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Molecular Properties and Conversion Factors These two papers are of general interest: A.D.Buckingham, J.Chem.Phys. 30, 1580-1585(1959). D.Neumann, J.W.Moskowitz J.Chem.Phys. 49, 2056-2070(1968). The first deals with multipoles, and the second with other properties such as electrostatic potentials. All units are derived from the atomic units for distance and the monopole electric charge, as given below. distance 1 au = 5.291771E-09 cm monopole 1 au = 4.803242E-10 esu 1 esu = sqrt(g-cm**3)/sec dipole 1 au = 2.541766E-18 esu-cm 1 Debye = 1.0E-18 esu-cm quadrupole 1 au = 1.345044E-26 esu-cm**2 1 Buckingham = 1.0E-26 esu-cm**2 octopole 1 au = 7.117668E-35 esu-cm**3 electric potential 1 au = 9.076814E-02 esu/cm electric field 1 au = 1.715270E+07 esu/cm**2 1 esu/cm**2 = 1 dyne/esu electric field gradient 1 au = 3.241390E+15 esu/cm**3 The atomic unit for the total electron density is electron/bohr**3, but 1/bohr**3 for an orbital density. The atomic unit for spin density is excess alpha spins per unit volume, h/4*pi*bohr**3. Only the expectation value is computed, with no constants premultiplying it. IR intensities are printed in Debye**2/amu-Angstrom**2. These can be converted into intensities as defined by Wilson, Decius, and Cross's equation 7.9.25, in km/mole, by multiplying by 42.255. If you prefer 1/atm-cm**2, use a conversion factor of 171.65 instead. A good reference for deciphering these units is A.Komornicki, R.L.Jaffe J.Chem.Phys. 1979, 71, 2150-2155. A reference showing how IR intensities change with basis improvements at the HF level is Y.Yamaguchi, M.Frisch, J.Gaw, H.F.Schaefer, J.S.Binkley, J.Chem.Phys. 1986, 84, 2262-2278. Raman activities in A**4/amu multiply by 6.0220E-09 for units of cm**4/g. One of the many sources explaining how activity relates to intensity is D.Michalska, R.Wysokinski Chem.Phys.Lett. 403, 211-217(2005) Polarizabilities Static polarizabilities are named alpha, beta, and gamma; and are called the polarizability, hyperpolarizability, and second hyperpolarizability. They are the 2nd, 3rd, and 4th derivatives of the energy with respect to applied uniform electric fields, with the 1st derivative being the dipole moment! It might be worth mentioning that a uniform field can be applied using $EFIELD, if you wish to develop custom usages, but $EFIELD is not to be given for any kind of run discussed below. The general approach to computing static polarizabilities is numerical differentiation, namely RUNTYP=FFIELD, which should work for any energy method provided by GAMESS. A sequence of computations with fields applied in the x, y, and/or z directions will generate the tensors. See $FFCALC for details. Analytic computation of all three tensors is available for SCFTYP=RHF only, see RUNTYP=TDHF and $TDHF. If you need to know just the alpha polarizability, see POLAR in $CPHF during any analytic hessian job. A break down of the static alpha polarizability in terms of contributions from individual localized orbitals can be obtained by setting POLDCM=.TRUE. in $LOCAL. Calculation will be by analytic means, unless POLNUM in that group is selected. This option is available only for SCFTYP=RHF. The keyword LOCHYP in $FFCALC gives a similar analysis for all three static polarizabilities, determined by numerical differentiation. Polarizabilities in a static electric field differ from those in an oscillating field such as a laser produces. For RHF cases only, a variety of frequency dependent polarizabilities alpha, beta, and gamma can be generated, depending on the experiment. A particularly easy one to describe is 'second harmonic generation', governed by a beta tensor describing the absorption of two photons and the emission of one at doubled frequency. See RUNTYP=TDHF, and papers listed under $TDHF, for many other non-linear optical experiments. Nuclear derivatives of the dipole moment and the various polarizabilities are also of interest. For example, knowledge of the derivative of the dipole with respect to nuclear coordinates yields the IR intensity. Similarly, the nuclear derivative of the static alpha polarizability gives Raman intensities: see RUNTYP=RAMAN. Analytic of 1st or 2nd nuclear derivatives of static or frequency dependent polarizabilities are available for SCFTYP=RHF only, see RUNTYP=TDHFX and $TDHFX, giving rise to experimental observations such as resonance Raman and hyper-Raman. Finally, instead of considering polarizabilities as a function of real frequencies, they can be considered to be dependent on the imaginary frequency. The imaginary frequency dependent alpha polarizability can be computed analytically for SCFTYP=RHF only, using POLDYN=.TRUE. in $LOCAL. Integration of this quantity over the imaginary frequency domain can be used to extract C6 dispersion constants. Polarizabilities are tensor quantities. There are a number of different ways to define them, and various formulae to extract "scalar" and "vector" quantites from the tensors. A good reference for learning how to compare the output of a theoretical program to experiment is A.Willetts, J.E.Rice, D.M.Burland, D.P.Shelton J.Chem.Phys. 97, 7590-7599(1992) Localized Molecular Orbitals Three different orbital localization methods are implemented in GAMESS. The energy and dipole based methods normally produce similar results, but see M.W.Schmidt, S.Yabushita, M.S.Gordon in J.Chem.Phys., 1984, 88, 382-389 for an interesting exception. You can find references to the three methods at the beginning of this chapter. The method due to Edmiston and Ruedenberg works by maximizing the sum of the orbitals' two electron self repulsion integrals. Most people who think about the different localization criteria end up concluding that this one seems superior. The method requires the two electron integrals, transformed into the molecular orbital basis. Because only the integrals involving the orbitals to be localized are needed, the integral transformation is actually not very time consuming. The Boys method maximizes the sum of the distances between the orbital centroids, that is the difference in the orbital dipole moments. The population method due to Pipek and Mezey maximizes a certain sum of gross atomic Mulliken populations. This procedure will not mix sigma and pi bonds, so you will not get localized banana bonds. Hence it is rather easy to find cases where this method give different results than the Ruedenberg or Boys approach. GAMESS will localize orbitals for any kind of RHF, UHF, ROHF, or MCSCF wavefunctions. The localizations will automatically restrict any rotation that would cause the energy of the wavefunction to be changed (the total wavefunction is left invariant). As discussed below, localizations for GVB or CI functions are not permitted. The default is to freeze core orbitals. The localized valence orbitals are scarcely changed if the core orbitals are included, and it is usually convenient to leave them out. Therefore, the default localizations are: RHF functions localize all doubly occupied valence orbitals. UHF functions localize all valence alpha, and then all valence beta orbitals. ROHF functions localize all valence doubly occupied orbitals, and all singly occupied orbitals, but do not mix these two orbital spaces. MCSCF functions localize all valence MCC type orbitals, and localize all active orbitals, but do not mix these two orbital spaces. To recover the invariant MCSCF function, you must be using a FORS=.TRUE. wavefunction, and you must set GROUP=C1 in $DRT, since the localized orbitals possess no symmetry. In general, GVB functions are invariant only to localizations of the NCO doubly occupied orbitals. Any pairs must be written in natural form, so pair orbitals cannot be localized. The open shells may be degenerate, so in general these should not be mixed. If for some reason you feel you must localize the doubly occupied space, do a RUNTYP=PROP job. Feed in the GVB orbitals, but tell the program it is SCFTYP=RHF, and enter a negative ICHARG so that GAMESS thinks all orbitals occupied in the GVB are occupied in this fictitous RHF. Use NINA or NOUTA to localize the desired doubly occupied orbitals. Orbital localization is not permitted for CI, because we cannot imagine why you would want to do that anyway. Boys localization of the core orbitals in molecules having elements from the third or higher row almost never succeeds. Boys localization including the core for second row atoms will often work, since there is only one inner shell on these. The Ruedenberg method should work for any element, although including core orbitals in the integral transformation is more expensive. The easiest way to do localization is in the run which generates the wavefunction, by selecting LOCAL=xxx in the $CONTRL group. However, localization may be conveniently done at any time after determination of the wavefunction, by executing a RUNTYP=PROP job. This will require only $CONTRL, $BASIS/$DATA, $GUESS (pick MOREAD), the converged $VEC, possibly $SCF or $DRT to define your wavefunction, and optionally some $LOCAL input. There is an option to restrict all rotations that would mix orbitals of different symmetries. SYMLOC=.TRUE. yields only partially localized orbitals, but these still possess symmetry. They are therefore very useful as starting orbitals for MCSCF or GVB-PP calculations. Because they still have symmetry, these partially localized orbitals run as efficiently as the canonical orbitals. Because it is much easier for a user to pick out the bonds which are to be correlated, a significant number of iterations can be saved, and convergence to false solutions is less likely. * * * The most important reason for localizing orbitals is to analyze the wavefunction. A simple example is to look at shapes of the orbitals with the MacMolPlt program. Or, you might read the localized orbitals in during a RUNTYP=PROP job to examine their Mulliken populations. Localized orbitals are a particularly interesting way to analyze MCSCF computations. The localized orbitals may be oriented on each atom (see option ORIENT in $LOCAL) to direct the orbitals on each atom towards their neighbors for maximal bonding, and then print a bond order analysis. The orientation procedure is newly programmed by J.Ivanic and K.Ruedenberg, to deal with the situation of more than one localized orbital occuring on any given atom. Some examples of this type of analysis are D.F.Feller, M.W.Schmidt, K.Ruedenberg J.Am.Chem.Soc. 104, 960-967 (1982) T.R.Cundari, M.S.Gordon J.Am.Chem.Soc. 113, 5231-5243 (1991) N.Matsunaga, T.R.Cundari, M.W.Schmidt, M.S.Gordon Theoret.Chim.Acta 83, 57-68 (1992). In addition, the energy of your molecule can be partitioned over the localized orbitals so that you may be able to understand the origin of barriers, etc. This analysis can be made for the SCF energy, and also the MP2 correction to the SCF energy, which requires two separate runs. An explanation of the method, and application to hydrogen bonding may be found in J.H.Jensen, M.S.Gordon, J.Phys.Chem. 99, 8091-8107(1995). Analysis of the SCF energy is based on the localized charge distribution (LCD) model: W.England and M.S.Gordon, J.Am.Chem.Soc. 93, 4649-4657 (1971). This is implemented for RHF and ROHF wavefunctions, and it requires use of the Ruedenberg localization method, since it needs the two electron integrals to correctly compute energy sums. All orbitals must be included in the localization, even the cores, so that the total energy is reproduced. The LCD requires both electronic and nuclear charges to be partitioned. The orbital localization automatically accomplishes the former, but division of the nuclear charge may require some assistance from you. The program attempts to correctly partition the nuclear charge, if you select the MOIDON option, according to the following: a Mulliken type analysis of the localized orbitals is made. This determines if an orbital is a core, lone pair, or bonding MO. Two protons are assigned to the nucleus to which any core or lone pair belongs. One proton is assigned to each of the two nuclei in a bond. When all localized orbitals have been assigned in this manner, the total number of protons which have been assigned to each nucleus should equal the true nuclear charge. Many interesting systems (three center bonds, back- bonding, aromatic delocalization, and all charged species) may require you to assist the automatic assignment of nuclear charge. First, note that MOIDON reorders the localized orbitals into a consistent order: first comes any core and lone pair orbitals on the 1st atom, then any bonds from atom 1 to atoms 2, 3, ..., then any core and lone pairs on atom 2, then any bonds from atom 2 to 3, 4, ..., and so on. Let us consider a simple case where MOIDON fails, the ion NH4+. Assuming the nitrogen is the 1st atom, MOIDON generates NNUCMO=1,2,2,2,2 MOIJ=1,1,1,1,1 2,3,4,5 ZIJ=2.0,1.0,1.0,1.0,1.0, 1.0,1.0,1.0,1.0 The columns (which are LMOs) are allowed to span up to 5 rows (the nuclei), in situations with multicenter bonds. MOIJ shows the Mulliken analysis thinks there are four NH bonds following the nitrogen core. ZIJ shows that since each such bond assigns one proton to nitrogen, the total charge of N is +6. This is incorrect of course, as indeed will always happen to some nucleus in a charged molecule. In order for the energy analysis to correctly reproduce the total energy, we must ensure that the charge of nitrogen is +7. The least arbitrary way to do this is to increase the nitrogen charge assigned to each NH bond by 1/4. Since in our case NNUCMO and MOIJ and much of ZIJ are correct, we need only override a small part of them with $LOCAL input: IJMO(1)=1,2, 1,3, 1,4, 1,5 ZIJ(1)=1.25, 1.25, 1.25, 1.25 which changes the charge of the first atom of orbitals 2 through 5 to 5/4, changing ZIJ to ZIJ=2.0,1.25,1.25,1.25,1.25, 1.0, 1.0, 1.0, 1.0 The purpose of the IJMO sparse matrix pointer is to let you give only the changed parts of ZIJ and/or MOIJ. Another way to resolve the problem with NH4+ is to change one of the 4 equivalent bond pairs into a "proton". A "proton" orbital AH treats the LMO as if it were a lone pair on A, and so assigns +2 to nucleus A. Use of a "proton" also generates an imaginary orbital, with zero electron occupancy. For example, if we make atom 2 in NH4+ a "proton", by IPROT(1)=2 NNUCMO(2)=1 IJMO(1)=1,2,2,2 MOIJ(1)=1,0 ZIJ(1)=2.0,0.0 the automatic decomposition of the nuclear charges will be NNUCMO=1,1,2,2,2,1 MOIJ=1,1,1,1,1,2 3,4,5 ZIJ=2.0,2.0,1.0,1.0,1.0,1.0 1.0,1.0,1.0 The 6th column is just a proton, and the decomposition will not give any electronic energy associated with this "orbital", since it is vacant. Note that the two ways we have disected the nuclear charges for NH4+ will both yield the correct total energy, but will give very different individual orbital components. Most people will feel that the first assignment is the least arbitrary, since it treats all four NH bonds equivalently. However you assign the nuclear charges, you must ensure that the sum of all nuclear charges is correct. This is most easily verified by checking that the energy sum equals the total SCF energy of your system. As another example, H3PO is studied in EXAM26.INP. Here the MOIDON analysis decides the three equivalent orbitals on oxygen are O lone pairs, assigning +2 to the oxygen nucleus for each orbital. This gives Z(O)=9, and Z(P)=14. The least arbitrary way to reduce Z(O) and increase Z(P) is to recognize that there is some backbonding in these "lone pairs" to P, and instead assign the nuclear charge of these three orbitals by 1/3 to P, 5/3 to O. Because you may have to make several runs, looking carefully at the localized orbital output before the correct nuclear assignments are made, there is an option to skip directly to the decomposition when the orbital localization has already been done. Use $CONTRL RUNTYP=PROP $GUESS GUESS=MOREAD NORB= $VEC containing the localized orbitals! $TWOEI The latter group contains the necessary Coulomb and exchange integrals, which are punched by the first localization, and permits the decomposition to begin immediately. SCF level dipoles can also be analyzed using the DIPDCM flag in $LOCAL. The theory of the dipole analysis is given in the third paper of the LCD sequence. The following list includes application of the LCD analysis to many problems of chemical interest: W.England, M.S.Gordon J.Am.Chem.Soc. 93, 4649-4657 (1971) W.England, M.S.Gordon J.Am.Chem.Soc. 94, 4818-4823 (1972) M.S.Gordon, W.England J.Am.Chem.Soc. 94, 5168-5178 (1972) M.S.Gordon, W.England Chem.Phys.Lett. 15, 59-64 (1972) M.S.Gordon, W.England J.Am.Chem.Soc. 95, 1753-1760 (1973) M.S.Gordon J.Mol.Struct. 23, 399 (1974) W.England, M.S.Gordon, K.Ruedenberg, Theoret.Chim.Acta 37, 177-216 (1975) J.H.Jensen, M.S.Gordon, J.Phys.Chem. 99, 8091-8107(1995) J.H.Jensen, M.S.Gordon, J.Am.Chem.Soc. 117, 8159-8170(1995) M.S.Gordon, J.H.Jensen, Acc.Chem.Res. 29, 536-543(1996) * * * It is also possible to analyze the MP2 correlation correction in terms of localized orbitals, for the RHF case. The method is that of G.Peterssen and M.L.Al-Laham, J.Chem.Phys., 94, 6081-6090 (1991). Any type of localized orbital may be used, and because the MP2 calculation typically omits cores, the $LOCAL group will normally include only valence orbitals in the localization. As mentioned already, the analysis of the MP2 correction must be done in a separate run from the SCF analysis, which must include cores in order to sum up to the total SCF energy. * * * Typically, the results are most easily interpreted by looking at "the bigger picture" than at "the details". Plots of kinetic and potential energy, normally as a function of some coordinate such as distance along an IRC, are the most revealing. Once you determine, for example, that the most significant contribution to the total energy is the kinetic energy, you may wish to look further into the minutia, such as the kinetic energies of individual localized orbitals, or groups of LMOs corresponding to an entire functional group.