Heinrich-Heine-Universita¨t, Du¨sseldorf, Germany and
M. ATANASOV
Bulgarian Academy of Sciences, Sofia, Bulgaria
2.52.1 INTRODUCTION 661
2.52.2 HAMILTONIAN, PARAMETERS, AND MATRICES 662
2.52.3 SYMMETRY CONSIDERATIONS 663
2.52.4 FEATURES OF AOMX 664
2.52.5 AOMX IN THE INTERNET 664
2.52.6 REFERENCES 664
2.52.1 INTRODUCTION
Since the 1980s, there has been a growing demand for a general ligand field program that is based on realdorbitals to help interpret highly resolved spectra of TM compounds in a straightforward manner. Increased spectral resolution at low temperatures together with sophisticated methods involving polarized light, time resolution, and magnetic fields have revealed electronic structures that could not be resolved previously. For the tedious work of assigning electronic states to absorption and emission spectra the angular overlap model (AOM) is preferred to crystal field theory because the AOM gives a more vivid view of chemical bonds in transition metal complexes and because it builds conceptual bridges to molecular orbital theory. An in-depth discussion of the AOM and its variants, as well as case studies of its applications to experimental data, can be found in Chapter 2.36.
The subject of this chapter is AOMX, a computer program that the authors decided to develop due to the lack of a general computational tool that allows for semiempirical calculations within the AOM parameter framework making extensive use of symmetry. The program was based on Hoggard’s d3 program (AOM1)1 and was generalized to treat arbitrary dn systems.2 Further extensions have led to the present version, called AOMX.
This chapter is organized as follows: after defining the Hamiltonian and the parameterization scheme, we will explain how the symmetry of the eigenstates is exploited by AOMX, then some technical details are presented, and, finally, how to obtain and how to use the program.
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2.52.2 HAMILTONIAN, PARAMETERS, AND MATRICES
AOMX computes ligand field states of dn complexes using the common tetragonally quantized dxy,dyz,dzx,dx2y2, dz2 orbital basis and a Hamiltonian of the general form
HẳVLF ỵ Vee ỵ VTrees ỵVSO ỵ Vmag ð1ị
which is diagonalized using antisymmetrized product functions (Slater determinants). Group theor- etical methods applied to the eigenfunctions of the system are utilized to compute the symmetry of the electronic states. The ligand field operatorVLFis described by parameters e, (ẳ,s,c,s,c) (where the subscripts c (cosine) and s (sine) denote the components of the two-fold degenerate bonding types and become important in cases of anisotropic bonds) and the angular position of eachligand k as defined by its Eulerian angles Yk, Fk, Ck, or Cartesian coordinates. In fact, AOMX always uses three Eulerian rotations as shown inFigure 1, which is the most general case.
(Note that any radial dependence is separated off into the VLFparameters.) The corresponding 55 transformation matrix Fi(Y,F,C) was described in detail by Scha¨ffer3 and Schmidtke.4 It decomposes eachmetal orbital Mi into functions that are symmetry-adapted to one of the ligands k:
jMiðx;y;zịi ẳX
jMiðx000;y000;z000ịihMiðx000;y000;z000ị jMiðx;y;zịi
ẳX
jMiðx000;y000;z000ịiFiðY;F;Cị ð2ị
Witheachligand having an effective potentialVL
k
eff and valence orbitalsL
k, we can write the general AOM matrix element as a second-order perturbation term:
HijAOMẳX
XnL
kẳ1
FiðYk;Fk;CkịFjðYk;Fk;Ckị hM VeffL
k
Mi þ hM VeffL
k
LkihLk VeffL
k
Mi/ð"M"Lkị
n o
ẳX
XnL
kẳ1
FiðYk;Fk;CkịFjðYk;Fk;Ckịeðkị
ð3ị
Thus, the AOM matrix element is separated into a sum of angular geometry factors Fi and electronic parameterse(k) that contain all the rest, including ionic as well as covalent contribu- tions. Note that we are dealing with independent local ligand potentials and that ẳ, ,. . . corresponds to the well-known types of chemical bonds.
The AOMX program makes use ofEquation (3)to compute the 55 matrix of the ligand field potential VLF. Higher coordination spheres can easily be incorporated by adding more ligators to the system. Further corrections like phase coupling, misdirected valency (to be included in AOMX in 2004), and s–d mixing can be added, the latter being particularly relevant in square planar complexes. The electron repulsion Vee is represented by Racahparameters A, B, C, and may optionally be corrected for anisotropic nephelauxetic effects by orbital reduction factorsi1 leading to electron-repulsion integrals of the formabcd< ab|r121
|cd>.5 As a further two-electron correction, the Trees operator VTreesẳL2(Trees parameter ) may
z z′
z″ y″
x″ x′
y′
y′′′z′′′
x′′′
yL zL xL
yL zL xL
yL zL xL
yL zL xL
y
x θ
Φ
θ
Ψ
R z R z′ R
z″
(Θ) (Ψ)
(Φ)
Figure 1 Rotation of the metal coordinate system by three Eulerian angles.
also be taken into account.2,6Matrix elements for including spin–orbit coupling are set up using the operatorVSOẳilisi withthe parameter. Here, too, spatial anisotropy may be taken into account by defining the components of l as kxlx, kyly and kzlz withreduction factors ki<1.7 Finally, the influence of an external magnetic field, described by the operator VmagẳBHðYmag;Fmagịiðli ỵ 2siị, can be incorporated into the calculation, again with the option of anisotropic orbital momentum reduction.
Once the system is parameterized, the many-electron matrix is set up by successively applying the operators ofEquation (1)to simple anti-symmetrized product functions (Slater determinants).
The size of the basis set is given by the number of possible distributions ofnd electrons over 10 spin orbitals,nDẳ 10n
d , which is quite moderate: Taking advantage of the electron/hole equivalence (in hole configurations, the sign of matrix elements of one-electron operators will be inverted by the program), it can adopt values between 10 (d1d9) and 252 (d5). Complex Hermitian matrices of this size can be diagonalized very rapidly and without numerical problems using the conventional Householder algorithm, as implemented in the EISPACK package,8which is part of the AOMX source code.
2.52.3 SYMMETRY CONSIDERATIONS
As the basic functions are not symmetrized, the eigenvectors of the system do not exhibit any symmetry and would be useless as such. But it is evident that each of the eigenvectors must belong to an irreducible representation of the symmetry group of the underlying Hamiltonian. Except in the trivial case of no symmetry (C1), AOMX automatically determines symmetries by applying group theoretical techniques as will be outlined here.
Symmetry information can be obtained if the characters of the state vector with respect to the symmetry operations of the group are found. ForCibeing the first of a gi-fold set of degenerate states, this can be achieved by evaluating
ðCiịðSị ẳiỵXgi1
kẳ1
hCkjSjCki ẳTrðSfCigị ð4ị
for each symmetry operator S and subsequent comparison of the results with the character table of the symmetry group of the complex compound. Special care has to be taken when rotating spin functions. In such cases the methods outlined in Wigner’s textbook9must be applied. Thorough inspection of the symmetry properties ofdnsystems (following Jứrgensen’s concept of holohedric symmetry (see Chapter 2.38, Section 2.38.3.1) shows that only a few representative symmetry operators need to be taken into account in order to identify the symmetry of adnstate.Table 1 shows the relevant cases unambiguously.
Beyond symmetry, the eigenstate vectors also contain information about orbital occupations withrespect to the one-electron basis functionsk, which can be very useful for chemists. AOMX computes orbital occupation numbersq(m) in a state functionCifrom the diagonal elements of the one-electron density operatorri:
qðmị hmjijmi ẳXnD
jẳ1
cjicjiXn
kẳ1
ðjk;mị ð5ị
wherenDis the number of Slater determinants,nis the number ofd electrons, andjkdenotes the orbital of the kthelectron in the jthdeterminant of state i. In other words: AOMX does a
Table 1 Rotations needed to identify the irreps ofdncomplexes.
O D4 D3 D2 C2 C1
Ordinary group or unique reps of the double group
Cz4,Cx2 Cz4,Cx2 Cx2 Cz2,Cx2 Cz2 a Double-valued reps of the
double group
Cz4 Cz4 Cz3 a a a
a distinction not required.
AOMX: Angular Overlap Model Computation 663
Mullikan-like population analysis, but without overlap densities because the orbital basis function are orthogonal to each other.
2.52.4 FEATURES OF AOMX
The program was designed with the aim in mind that it should be easy to use, allow for parameter fitting, and be fast and portable across systems. For this purpose, we developed a flexible input description language for defining the complex geometry, experimental state assignments, and parameter optimizations (using the Powell parallel subspace algorithm10) as well as multidimen- sional parameter scans. Parameters can also be constrained during optimizations or scans, and even alternative parameterization schemes (e.g., CF parameters) can be used as long as they can be related to the AOM parameter set. The output options include eigenvalues, wavefunctions, the symmetry of eigenstates, orbital occupation numbers, one-electron matrices of VLF, VSO, or Vmag, as well as interelectronic repulsion integrals.
The program is coded in standard Fortran 77 and is very fast; even multi-parameter fitting calculations are rapidly completed. Thanks to the widespread free GNU gcc/g77 compilers, it can be run on virtually any Unix/Linux system and also on Windows computers provided the GNU tools are installed on the machine.
2.52.5 AOMX IN THE INTERNET
AOMX is available in the Internet at the address http://www.aomx.de. This site will provide selected examples and an angular overlap model (AOM) parameter database, and will be further developed to a forum for scientific exchange about AOM applications to TM complexes.
2.52.6 REFERENCES
1. Hoggard, P. E.Coord. Chem. Rev.1986,37a, 85–120.
2. Adamsky, H. Thesis, University of Du¨sseldorf, 1995.
3. Scha¨ffer, C. E. Struct. Bonding1968,5, 68–95.
4. Schmidtke, H.-H.Z. Naturforsch.1964,19a, 1502–1510.
5. Schmidtke, H.-H.; Adamsky, H.; Scho¨nherr, T.Bull. Chem. Soc. Jpn.1988,61, 59–65.
6. Hoggard, P. E.Z. Naturforsch.1982,37a, 85–94.
7. Atanasov, M.; Scho¨nherr, T.Inorg. Chem.1990,29, 4545–4552.
8. Smith, B. T.; Doyle, J. M.; Dongarra, J. J.; Ikebe, Y.; Klema, V. C.; Moler, C. B.EISPACK Guide; Springer-Verlag:
Berlin, 1976.
9. Wigner, E.Group Theory and its Applications to the Quantum Mechanics of Atomic Spectra; Academic Press: New York, 1959.
10. Powell, M. J. D.Comp. J.1964,7, 155–162.
#2003, Elsevier Ltd. All Rights Reserved
No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers
Comprehensive Coordination Chemistry II ISBN (set): 0-08-0437486 Volume 2, (ISBN 0-08-0443249); pp 661–664
2.53
GAMESS and MACMOLPLT
M. W. SCHMIDT, B. M. BODE, and M. S. GORDON Iowa State University, Ames, IA, USA
2.53 REFERENCES 666
The General Atomic and Molecular Electronic Structure System (GAMESS)is an ab initio quantum chemistry code that has been developed by the Gordon group since 1982. GAMESS is in use at well over 9,000 sites worldwide. These sites range from high schools to undergraduate colleges to research universities in the USA and worldwide to government laboratories to the private sector. Unlike nearly all such widely used codes, GAMESS is distributed at no cost to all users, simply by accessingwww.msg.ameslab.govand signing a license agreement.
An important feature of GAMESS is that an increasing number of its functionalities can be run on scalable computers, ranging from clusters of low-cost Linux computers to clusters of high-end workstations using advanced switching technology to massively parallel computers. The scal- ability of GAMESS is facilitated by the distributed data interface (DDI)functionality, developed by Graham Fletcher, which allows large data arrays to be distributed across all available nodes.
Scalable features in GAMESS that make use of DDI include Hartree-Fock, MCSCF, density functional theory (DFT), closed shell second order perturbation theory (MP2) energies and gradients, multi-reference MP2 (MRMP2)energies, and RHF Hessians.
While GAMESS was originally developed as a research tool to provide computational under- standing of such disparate areas as drug design, materials development, and condensed phase effects, in recent years attention has turned to the use of GAMESS for educational purposes. To this end, the full GAMESS code was migrated to both the Apple Macintosh and the PC. As a result, GAMESS is not only free, but it can be used effectively on low-cost, affordable hardware, allowing GAMESS to be run on computers ranging from individual PCs up through world class supercomputers at national centers. GAMESS is an especially user-friendly system on the Macin- tosh, since it is mouse driven and no knowledge of UNIX is required.
The utility of GAMESS for educational purposes has been dramatically increased by the development of the graphical interface MacMolPlt. MacMolPlt includes a simple molecule builder and input generator for GAMESS to prepare basic GAMESS input files. However, the real power of MacMolPlt is its ability to visualize the results of GAMESS calculations.
MacMolPlt accepts output files from any version of GAMESS run on any platform. From the output files, MacMolPlt can illustrate structures including real time rotations, can animate molecular vibrations, minimum energy reaction paths, and dynamics trajectories, and can plot (in both two- and three-dimensions)molecular orbitals, localized orbitals, total electron densities, density differences, and electrostatic potential maps. Each of the surfaces is computed on the fly from the geometry, basis set, and orbital vectors at the request of the user and thus does not need to be pre-computed or stored. MacMolPlt includes the ability to produce output in a variety of formats including high-resolution images for full color printouts and QuickTime movies of any molecular animation. The combination of GAMESS and MacMolPlt have proved to be very useful in
665
classes ranging from freshman majors to physical chemistry laboratory to graduate computational chemistry. Both codes are continually being improved, both with regard to new features and ease of use, by the Gordon group. Future extensions will include the interface of MacMolPlt to other computational chemistry codes and the extension of the input generator to a large subset of GAMESS methods.
GAMESS has the capability to perform molecular orbital calculations at a wide variety of sophistication levels, ranging from semi-empirical and molecular mechanics calculations to the highest level ofab initiotheory.Ab initiowavefunctions can be of the RHF, high-spin or low-spin ROHF, GVB, UHF, or MCSCF types. Analytic gradients are available for all wavefunctions, and analytic hessians are available for RHF, ROHF, and one-pair GVB wavefunctions. Numer- ical hessians can be calculated for all wavefunction types. Post-SCF correlation treatments include configuration interaction (CI), perturbation theory, and coupled clusters. Energies at second order in perturbation theory can be calculated for RHF, UHF, ROHF, and MCSCF wavefunc- tions. CI computations may follow all of these except UHF. A recent addition to GAMESS is the capability to perform a variety of coupled cluster calculations as developed by the Piecuch group at Michigan State University. DFT, a very popular alternative to traditional means for achieving some measure of electron correlation, has recently been added to GAMESS. A number of functionals have been implemented for RHF, UHF, and ROHF type Kohn-Sham calculations.
GAMESS can perform a wide range of quantum chemical calculations, including:
Optimization of molecular geometries in Cartesian or internal coordinates.
Determination of saddle points on potential energy surfaces.
Tracing intrinsic reaction paths for chemical reactions.
Vibrational analysis including infrared and Raman intensities.
Determination ofab initioclassical trajectories without prior knowledge of the potential energy surface.
Calculation of properties, such as multipole moments, electrostatic potentials, electric fields and electric field gradients, polarizabilities and hyperpolarizabilities, electron and spin densities, population analyses, localized orbitals, and spin-orbit coupling.
The evaluation of the effects of solvation on chemical processes.
The analysis of reactions and dynamics occurring on surfaces.
The features for treating solvation and surface chemistry deserve special mention. Both methods fall into the general class of methods referred to as QM/MM (quantum mechanics interfaced with molecular mechanics). The surface chemistry approach, called surface integrated molecular orbital molecular mechanics or SIMOMM, is a traditional QM/MM method, in which the QM/MM boundary is implemented using ‘‘link atoms.’’ Any QM method in GAMESS may be used as the embedded QM cluster, while the MM program TINKER is used for the MM part. The effective fragment potential (EFP)discrete solvation method is a very sophisticated approach, in which terms that account for Coulomb interactions, polarizability and exchange repulsionþcharge transfer are added to the ab initioHamiltonian. An interface that couples EFP with continuum methods is also available in GAMESS.
For more details about GAMESS and MacMolPlt, the reader is referred to the web page cited above and to the following references.1–7
ACKNOWLEDGMENTS
The development of GAMESS has been supported for more than 20 years by grants from the Air Force Office of Scientific Research, and more recently (for both GAMESS and MacMolPlt)by software development grants from the Department of Defense and the Department of Energy.
Hardware grants from these agencies and from IBM have also aided the code developments.
2.53 REFERENCES
1. Schmidt, M. W.; Baldridge, K. K.; Boatz, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.;
Nguyen, K. A.; Su, S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A., Jr.J. Comp. Chem.1993,14, 1347.
2. Schmidt, M. W.; Gordon, M. S.Annu. Rev. Phys. Chem.1998,49, 233.
3. Shoemaker, J.; Burggraf, L. W.; Gordon, M. S.J. Phys. Chem. A1999,103, 3245.
4. Bode, B. M.; Gordon, M. S.J. Mol. Graph.1999,16, 133.
5. Fletcher, G. D.; Schmidt, M. W.; Gordon, M. S.Adv. Chem. Phys.1999,110, 267.
6. Schmidt, M. W.; Fletcher, G. D.; Bode, B. M.; Gordon, M. S.Comput. Phys. Commun.2000,128, 190.
7. Gordon, M. S.; Freitag, M. A.; Bandyopadhyay, P.; Kairys, V.; Jensen, J. H.; Stevens, W. J.J. Phys. Chem.2001,105, 293.
#2003, Elsevier Ltd. All Rights Reserved
No part of this publication may be reproduced, stored in any retrieval system or transmitted in any form or by any means electronic, electrostatic, magnetic tape, mechanical, photocopying, recording or otherwise, without permission in writing from the publishers
Comprehensive Coordination Chemistry II ISBN (set): 0-08-0437486 Volume 2, (ISBN 0-08-0443249); pp 665–667
GAMESS and MACMOLPLT 667
2.54