Young D.C. - Computational chemistry (2001)(en)

222 25 ELECTRONIC EXCITED STATES

R.Moccia, P. Spizzo, Computational Chemistry; Structure, Interactions and Reactivity Part B S. Fraga, Ed., 76, Elsevier, Amsterdam (1992).

Adv. Chem. Phys. J. W. McGowan, Ed. vol. 45 (1981).

S. D. Peyerimho¨, R. J. Buenker, Adv. Quantum Chem. 9, 69 (1975).

Excited states of solids and liquids are discussed in

W. L. Greer, S. A. Rice, Adv. Chem. Phys. 17, 229 (1970).

Excited states of polymers are discussed in

S.Basu, Adv. Quantum Chem. 11, 33 (1978).

For quantum Monte Carlo methods see

B.L. Hammond, W. A. Lester, Jr., P. J. Reynolds, Monte Carlo Methods in Ab Initio Quantum Chemistry World Scienti®c, Singapore (1994).

For density functional theory see

R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules Oxford, Oxford (1989).

For an introduction to excited states via path integral methods see

T. E. Sorensen, W. B. England, Mol. Phys. 89, 1577 (1996).

224 26 SIZE-CONSISTENCY

TABLE 26.1 Size Properties of Methods

Y ˆ yes. N ˆ no.

D ˆ true only if the reference space dissociates to the correct state for that system. A ˆ approximately.

Size-extensivity is of importance when one wishes to compare several similar systems with di¨erent numbers of atoms (i.e., methanol, ethanol, etc.). In all cases, the amount of correlation energy will increase as the number of atoms increases. However, methods that are not size-extensive will give less correlation energy for the larger system when considered in proportion to the number of electrons. Size-extensive methods should be used in order to compare the results of calculations on di¨erent-size systems. Methods can be approximately size±extensive. The size-extensivity and size-consistency of various methods are summarized in Table 26.1.

26.1CORRECTION METHODS

It is possible to make a method approximately size±extensive by adding a correction to the ®nal energy. This has been most widely used for correcting CISD energies. This is a valuable technique because a simple energy correction formula is easier to work with than full CI calculations, which require an immense amount of computational resources. The most widely used correction is the Davidson correction:

where DEDC is the energy lowering, ESD the CISD energy, and C0 the weight of the HF reference determinant in the CI expansion. This was designed to give the additional energy lowering from a CISD energy to a CISDTQ energy. This results in both a more accurate energy and in making the energy approximately

26.2 RECOMMENDATIONS 225

size-extensive. The method is based on perturbation theory and thus may perform poorly if the HF wave function has a low weight in the CI expansion.

A slightly improved form of this equation is the renormalized Davidson correction, which is also called the Brueckner correction:

This generally gives a slight improvement over the Davidson correction, although it does not reach the full CI limit.

A more detailed perturbation theory analysis leads to an improved correction formula. This method, known as the Davidson and Silver or Siegbahn correction, is

This correction does approximate the full CI energy, although it may overcorrect the energy.

There are many more error correction methods, which are reviewed in detail by Duch and Diercksen. They also discuss the correction of other wave functions, such as multireference methods. In their tests with various numbers of Be atoms, the correction most closely reproducing the full CI energy is

where N is the number of electrons in the system.

Another method for making a method size-extensive is called the selfconsistent dressing of the determinant energies. This is a technique for modifying the CI superdeterminant in order to make a size-extensive limited CI. The accuracy of this technique is generally comparable to the Davidson correction. It performs better than the Davidson correction for calculations in which the HF wave function has a low weight in the CI expansion.

26.2RECOMMENDATIONS

Size-consistency and size-extensivity are issues that should be considered at the outset of any study involving multiple molecules or dissociated fragments. As always, the choice of a computational method is dependent on the accuracy desired and computational resource requirements. Correction formulas are so simple to use that several of them can readily be tried to see which does best for

226 26 SIZE-CONSISTENCY

the system of interest. The Davidson, Brueckner, and Siegbahn corrections are commonly compared. There is not a large enough collection of results to make any general comments on the merits of the other correction methods.

BIBLIOGRAPHY

Introductory discussions can be found in

P.W. Atkins, R. S. Friedman, Molecular Quantum Mechanics Third Edition Oxford, Oxford (1997).

I. N. Levine, Quantum Chemistry Fourth Edition Prentice Hall, Englewood Cli¨s (1991). W. J. Hehre, L. Radom, P. v. R. Schleyer, J. A. Pople, Ab Initio Molecular Orbital

Theory John Wiley & Sons, New York (1986).

More detailed reviews are

W. Duch, G. H. F. Diecksen, J. Chem. Phys. 101, 3018 (1994).

A. Povill, R. Caballol, J. Rubio, J. P. Malrieu, Chem. Phys. Lett. 209, 126 (1993).

I.Shavitt, Advanced Theories and Computational Approaches to the Electronic Structure of Molecules C. E. Dykstra, Ed., 185, Reidel, Dordrecht (1984).

organic molecule calculations, is that the spin contamination is negligible if the value of hS2i di¨ers from s…s ‡ 1† by less than 10%. Although this provides a quick test, it is always advisable to doublecheck the results against experimental evidence or more rigorous calculations. Expected values of hS2i are listed in Table 27.1.

Spin contamination is often seen in unrestricted Hartree±Fock (UHF) calculations and unrestricted Mùller±Plesset (UMP2, UMP3, UMP4) calculations. UMP2 is often the most sensitive to spin contamination, followed by UHF, UMP3, and UMP4. Local MP2 (LMP2) usually has less spin contamination than MP2. It is less common to ®nd any signi®cant spin contamination in DFT calculations, even when unrestricted Kohn±Sham orbitals are used. Spin contamination has little e¨ect on CC and CI calculations, in which the variational principle will result in correcting for spin contamination in the reference wave function.

Unrestricted calculations often incorporate a spin annihilation step, which removes a large percentage of the spin contamination from the wave function. This helps minimize spin contamination but does not completely prevent it. The ®nal value of hS2i is always the best check on the amount of spin contamination present. In the Gaussian program, the option ``iop(5/14ˆ2)'' tells the program to use the annihilated wave function to produce the population analysis.

27.2RESTRICTED OPEN-SHELL CALCULATIONS

It is possible to run spin-restricted open-shell calculations (ROHF). The advantage of this is that there is no spin contamination. The disadvantage is that there is an additional cost in the form of CPU time required in order to correctly handle both singly occupied and doubly occupied orbitals and the interaction between them. As a result of the mathematical method used, ROHF calculations give good total energies and wave functions but the singly occupied orbital energies do not rigorously obey Koopman's theorem.

ROHF does not include spin polarization. Thus, it is not useful for some purposes, such as predicting EPR spectra. Also because of this, it cannot reliably predict spin densities.

ROHF calculations can also exhibit symmetry breaking. Symmetry breaking is due to the calculation converging to one of the resonance structures instead of the correct ground state, which is a superposition of possible resonance structures. This causes the molecule to distort to a lower-symmetry geometry. How much symmetry breaking a¨ects the geometry of a molecule is di½cult to determine because there are many examples in which a low symmetry is the correct shape of a molecule. In cases where the symmetry breaking is a nonphysical artifact of the calculation, the wave function often exhibits an abnormally pronounced localization of the spin density to one atom or bond.

Within some programs, the ROMPn methods do not support analytic gradients. Thus, the fastest way to run the calculation is as a single point energy calculation with a geometry from another method. If a geometry optimization must be done at this level of theory, a non-gradient-based method such as the Fletcher±Powell optimization should be used.

When it has been shown that the errors introduced by spin contamination are unacceptable, restricted open-shell calculations are often the best way to obtain a reliable wave function.

27.3SPIN PROJECTION METHODS

Another approach is to run an unrestricted calculation and then project out the spin contamination after the wave function has been obtained (PUHF, PMP2). This gives a correction to the energy but does not a¨ect the wave function. Spin projection nearly always improves ab initio results, but may seriously harm the accuracy of DFT results.

A spin projected result does not give the energy obtained by using a restricted open-shell calculation. This is because the unrestricted orbitals were optimized to describe the contaminated state, rather than the spin-projected state. In cases of very-high-spin contamination, the spin projection may fail, resulting in an increase in spin contamination.

A similar e¨ect is obtained by using the spin-constrained UHF method (SUHF). In this method, the spin contamination error in a UHF wave function is constrained by the use of a Lagrangian multiplier. This removes the spin contamination completely as the multiplier goes to in®nity. In practice, small positive values remove most of the spin contamination.

27.4HALF-ELECTRON APPROXIMATION

Semiempirical programs often use the half-electron approximation for radical calculations. The half-electron method is a mathematical technique for treating a singly occupied orbital in an RHF calculation. This results in consistent total energy at the expense of having an approximate wave function and orbital energies. Since a single-determinant calculation is used, there is no spin contamination.

230 27 SPIN CONTAMINATION

The consistent total energy makes it possible to compute singlet-triplet gaps using RHF for the singlet and the half-electron calculation for the triplet. Koopman's theorem is not followed for half-electron calculations. Also, no spin densities can be obtained. The Mulliken population analysis is usually fairly reasonable.

27.5RECOMMENDATIONS

If spin contamination is small, continue to use unrestricted methods, preferably with spin-annihilated wave functions and spin projected energies. Do not use spin projection with DFT methods. When the amount of spin contamination is more signi®cant, use restricted open-shell methods. If all else fails, use highly correlated methods.

BIBLIOGRAPHY

Some discussion and results are in

F.Jensen, Introduction to Computational Chemistry John Wiley & Sons, New York (1999).

W. J. Hehre, L. Radom, P. v.R. Schleyer, J. A. Pople, Ab Initio Molecular Orbital Theory John Wiley & Sons, New York (1986).

T. Clark, A Handbook of Computational Chemistry Wiley, New York (1985).

Review articles are

T. Bally, W. T. Borden, Rev. Comput. Chem. 13, 1 (1999).

H.B. Schlegel, Encycl. Comput. Chem. 4, 2665 (1998).

A more mathematical treatment can be found in the paper

J.S. Andrews, D. Jayatilake, R. G. A. Bone, N. C. Handy, R. D. Amos, Chem. Phys. Lett. 183, 423 (1991).

Articles that compare unrestricted, restricted and projected results are

M. W. Wong, L. Radom, J. Phys. Chem. 99, 8582 (1995).

J.Baker, A. Scheiner, J. Andzelm, Chem. Phys. Lett. 216, 380 (1993).

SUHF results are examined in

P.K. Nandi, T. Kar, A. B. Sannigrahi, Journal of Molecular Structure (Theochem) 362, 69 (1996).

Spin projection is addressed in

J. M. Wittbrodt, H. B. Schlegel, J. Chem. Phys. 105, 6574 (1996).

I. Mayer, Adv. Quantum Chem. 12, 189 (1990).