.18). Note that the first term on the r.h.s. of the last equality of Eq. (C.34) is simply the energy of the spin-contaminated wave function, so the second term may be considered the ‘correction’ associated with spin annihilation. Note also that the only Hamiltonian matrix elements that will be non-zero will be for excited states differing from the ground state by double excitations (the singles are zero by Brillouin’s theorem, and triples and higher are zero because the Hamiltonian is a two-electron operator). The Hamiltonian matrix elements required to compute the correction term are exactly those needed for an MP2 calculation (cf. Eq. (7.47)), so the computational effort required for a PUHF calculation is essentially that for an MP2 calculation.

While the PUHF energy is better than the more highly contaminated UHF energy, there are many potential problems associated with it. To begin, the molecular orbitals were optimized for the contaminated wave function, not the annihilated wave function, and as such they may be considerably less than ideal for the single Slater determinant describing the pure spin state. In addition, the geometry of the contaminated state may not be a good representation of the geometry of the pure state. As analytic derivatives are not available for the PUHF energy, reoptimization is tedious.

Another potential problem is that the wave function may be contaminated not only by state s + 1, but also states s + 2, s + 3, etc. The s + 1 annihilation operator will reduce the weights of these states in the annihilated wave function, but it will not eliminate them. Inspection of Eq. (C.33) should make clear that the higher states will contribute to the PUHF energy if they appear on both the left and right sides of the Hamiltonian expectation values with non-zero coefficients. When such contamination is important, recourse to a more complete projection operator, that annihilates an arbitrary number of spin states is available, but the computational cost increases to essentially that of an MP4 calculation. Note that the problems of the orbitals being non-ideal for the pure lowest spin state persist in this instance.

In order to account for electron correlation, projection operator methods within the MPn perturbation theory formalism have also been described (Schlegel 1988). Such PMPn methods can be valuable in assessing convergence of projected pure-spin-state energies, but it should be recalled that perturbation theory is most successful when the true wave function differs from the HF determinant by only a small amount. Since the HF determinant starts with contamination, it is a given that it is potentially too far removed from the true wave function to be of much use in perturbation theory, and some caution should be exercised. Krylov (2000) has demonstrated that coupled-cluster theory can be more robust than perturbation theory in relying on spin-contaminated UHF wave functions so long as the reference orbitals are chosen properly, but this method too ultimately breaks down as the spin contamination becomes especially severe. As a final resort, MCSCF methods can always be used to construct multideterminantal, pure spin states, although this may not be the most convenient choice for a particular problem.

References

Krylov, A. I. 2000. J. Chem. Phys., 113, 6052.

Schlegel, H. B. 1988. J. Phys. Chem., 92, 3075.

Appendix D

Orbital Localization

D.1 Orbitals as Empirical Constructs

It is both flattering and vexing to quantum chemists how ubiquitous it has become to rationalize chemical behavior using molecular-orbital-based arguments. It is flattering to the extent that it indicates how large an impact quantum mechanics has had on modern chemistry, but it is vexing because orbitals themselves, unlike the wave function, are not really a rigorous part of quantum mechanics.

Indeed, there are other formulations of quantum mechanics, all of which have been shown to be entirely equivalent in a formal sense to the matrix-algebraic-molecular-orbital version, that do not in any way require an invocation of orbitals. However, the matrix-algebraic method lends itself most readily to implementation on the architecture of a digital computer, and thus it has come to overwhelmingly dominate modern computational chemistry. As a result, the orbitals that are part of the computational machinery for approximately solving the matrix algebraic equations have taken on the character of unassailable parts of the quantum mechanical formalism, but that status is undeserved.

Consider, for example, two orbitals which might be obtained from a HF calculation on ethylene, namely the orthonormal σ and π bonding orbitals, both of which are doubly occupied (Figure D.1). If we restrict our consideration to only these two orbitals, and moreover we use restricted HF theory so that we can ignore the details of spin orbitals, we can write the properly antisymmetric HF wave function for this system of two orbitals as

Now consider a different wave function, formed from different orbitals. In particular, we will take the positive and negative linear combinations of the σ and π orbitals shown in

Essentials of Computational Chemistry, 2nd Edition Christopher J. Cramer

2004 John Wiley & Sons, Ltd ISBNs: 0-470-09181-9 (cased); 0-470-09182-7 (pbk)

Figure D.1 The usual σ and π orbitals of a doubly bonded system (left) and the banana bonds formed by their linear combination (right)

Figure D.1 to create so-called ‘banana-bond’ orbitals. We define these as

It is a simple exercise to show that if σ and π are orthonormal then a and b are too. Let us now consider the antisymmetric wave function

which may be rewritten using Eqs. (D.3) and (D.4) as

|H | = 18 [σ (1) + π(1)][σ (2) − π(2)]|H |[σ (1) + π(1)][σ (2) − π(2)]

−18 [σ (1) + π(1)][σ (2) − π(2)]|H |[σ (2) + π(2)][σ (1) − π(1)]

−18 [σ (2) + π(2)][σ (1) − π(1)]|H |[σ (1) + π(1)][σ (2) − π(2)]

+ 18 [σ (2) + π(2)][σ (1) − π(1)]|H |[σ (2) + π(2)][σ (1) − π(1)] (D.7)

A somewhat tedious separation and collection of the 64 individual integrals in Eq. (D.7) (an exercise left for the industrious reader) leads to

which has the same r.h.s. as Eq. (D.2). That is, the expectation value of the energy for the wave function is the same as that for the wave function . Since the HF process was variational, this implies that is an ‘equally optimal’ wave function as , and there is no obvious reason to prefer one over the other.

This particular example illustrates what can be shown more formally to be true in general: the energy of the wave function is invariant to expressing the wave function using any normalized linear combination of the occupied HF orbitals, as are the expectation values of all other quantum mechanical operators. Since all such choices of linear combinations of orbitals satisfy the variational criterion, one may legitimately ask why the HF orbitals should be assigned any privileged status of their own as chemical entities.

The answer to that question is that, empirically, the HF orbitals have proven to be useful models for rationalizing certain chemical phenomenon. To that extent, like many other chemical models that do not have rigorous quantum mechanical foundations, they are useful tools that now find widespread use in the chemical community. It is worthwhile to consider briefly the advantages and disadvantages of various schemes for formulating MOs that have appeared in the chemical literature.

The persistence of the HF orbitals as objects for chemical discussion stems from three features. First, they are the fundamental products of the HF SCF process, and thus immediately at hand following an HF calculation. Secondly, and most importantly, they have associated with them specific orbital energies. The HF MOs are precisely those MOs that diagonalize the Fock operator, and so they have associated energy eigenvalues. It is known from photoelectron spectroscopy that electrons do reside at distinct energy levels, and from an experimental standpoint, a molecular orbital is ‘defined’ by the energy of an electron and the instantaneous difference in electron density between the preand post-ionized state (of course, measuring the density difference on a timescale faster than electronic relaxation is impossible, but the relaxed difference may in some cases be reasonably close to the pre-relaxed difference). Thus, the HF orbitals are natural objects for the discussion of spectroscopic quantities. Finally, the HF orbitals can be assigned to individual irreps of the molecular point group; this will be true for other orbitals only if linear combinations of the HF orbitals are restricted to be exclusively within their individual irreps.

Other orbital localization schemes create MOs that do not diagonalize the Fock operator, and thus it is more difficult to assign orbital energies to them. However, the canonical HF orbitals have certain features that are occasionally regarded as undesirable, and this has motivated the development of alternative localization methods. Thus, for example, many of the HF orbitals in large systems tend to be highly delocalized, but most chemical reactivity concepts are local in nature. For the discussion of such concepts, it is desirable to have highly localized MOs. In general, localization schemes impose some sort of critical constraint on the final orbitals. For instance, one constraint might be that the repulsion between electrons in the same orbitals be maximized – this results in very compact orbitals. An alternative is that interactions between electrons in different orbitals might be minimized – this results in