Theory and range of modern semiempirical molecular orbital methods

Semiempirical molecular orbital methods have a long history. They serve to tackle large systems and complicated processes beyond the reach of ab initio or density functional methods. Although their setup is derived from Hartree–Fock theory, the design of approximate energy expressions and the empirical parameters are used to achieve higher accuracy than the underlying ab initio theory. In this way the effect of larger basis sets or correlation can be partially simulated. All widely used semiempirical methods establish their accuracy by error statistics for molecular properties with experimental and high-level ab initio or density functional theory calculations as a reference. Their computational efficiency makes them suitable for the study of biochemical systems and solid materials. The present review presents a variety of applications which demonstrate the need for and success of semiempirical methods.     

Accurate calculation of anharmonic vibrational frequencies of medium sized molecules using local coupled cluster methods

Local coupled cluster methods were applied for the automated generation of accurate multidimensional potential energy surfaces for a set of test molecules ranging from six to nine atoms. Based on these surfaces anharmonic fundamental frequencies were computed using vibrational self-consistent field and configuration interaction methods. The computed vibrational frequencies are compared to those obtained from similar calculations using conventional coupled cluster methods and to experimental values. The results from local and conventional methods are found to be of similar accuracy and in close agreement with experimental values. In addition, an efficient parallelization of the fully automated surface generation code is presented.     

Approximating coupled cluster level vibrational frequencies with composite methods

An extensive study of the harmonic frequencies of a large set of small polyatomic closed-shell molecules computed at both single level ab initio and composite approximations is presented here. Using various combinations of basis sets, composite methods are capable of predicting single level ab initio CCSD(T) harmonic frequencies to within 5 cm(-1) on average, which suggests a computationally affordable means of obtaining highly accurate vibrational frequencies compared to the CCSD(T) level. A general approach for calculating the composite level equilibrium geometries and harmonic frequencies for polyatomic systems that uses the Collin's method of interpolating potential energy surfaces is also described here. This approach is further tested on tetrafluoromethane, and an estimation of the potential CPU time savings that may be obtained is also presented. It is envisaged that the findings here will enable theoretical studies of fundamental frequencies and energetics of significantly larger molecular systems.     

Calculations of vibrational energy transfer using semi-classicals matrix theory

We utilize an extension of Miller's semi-classical S matrix theory to calculate resonant and non-resonant vibrational energy transfer probabilities. The collisions under study are collinear D₂D₂ interactions at energies below and above the classical dynamic transition threshold and collinear H₂D₂ interactions above the dynamic threshold. Below threshold we employ an initial angle representation. We find that the computed probabilities are in substantial agreement with exact quantum mechanical computations and represent a major improvement over quasi-classical results. At energies above threshold we apply the first order and the classical semi-classical versions of the theory. The results indicate fair agreement with quantum mechanical calculations, but no significant improvement over quasi-classical results.     

Circular dichroism of helical structures using semiempirical methods

A general semiempirical scheme has been elaborated to simulate circular dichroism (CD) spectra of supramolecular systems. This approach adopts the analytical method of Beck and Hohlneicher [Theor. Chem. Acc. 101, 297 (1999).] to evaluate the one- and two-center integrals over Slater atomic orbitals. The performance of the method, employing INDO/S and CNDO/S semiempirical parametrizations, has been assessed by considering (i) the effect of the size of the singly excited states manifold, (ii) the origin invariance, and (iii) comparisons with the experimental and other theoretical spectra of several helicenes as well as pyridine-pyrimidine oligomers, which can adopt helical conformations. The main results are (i) the INDO/S parametrization with rather small excitation manifolds is able to reproduce, at low computational costs, the experimental CD spectra of several helicenes as well as CD simulations performed at ab initio and time-dependent density functional theory level of approximation; (ii) in the series of homohelicenes, the rotatory strength of the lowest-energy band increases almost linearly with the size of the helix; (iii) as evidenced by the study of tetradodecyloxy helicene bisquinone, packing effects can change the sign of remarkable CD bands, which are used to assign the structure configuration.     

Molecular orbital predictions of the vibrational frequencies of some molecular ions

Recent spectroscopic advances have led to the first determinations of infrared vibration-rotation bands of polyatomic molecular ions. These initial detections were guided by ab initio predictions of the vibrational frequencies. The calculations reported here predict the vibrational frequencies of additional ions which are candidates for laboratory analysis. Vibrational frequencies of neutral molecules computed at three levels of theory, HF/3-21G, HF/6-31G*, and MP2/6-31G*, were compared with experiment and the effect of scaling was investigated to determine how accurately vibrational frequencies could be predicted. For 92% of the frequencies examined, uniformly scaled HF/6-31G* vibrational frequencies were within 100 cm-1 of experiment with a mean absolute error of 49 cm-1. This relatively simple theory thus seems suitable for predicting vibrational frequencies to guide laboratory spectroscopic searches for ions in the infrared. Hence, the frequencies of 30 molecular ions, many with astrochemical significance,were computed. They are CH2+, CH3+, CH5+, NH2+, NH4+, H3O+, H2F+, SiH2+, PH4+, H3S+, H2Cl+, C2H+, classical C2H3+, nonclassical C2H3+, nonclassical C2H5+, HCNH+, H2CNH2+, H3CNH3+, HCO+, HOC+, H2CO+, H2COH+, H3COH2+, H3CFH+, HN2+, HO2+, C3H+, HOCO+, HCS+, and HSiO+.     

Effects of rapid intramolecular vibrational redistribution on molecular spectra at microwave frequencies

Some molecules with more than 10 atoms and more than two torsional degrees of freedom have state densities sufficient for rapid (10¹⁰ s^?1) intramolecular vibrational redistribution at energies as low as 0.25 kcal/mol. Predicted features of low-resolution microwave (LRMW) band spectra of rapidly relaxing polar prolate molecules are discussed and compared with LRMW spectra of ethyl esters.     

Comparison of algorithms for conical intersection optimisation using semiempirical methods

We present a comparison of three previously published algorithms for optimising the minimum energy crossing point between two Born–Oppenheimer electronic states. The algorithms are implemented in a development version of the MNDO electronic structure package for use with semiempirical configuration interaction methods. The penalty function method requires only the energies and gradients of the states involved, whereas the gradient projection and Lagrange–Newton methods also require the calculation of non-adiabatic coupling terms. The performance of the algorithms is measured against a set of well-known small molecule conical intersections. The Lagrange–Newton method is found to be the most efficient, with the projected gradient method also competitive. The penalty function method can only be recommended for situations where non-adiabatic coupling terms cannot be calculated. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

The vibrational frequencies of HNO

Electronic and vibrational emission spectra have been produced by the reaction of H atoms with NO, and studied under moderate resolving power. Preliminary rotational analyses of some of the bands give ν″₁=2684_.7 cm^?1, ν″₂=1565_.3 cm^?1 and ν″₃=1500_.4 cm^?1. The values for ν″₁ and ν″₃ are considerably different from early published values.     

Van der Waals interaction and hydrogen bond effects on molecular vibrational frequencies

The frequency shift of an AZ two-atomic fragment upon molecule transfer from gas to solution or matrix is investigated. The nearest neighbour ligands are considered to form a cluster. The coupling of the ν_s(AZ) mode with the cluster low-frequency oscillators (ν_Q) is taken into account. One such oscillator can be the ν_σ(AZ…B) intermolecular stretching frequency oscillator in the H-bonded or charge transfer complex. A general expression for the ν_s(AZ) frequency shift (Δω) is obtained, which allows for both the usual van der Waals interaction and the (ν_s, ν_σ) and (ν_s, ν_Q) mode couplings. When the former prevails, the relative frequency shift Δω/ω_f is invariant to the Z → Z* (e.g., H → D) isotopic substitution. When the mode coupling prevails, the quantity Δω/ω²_f is invariant what is typical for H bonds. If the H-bonded or charge transfer complexes are absent, the frequency shift Δω is proportional to α(ϱ_AZ + ϱ_L)⁻⁶ where α is the polarizability of the ligand, ϱ_AZ and ϱ_L are the van der Waals radii of AZ and ligand, respectively. The additional ν_s(AH) frequency shift upon transfer of the AH…B complex from the gas to solution seems to be caused by (ν_s, ν_Q) mode coupling.     

Comparison of SCC-DFTB and NDDO-based semiempirical molecular orbital methods for organic molecules

Extensive testing of the SCC-DFTB method has been performed, permitting direct comparison to data available for NDDO-based semiempirical methods. For 34 diverse isomerizations of neutral molecules containing the elements C, H, N, and O, the mean absolute errors (MAE) for the enthalpy changes are 2.7, 3.2, 5.0, 5.1, and 7.2 kcal/mol from PDDG/PM3, B3LYP/6-31G(d), PM3, SCC-DFTB, and AM1, respectively. A more comprehensive test was then performed by computing heats of formation for 622 neutral, closed-shell H, C, N, and O-containing molecules; the MAE of 5.8 kcal/mol for SCC-DFTB is intermediate between AM1 (6.8 kcal/mol) and PM3 (4.4 kcal/mol) and significantly higher than for PDDG/PM3 (3.2 kcal/mol). Similarly, SCC-DFTB is found to be less accurate for heats of formation of ions and radicals; however, it is more accurate for conformational energetics and intermolecular interaction energies, though none of the methods perform well for hydrogen bonds with strengths under ca. 7 kcal/mol. SCC-DFTB and the NDDO methods all reproduce MP2/cc-pVTZ molecular geometries with average errors for bond lengths, bond angles, and dihedral angles of only ca. 0.01 A, 1.5 degrees , and 3 degrees . Testing was also carried out for sulfur containing molecules; SCC-DFTB currently yields much less accurate heats of formation in this case than the NDDO-based methods due to the over-stabilization of molecules containing an SO bond.     

The performance of explicitly correlated methods for the computation of anharmonic vibrational frequencies

Anharmonic vibrational frequencies for closed-shell molecules computed with CCSD(T)-F12b/aug-cc-pVTZ differ from significantly more costly composite energy methods by a mean absolute error (MAE) of 7.5 cm⁻¹ per fundamental frequency. Comparison to a few available gas phase experimental modes, however, actually lowers the MAE to 6.0 cm⁻¹. Open-shell molecules have an MAE of nearly a factor of six greater. Hence, open-shell molecular anharmonic frequencies cannot be as well-described with only explicitly correlated coupled cluster theory as their closed-shell brethren. As a result, the use of quartic force fields and vibrational perturbation theory can be opened to molecules with six or more atoms, whereas previously such computations were limited to molecules of five or fewer atoms. This will certainly assist in studies of more chemically interesting species, especially for atmospheric and interstellar infrared spectroscopic characterization.     

Calculations of silicooxygen ring vibration frequencies

In the work model calculations of the vibrations of ideally isolated silicooxygen rings (using PM3 method) have been carried out. three-, four-, and six-membered rings have been considered. It has been found that that the three-membered silicooxygen rings are flat and practically undeformed showing D_3h symmetry. The rings of higher number of ring members (i.e. n>3) are deformed to some extent. The deformation reveals itself most significantly in the Si–O–Si bond angles distribution. In the case of all the rings the bridging Si–O–Si bonds are ca. 0.02–0.04 Å shorter than the non-bridging Si–O⁻ bonds. Hypothetical IR spectra for all the rings considered have been also calculated. Analysis of these hypothetical spectra leads to the conclusion that the whole spectrum can be divided into four wavenumbers regions, 1200–1100 cm⁻¹ stretching Si–O(Si) vibrations; 1000–800 cm⁻¹ stretching Si–O⁻ vibrations; 800–600 cm⁻¹; the region in which a band characteristic of silicooxygen rings appears, and below 600 cm⁻¹ bending ⁻O–Si–O⁻ and (Si)O–Si–O(Si). It has been also found that as the number of ring members increases the ‘ring band’ shifts to lower wavenumbers: 725 cm⁻¹ for three-membered rings, 650 cm⁻¹ for four-membered rings and 610 cm⁻¹ for six-membered rings. Calculated spectra have been compared with the experimental spectra of cyclosilicates. They showed good agreement in the 1200–600 cm⁻¹ region. In the experimental spectra as well as in the calculated ones, with increasing the number of ring members the ‘ring band’ shifts towards lower wavenumbers.     

Molecular equilibrium geometries and vibrational frequencies by maximum overlap symmetry molecular orbital method

It is demonstrated that the maximum overlap symmetry molecular orbital method can be used for optimization of molecular geometries and calculation of vibrational frequencies by adding a two-body repulsive energy term and a modification of the Wolfsberg–Helmholz formula. The obtained equilibrium geometries and vibrational frequencies are on the whole in good accordance with experimental data, which shows that the basic idea using the method to optimize molecular geometries and to calculate vibrational frequencies is reasonable.     

Faster gradients for semiempirical methods

A new algorithm for the numerical evaluation of gradients in semiempirical methods is described. The method is approximately twice as fast as the schemes currently employed and produces gradients of comparable accuracy. This method has been tested by comparing the results obtained by the new method with those of the previous numerical scheme, and also with those calculated analytically. The results of using the new gradients in geometry optimizations are also presented. ©1999 John Wiley & Sons, Inc. J Comput Chem 20: 629–635, 1999     

Consistent parametrization of semiempirical MO methods

The parametrization of semiempirical molecular orbital methods is reviewed. The parametrization procedures are classified in three categories. The newly proposed category of consistency of parameters in a row of elements is applied in SINDO 1 to the Si atom. The calculations on test molecules and silicon clusters demonstrate that significant improvements in accuracy can be obtained in this way. © 1992 John Wiley & Sons, Inc.