Young - Computational chemistry

90 10 USING EXISTING BASIS SETS

year is printed in the Journal of Molecular Structure and indexed by molecular formula. Using this bibliography, it will be fairly easy to ®nd all previous computational work on a given compound.

BIBLIOGRAPHY

References to the individual basis sets have not been included here. Most can be readily found in the review articles.

Some books discussing basis sets

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

P. W. Atkins, R. S. Friedman, Molecular Quantum Mechanics Oxford, Oxford (1997). J. Simons, J. Nichols, Quantum Mechanics in Chemistry Oxford, Oxford (1997).

A. R. Leach, Molecular Modelling Principles and Applications Longman, Essex (1996). I. N. Levine, Quantum Chemistry Prentice Hall, Englewood Cli¨s (1991).

L.Szasz, Pseudopotential Theory of Atoms and Molecules John Wiley & Sons, New ork (1985).

R.Poirer, R. Kari, I. G. Csizmadia, Handbook of Gaussian Basis Sets: A Compendium for Ab Initio Molecular Orbital Calculations Elsevier Science Publishing, New York (1985).

Gaussian basis sets for molecular calculations S. Huzinaga, Ed., Elsevier, Amsterdam (1984).

Review articles covering basis set construction and performance are

T. H. Dunning, Jr., K. A. Peterson, D. E. Woon, Encycl. Comput. Chem. 1, 88 (1998). S. Wilson D. Moncrie¨, Adv. Quantum Chem. 28, 47 (1997).

C. W. Bauschlicker, Jr., Theor. Chim. Acta 92, 183 (1995).

J.R. Thomas, B.-J. DeLeeuw, G. Vacek, T. D. Crawford, Y. Yamaguchi, H. F. Schaefer III, J. Chem. Phys. 91, 403 (1993).

T.L. Sordo, C. Barrientos, J. A. Sordo, Computational Chemistry: Structure, Interactions and Reactivity S. Fraga, Ed., 23, Elsevier, Amsterdam (1992).

J. AlmloÈf, P. R. Taylor, Adv. Quantum Chem. 22, 301 (1991).

D.Feller, E. R. Davidson, Rev. Comput. Chem. 1, 1 (1990).

E.R. Davidson, D. Feller, Chem. Rev. 86, 681 (1986).

S. Huzinaga, Comp. Phys. Reports 2, 279 (1985).

S.Wilson, Methods in Computational Molecular Physics 71 G. H. Diercksen, S. Wilson, Eds., Dordrecht, Reidel (1983).

T. H. Dunning, P. J. Hay, Modern Theoretical Chemistry vol. 4 H. F. Schaefer III Ed., 1, Plenum, New York (1977).

K. A. R. Mitchell, Chem. Rev. 69, 157 (1969).

J. D. Weeks, A. Hazi, S. A. Rice, Adv. Chem. Phys. 16, 283 (1969).

BIBLIOGRAPHY 91

L. A. Leland, A. M. Caro, Rev. Mod. Phys. 32, 275 (1960).

J. K. Labanowski, http://ccl.osc.edu/pub/documents/basis-sets/

Reviews of ECP basis sets are

M. Klobukowski, S. Huzinaga, Y. Sakai, Computational Chemistry Reviews of Current Trends Volume 3 49, J. Leszczynski, Ed., World Scienti®c, Singapore (1999).

M. Dolg, H. Stoll, Handbook on the Physics and Chemistry of Rare Earths K. A. Gschneidner, Jr., L. Eyring, Eds., 22, 607, Elsevier, Amsterdam (1994).

W. C. Ermler, R. B. Ross, P. A. Christiansen, Adv. Quantum Chem. 19, 139 (1988). K. Balasubramanian, K. S. Pitzer, Adv. Chem. Phys. 67, 287 (1987).

P. A. Christiansen, W. C. Ermler, K. S. Pitzer, Annu. Rev. Phys. Chem. 36, 407 (1985). M. Krauss, W. J. Stevens, Ann. Rev. Phys. Chem. 35, 357 (1984).

Numerical basis sets are discussed in

E.A. McCullough, Jr., Encycl. Comput. Chem. 3, 1941 (1998).

Comparisons of results are in

W.J. Hehre, Practical Strategies for Electronic Structure Calculations Wavefunction, Irvine (1995).

J. W. Ochterski, G. A. Petersson, K. B. Wiberg, J. Am. Chem. Soc. 117, 11299 (1995). W. J. Hehre, L. Radom, P. v.R. Schleyer, J. A. Pople, Ab Initio Molecular Orbital

Theory Wiley-Interscience, New York (1986).

Web based forms for obtaining basis sets

http://www.emsl.pnl.gov:2080/forms/basisform.html

http://www.theochem.uni-stuttgart.de/pseudopotentiale/

Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems. David C. Young Copyright ( 2001 John Wiley & Sons, Inc.

ISBNs: 0-471-33368-9 (Hardback); 0-471-22065-5 (Electronic)

11 Molecular Vibrations

The vibrational states of a molecule are observed experimentally via infrared and Raman spectroscopy. These techniques can help to determine molecular structure and environment. In order to gain such useful information, it is necessary to determine what vibrational motion corresponds to each peak in the spectrum. This assignment can be quite di½cult due to the large number of closely spaced peaks possible even in fairly simple molecules. In order to aid in this assignment, many workers use computer simulations to calculate the vibrational frequencies of molecules. This chapter presents a brief description of the various computational techniques available.

Di¨erent motions of a molecule will have di¨erent frequencies. As a general rule of thumb, bond stretches are the highest energy vibrations. Bond bends are somewhat lower energy vibrations and torsional motions are even lower. The lowest frequencies are usually torsions between substantial pieces of large molecules and breathing modes in very large molecules.

11.1HARMONIC OSCILLATOR APPROXIMATION

The simplest description of a vibration is a harmonic oscillator, which describes springs exactly and pendulums with small amplitudes fairly well. A harmonic oscillator is de®ned by the potential energy proportional to the square of the distance displaced from an equilibrium position. In a classical treatment of a vibrating object, the motion is fastest at the equilibrium position and comes to a complete stop for an instant at the turning points, where all the energy is potential energy. The probability of ®nding the object is highest at the turning point and lowest at the equilibrium point.

A quantum mechanical description of a harmonic oscillator uses the same potential energy function, but gives radically di¨erent results. In a quantum description, there are no turning points. There is some probability of ®nding the object at any displacement, but it becomes very small (decreasing exponentially) at large distances. The energy is quantized, with a quantum number describing each possible energy state and only certain possible energies. Very small objects, such as atoms, behave according to the quantum description with low quantum numbers.

The vibration of molecules is best described using a quantum mechanical approach. A harmonic oscillator does not exactly describe molecular vibra-

92

11.1	HARMONIC OSCILLATOR APPROXIMATION 93
TABLE 11.1 Vibrational Frequency Correction Factors
Correction	Method
0.86±0.9	HF
0.9085	HF/3ÿ12G
0.8953	HF/6ÿ31G*
0.893	ROHF/6ÿ31G*
0.8970	HF/6ÿ31‡G*
0.8992	HF/6ÿ31G**
0.9051	HF/6ÿ311G**
0.9	HF/augÿccÿpVTZ
0.9	MNDO
0.95	AM1
0.993	SAM1
0.92	MP2
0.9434, 0.95	MP2/6ÿ31G*
0.9427, 0.9370	MP2/6ÿ31G**
0.9496	MP2/6ÿ311G**
0.96	DFT
0.9945	BLYP/6ÿ31G*
0.9914	BP86/6ÿ31G*
0.9558	B3P86/6ÿ31G*
0.9573	B3PW91/6ÿ31G*
0.9614, 1.0	B3LYP/6ÿ31G*
0.976	B3LYP/ccÿpVDZ
None	CC with large basis sets

tions. Bond stretching is better described by a Morse potential and conformational changes have sine-wave-type behavior. However, the harmonic oscillator description is very useful as an approximate treatment for low vibrational quantum numbers. A harmonic oscillator approximation is most widely used for computing molecular vibrational frequencies because more accurate methods require very large amounts of CPU time.

Frequencies computed with the Hartree±Fock approximation and a quantum harmonic oscillator approximation tend to be 10% too high due to the harmonic oscillator approximation and lack of electron correlation. The exception is the very low frequencies, below about 200 cmÿ1, which are often quite far from the experimental values. Many studies are done using ab initio methods and multiplying the resulting frequencies by about 0.9 to obtain a good estimate of the experimental results. A list of correction factors is given in Table 11.1. Some researchers take this idea one step further by using di¨erent correction factors for stretching modes, angle bends, and so on.

Vibrational frequencies from semiempirical calculations tend to be qualitative in that they reproduce the general trend mentioned in the introduction here. However, the actual values are erratic. Some values will be close, whereas others are often too high. SAM1 is generally the most accurate semiempirical

94 11 MOLECULAR VIBRATIONS

method for predicting frequencies. PM3 is generally more accurate than AM1 with the exception of SaH and PaH bonds, for which AM1 is superior.

Some density functional theory methods occasionally yield frequencies with a bit of erratic behavior, but with a smaller deviation from the experimental results than semiempirical methods give. Overall systematic error with the better DFT functionals is less than with HF.

A molecular mechanics force ®eld can be designed to describe the geometry of the molecule only or speci®cally created to describe the motions of the atoms. Calculation of the vibrational frequencies using a harmonic oscillator approximation can yield usable results if the force ®eld was designed to reproduce the vibrational frequencies. Note that many of the force ®elds in use today were not designed to reproduce vibrational frequencies in this manner. When using this method, there is not necessarily a systematic error between the results and the experiments. This is because the parameters may have been created by determining what harmonic parameters would reproduce the experimental results, thus building in the correction. As a general rule of thumb, mechanics methods give qualitatively reasonable frequencies if the compound being examined is similar to those used to create the parameters. Molecular mechanics does not do so well if the structure is signi®cantly di¨erent from the compounds in the parameterization set.

Some computer programs will output a set of frequencies containing six values near zero for the three degrees of translation and three degrees of rotation of the molecule. Any number within a range of about ÿ20 to 20 cmÿ1 is essentially zero within the numerical accuracy of typical software packages. This range is larger if second derivatives are computed numerically. Other programs will use a more sophisticated technique to avoid computing these extra values, thus reducing the computation time.

Before frequencies can be computed, the program must compute the geometry of the molecule because the normal vibrational modes are centered at the equilibrium geometry. Harmonic frequencies have no relevance to the vibrational modes of the molecule, unless computed at the exact same level of theory that was used to optimize the geometry.

Orbital-based methods can be used to compute transition structures. When a negative frequency is computed, it indicates that the geometry of the molecule corresponds to a maximum of potential energy with respect to the positions of the nuclei. The transition state of a reaction is characterized by having one negative frequency. Structures with two negative frequencies are called secondorder saddle points. These structures have little relevance to chemistry since it is extremely unlikely that the molecule will be found with that structure.

11.2ANHARMONIC FREQUENCIES

For very-high-accuracy ab initio calculations, the harmonic oscillator approximation may be the largest source of error. The harmonic oscillator frequencies

11.3 PEAK INTENSITIES

95

are obtained directly from the Hessian matrix, which contains the second derivative of energy with respect to the movement of the nuclei. One of the most direct ways to compute anharmonic corrections to the vibrational frequencies is to compute the higher-order derivatives (3rd, 4th, etc.). The frequency for a potential represented by a polynomial of this order can then be computed. This requires considerably more computer resources than harmonic oscillator calculations; thus, it is much more seldom done.

Both harmonic oscillator and higher-order derivative calculations represent the potential energy surface near the optimized geometry only. It is possible to compute vibrational frequencies taking the entire potential energy surface into account. For a diatomic molecule, this requires computing the entire bond dissociation curve, which requires far more computer time than computing higherorder derivatives. Likewise, computing anharmonic frequencies for any molecule requires computing at least a sampling of all possible nuclear positions. Due to the enormous amount of time necessary to compute all these energies, this sort of calculation is very seldom done. The advantage of this technique is that it is applicable to very anharmonic vibrational modes and also high-energy modes.

A technique built around molecular mechanics is a dynamics simulation. The vibrational motion seen in molecular dynamics is a superposition of all the normal modes of vibration so frequencies cannot be determined directly from this simulation. However, the spectrum can be determined by applying a Fourier transform to these motions. The motion corresponding to a peak in this spectrum is determined by taking just that peak and doing the inverse Fourier transform to see the motion. This technique can be used to calculate anharmonic modes, very low frequencies, and frequencies corresponding to conformational transitions. However, a fairly large amount of computer time may be necessary to obtain enough data from the dynamics simulation to get a good spectrum.

11.3PEAK INTENSITIES

Another related issue is the computation of the intensities of the peaks in the spectrum. Peak intensities depend on the probability that a particular wavelength photon will be absorbed or Raman-scattered. These probabilities can be computed from the wave function by computing the transition dipole moments. This gives relative peak intensities since the calculation does not include the density of the substance. Some types of transitions turn out to have a zero probability due to the molecules' symmetry or the spin of the electrons. This is where spectroscopic selection rules come from. Ab initio methods are the preferred way of computing intensities. Although intensities can be computed using semiempirical methods, they tend to give rather poor accuracy results for many chemical systems.

There have been a few studies comparing ab initio intensities. In nearly all

96 11 MOLECULAR VIBRATIONS

cases, the addition of polarization functions to the basis set leads to a signi®cant improvement in results. HF-computed intensities are usually scaled by a constant factor prior to making comparisons. In general, results from scaled HF, DFT, and MP2 all give similar accuracies. Out of this group, some of the hybrid DFT functionals seem to perform best. Higher-level correlated calculations, such as CISD and CCSD, give an improvement in results. Intensities also improve with higher orders of perturbation theory.

11.4 ZERO-POINT ENERGIES AND THERMODYNAMICS CORRECTIONS

The total energy computed by a geometry optimization is the minimum on the potential energy curve. However, a molecule can never actually have this energy because it must always have some vibrational motion. Many programs compute the zero-point energy correction due to being in the lowest-energy vibrational mode along with the vibrational frequencies. For accurate work, the zero-point energy correction will be added to the total energy for the optimized geometry. This corrected value can then be used for computing the relative energies of various conformers, isomers, and the like and should be slightly closer to the experimental results.

Molecular enthalpies and entropies can be broken down into the contributions from translational, vibrational, and rotational motions as well as the electronic energies. These values are often printed out along with the results of vibrational frequency calculations. Once the vibrational frequencies are known, a relatively trivial amount of computer time is needed to compute these. The values that are printed out are usually based on ideal gas assumptions.

11.5RECOMMENDATIONS

It is possible to use computational techniques to gain insight into the vibrational motion of molecules. There are a number of computational methods available that have varying degrees of accuracy. These methods can be powerful tools if the user is aware of their strengths and weaknesses. The user is advised to use ab initio or DFT calculations with an appropriate scale factor if at all possible. Anharmonic corrections should be considered only if very- high-accuracy results are necessary. Semiempirical and molecular mechanics methods should be tried cautiously when the molecular system prevents using the other methods mentioned.

BIBLIOGRAPHY

For introductory discussions see the texts

J. Simons, J. Nichols, Quantum Mechanics in Chemistry Oxford, Oxford (1997).

BIBLIOGRAPHY 97

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

I. N. Levine, Quantum Chemistry Prentice Hall, Englewood Cli¨s (1991).

D. A. McQuarrie, Quantum Chemistry University Science Books, Mill Valley (1983). M. Karplus, R. N. Porter, Atoms & Molecules: An Introduction for Students of Physical

Chemistry W. A. Benjamin, Menlo Park (1970).

Books on molecular vibrations are

C.E. Dykstra, Quantum Chemistry & Molecular Spectroscopy Prentice Hall, Englewood Cli¨s (1992).

I.B. Bersuker, V. Z. Polinger, Vibronic Interactions in Molecules and Crystals SpringerVerlag, Berlin (1989).

G.Fischer, Vibronic Coupling Academic Press, London (1984).

I.B. Bersuker, The Jahn-Teller E¨ect and Vibronic Interactions in Mondern Chemistry

Plenum, New York (1984).

E.B. Wilson Jr., J. C. Decius, P. C. Cross, Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra Dover, New York (1980).

Reviews are

S. Krimm, K. Palmo, Encycl. Comput. Chem. 2, 1360 (1998).

W.Cornell, S. Louise-May, Encycl. Comput. Chem. 3, 1904 (1998).

J.J. P. Stewart, Encycl. Comput. Chem. 4, 2579 (1998).

T.Carrington, Jr., Encycl. Comput. Chem. 5, 3157 (1998).

H.KoÈppel, W. Domcke, Encycl. Comput. Chem. 5, 3166 (1998).

J.Cremer, J. A. Larsson, E. Kraka, Theoretical Organic Chemistry C. PaÂrkaÂnyi, ed., 259, Elsevier, Amsterdam (1998).

J.Tennyson, S. Miller, Chem. Soc. Rev. 21, 91 (1992).

Ê

H. Agren, A. Cesar, C.-M. Liegener, Adv. Quantum Chem. 23, 4 (1992). C. W. Bauschlicker, Jr., S. R. Langho¨, Chem. Rev. 91, 701 (1991).

C. B. Harris, D. E. Smith, D. J. Russell, Chem. Rev. 90, 481 (1990). K. Balasubramanian, Chem. Rev. 90, 93 (1990).

Z. Bai, J. C. Light, Ann. Rev. Phys. Chem. 40, 469 (1989). R. D. Amos, Adv. Chem. Phys. 67, 99 (1987).

Ï	R. Zahradnõk, Chem. Rev. 86, 709 (1986).
B. A. Hess, Jr., L. J. Schaad, P. Carsky,Â

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

H.KoÈppel, W. Domcke, L. S. Ceederbaum, Adv. Chem. Phys. 57, 59 (1984).

I.B. Bersuker, V. Z. Polinger, Adv. Quantum Chem. 15, 85 (1982).

D. W. Oxtoby, Ann. Rev. Phys. Chem. 32, 77 (1981).

R. T. Bailey, F. R. Cruickshank, Gas Kinetics and Energy Transfer vol. 3 127, P. G. Ashmore, R. J. Donovan (Eds.) Chemical Society, London (1978).

D. Rapp, T. Kassal, Chem. Rev. 69, 61 (1969). Adv. Chem. Phys. vol. 85, part 3. (1994).

98 11 MOLECULAR VIBRATIONS

Papers comparing the accuracy of computed intensities are

M. D. Halls, H. B. Schlegel, J. Chem. Phys. 109, 10587 (1998).

J.R. Thomas, B. J. DeLeeuw, G. Vacek, T. D. Crawford, Y. Yamaguchi, H. F. Schaefer III, J. Chem. Phys. 99, 403 (1993).

J.F. Stanton, W. N. Lipscomb, D. H. Magers, R. J. Bartlett, J. Chem. Phys. 90, 3241 (1989).

A. Holder, R. Dennington, Theochem 401, 207 (1997).

Correction factors are published in

http://srdata.nist.gov/cccbdb/

J. Baker, A. A. Jarzecki, P. Pulay, J. Phys. Chem. A 102, 1412 (1998). J. R. Wright, Jaguar User's Guide SchroÈdinger, Portland OR (1998). A. P. Scott, L. Radom, J. Phys. Chem. 100, 16502 (1996).

J. M. L. Martin, J. El-Yazal, J.-P. Francois, J. Phys. Chem. 100, 15358 (1996). D. Bakowies, W. Thiel, Chem. Phys. 151, 309 (1991).

Harmonic and anharmonic vibrations are discussed at

http://suzy.unl.edu/bruno/CHE484/irvib.html

Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems. David C. Young Copyright ( 2001 John Wiley & Sons, Inc.

ISBNs: 0-471-33368-9 (Hardback); 0-471-22065-5 (Electronic)

12 Population Analysis

Chemists are able to do research much more e½ciently if they have a model for understanding chemistry. Population analysis is a mathematical way of partitioning a wave function or electron density into charges on the nuclei, bond orders, and other related information. These are probably the most widely used results that are not experimentally observable.

Atomic charges cannot be observed experimentally because they do not correspond to any unique physical property. In reality, atoms have a positive nucleus surrounded by negative electrons, not partial charges on each atom. However, condensing electron density and nuclear charges down to partial charges on the nucleus results in an understanding of the electron density distribution. These are not integer formal charges, but rather fractions of an electron corresponding to the percentage of time an electron is near each nucleus. Although this is an arti®cial assignment, it is very e¨ective for predicting sites susceptible to nucleophilic or electrophilic attack and other aspects of molecular interaction. These partial charges correspond well to the chemist's view of ionic or covalent bonds, polarity, and so on. Only the most ionic compounds, such as alkali metal halides, will have nearly whole number charges. Organometallics typically have charges on the order of G0.5. Organic compounds often have charges around G0.2 or less.

12.1MULLIKEN POPULATION ANALYSIS

One of the original and still most widely used population analysis schemes is the Mulliken population analysis. The fundamental assumption used by the Mulliken scheme for partitioning the wave function is that the overlap between two orbitals is shared equally. This does not completely re¯ect the electronegativity of the individual elements. However, it does give one a means for partitioning a wave function and has been found to be very e¨ective for small basis sets. For large basis sets, results can be very unreasonable. This is due to di¨use functions describing adjacent atoms more than they describe the atom on which they are centered. In some cases, Mulliken analysis can assign an electron population to an orbital that is negative or more than two electrons. It also tends to underestimate the charge separation in ionic bonded systems.

In spite of its de®ciencies, the Mulliken population scheme is very popular. One reason is that it is very easy to implement so it is available in many software packages. Probably the most important reason for its popularity is the fact

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