Abstract
AMPAC 10 is the latest product of a continuing research effort spanning some three decades. It is a general purpose semiempirical package designed for an audience of both theoreticians and experimentalists. AMPAC offers some of the most advanced computational capabilities available in any computational chemistry software package, yet is simple enough in application that occasional users can obtain important information from these calculations.
Table of Contents
With each new version, AMPAC continues to expand its functionality. Here we highlight some of these new capabilities:
Large Molecule Capacity (SPARSE, PSOLVE). The most significant and dramatic improvement involves the ability to handle very large chemical systems. The new keyword, SPARSE, instructs the program to ignore certain small matrix elements allowing for a great reduction in memory and CPU cost for many large systems. For extended an discussion of this new capability, see Chapter 7, Sparse Matrix Methods. Many additional changes have been made to the regular (non-sparse) jobs to make them more efficient for large molecules.
Large Molecule Optimization (TRUSTE=LARGE). In fitting with the move toward larger molecules, improvements have been made AMPAC's primary geometry optimizer to improve its performance and reduce memory for large molecules. This is invoked using the new TRUSTE=LARGE keyword.
Restricted Open-shell Hartree-Fock (ROHF). The new ROHF feature extends AMPAC's ability to handle open-shell systems. The extensive C.I. support is a very powerful way to model these types of systems but there are times where non-C.I. approach is needed. ROHF now joins UHF as tools for open-shell systems.
Natural Population Analysis and Natural Bond Orbitals (NBO). The NBO package provides an analysis of the molecule's wavefunction and density in terms of "natural orbitals." The natural bond orbitals are localized orbitals and so allow for greater chemical insight than the standard canonical orbitals that are diffused over the whole molecule.
Initial Guess Orbitals and Densities (LEWIS). The Lewis dot structure gives chemists an approximate picture of how eletrons distribute themselves in a molecule. AMPAC is now capable of analyzing a molecules Lewis dot structure and based on that information, generate a molecular orbitals and density matrix that can be used to start of the SCF iterations. For documentation, see the LEWIS keyword.
Hydrogen Bonding and Dispersion Corrections (PM6-D3H4, PM3-D3H4, AM1-D3H4, RM1-D3H4, AM1-FS2). Semi-empirical methods generally do poorly at modeling hydrogen bonding and dispersion in molecules but these forces are important for certain types of systems. There are two different correction models that add extra terms to better describe these intermolecular forces. First is the D3H4 correction that is implemented with four different models: AM1-D3H4, PM3-D3H4, PM6-D3H4, and RM1-D3H4. Second, AM1-FS2 that only works with the AM1 model.
AM1, RM1, PM3, PM6, MINDO3, MNDO, MNDOC, MNDO/d, SAM1, SAM1D Hamiltonians
RHF, ROHF, and UHF methods
Extensive Configuration Interaction (C.I.)
Selected configurations from a 10x10 manifold
Spin states up to 20 unpaired electrons
Excited states
Geometry optimization, etc., on specified states
Single SCF calculation
Geometry optimization
Gradient minimization
Transition state location
Reaction path calculation
Reaction grid calculation
Intrinsic Reaction Coordinate calculation
Annealing for multiple minima searches
Force constants and vibrational frequency analysis
Normal mode analysis
Sparse matrix solvers (PSOLVE, SPARSE) to handle very large molecules
Initial guess orbitals and density based on Lewis dot structure analysis (LEWIS)
Solvated molecular properties (AMSOL, COSMO)
Thermodynamic properties
Localized orbitals
Covalent bond orders and charge distributions (Mulliken, ESP, NBO)
Unpaired spin densities
Corrections for dispersion and hydrogen bonding (PM6-D3H4, PM3-D3H4, AM1-D3H4, RM1-D3H4, AM1-FS2)
Computational chemistry is one of today’s most rapidly expanding and exciting areas of scientific endeavor. New computer technologies have made the purchase and maintenance of computational resources less expensive than most other major chemical instrumentation, and a trained researcher can usually perform an extensive computational study in much less time than is required for the complete experimental counterpart. Given the rising cost of maintaining laboratories and trained personnel, computational chemistry often becomes a cost-effective alternative and companion to traditional bench chemistry. As with any other scientific method, the results from computational chemistry require careful interpretation. The limitations and results of each type of technique must be considered in the broader context of all the information available for a particular system.
Computational chemistry finds one of its most important applications in the support of experimental efforts. As noted in the table below, many types of data that are directly measured experimentally can be computed as well. An example of one of the most useful of these is vibrational spectra (IR and Raman). Detailed descriptions of vibrational motions and fairly accurate frequency predictions are sometimes critical for explaining spectra that are the result of complicated experimental situations, such as a mixture of conformers. Extensive calculations can be performed on the specific conformers of a particular compound, yielding such information as a particular conformer’s dipole moment, its relative energy as compared to other conformers, and its unique spectra without the interference of the other conformers. Band assignment can be facilitated by the ability to subtract calculated spectra from experimental and to assign, based on the computational description, particular motions to spectral bands.
As well as supporting experimental studies, electronic structure methods can also produce data that is not generally available from experiment. Electron densities on specific atoms is one of the most important examples of this type of data. Although recent advances^{[1]} indicate that more regular determinations of experimental charge densities may be on the horizon, these techniques are not routine. Information about charge densities can be used to predict relative reactivities of different substrates as well as sites of reaction. Also available from these methods are electronic bond orders. The bond orders are measures of the net bonding between pairs of atoms. Many qualitative assessments of a species’ bonding patterns can be made using bond orders. Such information can also be correlated with other observables like bond lengths, vibrational frequencies, and NMR coupling constants.^{[2]} ^{[3]}
Electronic structure methods can also be used to study reaction mechanisms. This is particularly true for the semiempirical techniques. Because of their fast computational speed, they can be used to extensively examine reaction potential surfaces. Indeed, an impressive arsenal is available for the virtually automated location and characterization of transition states. These calculations can then rapidly differentiate between a set of mechanisms postulated from the available experimental data. In short, the application of computational methods is only limited to the investigator’s ability to define a problem into a proper set of questions. Indeed, a recent study^{[4]} on hexuloses has shown that such a nebulous property as sweetness can be predicted computationally!
(As computational programs have developed, the use of graphical tools for interpretation has become common place. Indeed, recent graduates in computational chemistry may have never prepared a z-matrix or Cartesian coordinate definition for a molecule by hand, except as part of a class exercise! Further, they are able to utilize display methodologies that were unknown just a few years ago to extract information from the sometimes bewildering mass of data that most computational chemistry programs produce. This results in a more intuitive and natural grasp of the results of the calculations and concomitant greater utility of the methods. GUIs are now almost as important a component in computational chemistry efforts and the underlying theories on the chemical models themselves!)
So, then, what sort of results are expected from computational methods and how are they useful to the experimental chemist? Some of the information obtained from different types of methods is listed in the Table below. One group of popular computational methods is based on the approximation of the various interactions present in molecular systems in terms of a force field. These procedures fall in the general category of Molecular Mechanics (MM) and Molecular Dynamics (MD). They are parameterized to fit a large body of experimental data by defining bond force constants along with angle and torsional stretching potentials for different chemical environments. (For example, there are different MM parameters for the C-N bond of an amine and the C-N bond of an amide.) Molecular Mechanics techniques do not, however, treat electrons, limiting the relevance of the MM/MD model in many situations. MM/MD finds its greatest utility in the prediction of conformational energies and structures, and is often applied in the study of biological systems due to their large size and MM/MD’s computational speed and efficiency. Calculations using MM models are the fastest of the computational chemistry approaches.
Table 1.1. Information Available from Computational Methods
Data Item | Molecular Mechanics | Semiempirical | ab initio |
---|---|---|---|
Heat of Formation | • | • | • |
Entropy of Formation | • | • | • |
Free Energy of Formation | • | • | • |
Heat of Activation | • | • | |
Entropy of Activation | • | • | |
Free Energy of Activation | • | • | |
Heat of Reaction | • | • | • |
Entropy of Reaction | • | • | • |
Free Energy of Reaction | • | • | • |
Strain Energy | • | † | † |
Vibrational Spectra | • | • | |
Dipole Moments | • | • | |
Optimized Geometries | • | • | • |
Electronic Bond Order | • | • | |
Electronic Distribution | • | • | |
Mulliken Pop. Analysis | • | • | |
Transition State Location | • | • |
• Property Predicted † Indirectly Available
Two other types of procedures, the semiempirical and ab initio molecular orbital approaches, are examples of Electronic Structure methods applying quantum mechanics. They implicitly consider the electron distribution in chemical systems as a function of nuclear position. Electronic structure theories vary greatly in complexity and accuracy. Some models are essentially conceptual, requiring no computations and are best suited to paper and pencil. Others only become usable when supercomputers are applied to their solution. At present, the best and most flexible chemical models are derived from quantum mechanics and are based on the Schrödinger Equation. The Schrödinger Equation relates the properties of electrons to those of waves and permits a mathematical description of atomic, and by extension, molecular characteristics. The standard form of the Schrödinger Equation can be solved exactly, however, for only the simplest case, the hydrogen atom. Significant approximations are required to apply the Schrödinger wave function approach to problems of interest. A commonly used and essentially standard set of approximations and assumptions has come to be known as Hartree-Fock (HF) Theory and is the basis for the majority of work done in electronic structure chemistry at present.
A rigorous execution of Hartree-Fock Theory is called an ab initio (“from first principles”) approach. These calculations involve a near complete mathematical treatment of the theoretical model underlying Hartree-Fock Theory. Comprehensive calculations of this type result in a potentially enormous number of integrations and differentiations of complex algebraic formulae. The sheer number of separate computations can easily become so vast that only a supercomputer has the requisite speed, memory, and disk storage space for even moderately sized systems ( i.e., <100 non-hydrogen atoms). Ab initio methods, by virtue of being derived from the Hartree-Fock assumptions and approximations, have theoretical inaccuracies that cause difficulties in certain cases. Perhaps the most important of these is the neglect of the dynamic electron correlation effects in the motion of electrons within the self-consistent field used in the iterative solution process. Ab initio corrections are available for correlation, (e.g. Møller-Plesset (MP) Perturbation Theory, configuration interaction (CI), or multi-configuration-SCF (MCSCF)), but are very expensive in terms of computing time and disk space. It should be emphasized that in the great majority of cases where ab initio methods have been rigorously applied, the results have been very good. However, the constraints listed above strictly limit the size and complexity of the systems feasible for a full ab initio treatment.
A computational method which is similar to, yet still separate from, other HF molecular orbital techniques is density functional theory (DFT).^{[5]} The basic principle behind DFT is that the electron density is a fundamental quantity that can be used to develop a rigorous many-body theory, applicable to any atomic, molecular, or solid state system. In the mid-1960s Kohn, Hohenberg, and Sham derived a formal proof of this principle as well as a set of equations (the Kohn‑Sham equations) which are similar in form and function to the Hartree‑Fock equations of molecular orbital theory. This formalism is such that electron correlation is inherently included in the method at no extra cost in computational efficiency. The basic difference between HF theory and DFT can be summarized in the following way: for a given set of atomic positions, HF theory expresses the total energy of the system of nuclei and electrons as a function of the total wave function, whereas DFT expresses this total energy as a functional of the total electron density. Current DFT methods contain no empirical or adjustable parameters and are thereby acknowledged to be “ ab initio ” approaches.
Acceptance of DFT methods by chemists is growing now that general purpose computational packages exist containing them. This acceptance is also based on literature reports of systematic comparisons of DFT methods with experiment as well as with HF and post-HF methods. Chemists are also quickly learning that DFT methods are slightly computationally more efficient than ab initio HF methods. HF calculations formally scale as N^{4} for N basis functions but, in reality, scale N^{2.5} - N^{3}, depending on the size of the molecular system. Post-HF methods, which include electron correlation in some manner, scale as N^{m} where m > 4. DFT methods, however, scale as N^{2} - N^{3}, including some electron correlation. Therefore, DFT calculations are practical for molecular systems which may be too large and/or troublesome for ab initio HF techniques. For example, DFT methods have been particularly successful in predicting the properties of transition metal systems which have been notoriously difficult for HF and post-HF techniques.
Semiempirical quantum mechanical methods are also based on Hartree-Fock Theory. Semiempirical calculations ignore some of the less important aspects of HF theory that full ab initio treatments explicitly compute, so that fewer actual operations are performed. Also, the semiempirical approach uses empirically determined parameters and parameterized functions to replace some sections of a more complete HF treatment. The approximations in semiempirical theory result in much more rapid single energy calculations than in either HF or DFT ab initio methodologies. The advantage gained in the energy calculations leads to semiempirical methods being some 100-1000 times faster overall than ab initio HF or DFT methods of comparable predictive quality. The most popular of the semiempirical methods are those developed by or derived from work by Michael J. S. Dewar.^{[6]} They include MINDO3,^{[7]} MNDO,^{[8]} PM3,^{[9]} ^{[10]} AM1,^{[11]} and SAM1.^{[12]} Some older semiempirical programs and methods are still used successfully for specific purposes (e.g. ZINDO for spectroscopy on a wide range of elements). In summary, semiempirical methods offer chemical accuracy as well as computational efficiency. As computational power increases, so will the size of systems to which semiempirical methods are applied.
^{[1] }J. Phys. Chem.. 1989. 7989.
^{[2] }J. Phys. Chem.. 1990. 7418.
^{[3] }J. Am. Chem. Soc.. 1990. 7537.
^{[4] }J. Am. Chem. Soc.. 1990. 4732.
^{[5] }These two paragraphs have been distilled from:
^{[6] }Most of this work was accomplished at the University of Texas at Austin.
^{[7] }J. Am. Chem. Soc.. 1975. 1285.
^{[8] }J. Am. Chem. Soc.. 1977. 4899.
^{[9] }J. Comput.Chem.. 1989. 209.
^{[10] }J. Comput.Chem.. 1989. 221.
^{[11] }J. Am. Chem. Soc.. 1985. 3902-3909.
^{[12] }Tetrahedron. 1993. 5003.
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