Currently the INDO- and NDDO-based methods are most frequently used for examining TM species. The CNDO-based methods, which were popular in the 1970s and 1980s to study molecular systems containing TM elements, are rarely used nowadays. More accurate INDO and NDDO methods have replaced CNDO methods at small cost in computing time.
The availability of TM and lanthanide elements in semiempirical SCF MO methods is shown in Table 2. See Chapter 2.58 for information about different quantum mechanical program packages.
Note that atomic parameters and element availability may differ in different program packages.
2.38.3.1 INDO/1
This semiempirical method (often referred to as ZINDO/1), although not specifically parameter- ized to favor any experimental quantity, can be used for calculation of binding energies, geometries, ionization energies, and dipole moments of TM and lanthanide species.76–82 In INDO/1, the one- center TERIs are calculated theoretically over Slater-type AOs and scaled (if necessary).
Table 2 Parameter availability for TM elements and the lanthanides in modern semiempirical SCF MO methods.
TM elements
Method First row Second row Third row Lanthanides
AM1 Zna Hga Eu,bGdb
AM1/dc Sc–Cr, Fe–Cu Zr, Mo, Pd, Ag Pt
CINDO-E/S Ti–Ni Zr, Mo, Ru–Pd
CNDO-S2 Fe,Ni Pd
GRINDOL Ti–Ni Y–Cd
MINDO/3 Zna
MINDO/SR Sc–V, Fe–Ni
MNDO Zna Hga
MNDO/dd Sc–Cr, Fe–Cu, Zna Zr, Mo, Pd, Ag, Cda Hf, Pt, Hga
MSINDO Sc–Zn
NDDO/MC Cr–Ni Ag
PM3 Zna Cda Hga
PM3d Cr
PM3(tm) Ti–Cu Zr, Mo–Pd Hf–Pt Gd
PM5 Sc–Cr, Fe–Cu, Zna Zr, Mo, Pd, Ag, Cda Pt, Hga
SAM1 Fe, Cu, Zn
SINDO1 Sc–Zn
(Z)INDO1 and (Z)INDO/S
Sc–Zn Y–Cd Au Ce–Lu
a Aspbasis is employed for these elements. b Using the ‘‘sparkle’’ model.150,151 c Only parameters for Mo are published and recommended for general use; parameters for other TM elements are preliminary.290 d Only parameters for Zn, Cd, and Hg are recommended for general use; parameters for other TM elements are preliminary.286,290
Semiempirical SCF MO Methods, Electronic Spectra, and Configurational Interaction 475
In general, INDO/1-predicted geometries agree with experimental data reasonably well with the average absolute error in bond lengths around 5%.80,81,277 The calculated stretching frequencies are overestimated by approximately 40%.81 INDO/1 can also reliably predict spin states in a variety of different coordination environments.225 INDO/1 is available in HyperChem and the ZINDO semiempirical package.
2.38.3.2 INDO/S
This is a popular semiempirical method (often referred to as ZINDO/S or ZINDO) for calcula- tion of electronic spectra of both organic molecules and TM species.60–75,277,278,320
The INDO/S parametrization was carried out at the CIS level (see Section 2.38.4.2). The Slater–Condon integrals, which are used to evaluate the TERIs, were taken from atomic spectroscopy data.
The calculated transition energies are chosen to match energies of absorption maxima, !max, as opposed to absorption band origins.
Instead of standard overlap integrals S, the ‘‘weighted’’ overlap SS is adopted in INDO/S:11,60–62,64
S SẳXl
mẳ0
fl;mgSðmịðmị ð42ị
wheregare the Eulerian rotation factors required to convert from the local diatomic coordinate system to the molecular system, S(m)(m)are theororcomponents of the overlap integrals in the local diatomic system, andfl,mare empirical weighting factors for(mẳ0), (mẳ1), and (mẳ2) types of interactions (lẳ0, 1, 2 for s-, p-, and d-type orbitals respectively). They are chosen to reproduce the MO energy spread of model abinitio calculations.60–62 Usually, f1,0ẳ1.267,f1,1ẳ0.585–0.68, and all other fl,mẳ1.
Generally, INDO/S reproduces the energies of electronic transitions below 45,000 cm1within 2,000 cm1.11,60–66,69–75,277,278
Charge transfer (CT) bands in non-solvatochromic systems andd–d transitions are particularly well reproduced. The oscillator strengths calculated using the dipole length approximation at the CIS level are usually overestimated by a factor of 2–3.11,60,69,277,278
The INDO/S calculations with the random-phase approximation (RPA) (see Section 2.38.4.3.1) produce more accurate transition intensities.69,72 Solvent effects on absorption spectra can be estimated by using the self-consistent reaction field (SCRF) method279,280 or other approaches.71,74,280–284
INDO/S is available in Gaussian 98, HyperChem, and the ZINDO and CNDO/INDOsemiempirical packages.
2.38.3.3 CINDO-E/S
This is another method (also referred to as CINDO/S) for calculation of electronic spectra of TM complexes.180–197 The method uses a mixed INDO/CNDO approximation (INDO/S for TM elements and CNDO/S for the main-group elements). Generally, CINDO-E/S results are similar to Zerner’s INDO/S.
CINDO-E/S parameters are available for first- and second-row TM elements.183,184 User- friendly software for performing CINDO-E/S calculations has never been distributed.
2.38.3.4 MINDO/SR
MINDO/SR158–172is an extension of MINDO/344–50that permits the treatment of TM compounds.
One of the features of the method is that the matrix elementsHcore involving an orbital of a TM atom are assumed to be proportional to S/RAB, where S is the overlap integral between the AOsandcentered on the atomsAandBrespectively, andRABis their internuclear distance.158 MINDO/SR parameters are available for a majority of first-row TM elements. The MINDO/SR code has not been distributed.
2.38.3.5 SINDO1
SINDO189–100is a semiempirical method based on the INDO model and derives its name from an approximate transformation to symmetrically orthogonalized orbitals. Since the new version of this method, MSINDO (see Section 2.38.3.6), is more accurate,105 the newly developed version should be used instead of SINDO1.
2.38.3.6 MSINDO
MSINDO101–105 is a new semiempirical method and is based on SINDO1. Two sets of orbital exponents are employed in MSINDO: one for calculating one-center integrals and another for calculating two-center integrals. Orthogonalization corrections are restricted to the core Hamiltonian matrix elements. The set of atomic parameters is increased (comparing with SINDO1), however bond parameters97 are no longer used.101MSINDO parameters are available for first-row TM elements.105 The parameter development for second-row TM elements is to begin in 2002.285
MSINDO provides significant improvements over SINDO1 for predicting ground-state proper- ties (heats of formation, geometries, ionization energies, and dipole moments) of molecules. Mean errors for heats of formation and bond lengths for a reference molecule set, which includes all 10 first-row transition elements, are 7.4 kcal mol1 and 0.03 A˚ respectively. For SINDO/1, the corresponding mean errors are 10.7 kcal mol1and 0.07 A˚. The MSINDO code and documenta- tion is available from the authors.
2.38.3.7 MNDO/d MNDO/d228–234,157,158
is an extension of the MNDO method.112–131 The implementation of the method is analogous to MNDO, with minor variations. For the first- and second-period elements, MNDO/d uses the same parameters as MNDO. MNDO/d parameters have been published for Zn, Cd, and Hg.233 Currently, these are the only TM parameters that are recommended for general use.286There are preliminary parameter sets for other TM elements (e.g., Ti, Fe, Ni, Cu, Zr, Ag, Hf ) which are listed in the MNDO97 code.234 The parameter development for Cr, Mo, W, Ru, and Rh is in progress in Thiel’s research group.286,287 The MNDO/d parameters for 16 TM elements have been independently developed by J. J. P. Stewart.290MNDO/d is available in AMPAC, HyperChem 7, MOPAC, Spartan, Titan,and UniChem.
2.38.3.8 AM1/d
AM1/d239 is an extension of the AM1 method137–153 to a spd basis. The AM1 formalism and parametrization are unchanged for all main-group elements. The method uses the multipole model228 to calculate the TERIs and was parametrized to reproduce the heats of formation and equilibrium geometries. AM1/d parameters are available for 16 TM elements.244,290 Cur- rently, only the Mo parameters have been published and are recommended for general use.239 Parameter development for other TM elements is in progress.288,290 Mean absolute errors for heats of formation and bond lengths for a reference set, which includes Mo species, are 6.5 kcal mol1(50 comparisons) and 0.044 A˚ (83 comparisons) respectively.239AM1/d is available inMOPAC 2000or higher.
2.38.3.9 PM3(tm)
PM3(tm)217–227 is an extension of the PM3 method.152–157 The PM3 parametrization are unchanged for all main-group elements, Zn, Cd, and Hg. PM3(tm) uses the multipole model228 to calculate the TERIs and was parametrized to reproduce equilibrium and transi- tion-state geometries of TM and organometallic species. Thus, this method focuses on structural predictions.
In general, PM3(tm)-predicted geometries agree with experimental data well with the average error in bond lengths around 3%. PM3(tm) parameters are available for most TM elements.219 Semiempirical SCF MO Methods, Electronic Spectra, and Configurational Interaction 477
It appears that the predictive ability is excellent for the group 4–9 TMs but diminishes for the other TMs.223The method can also reliably predict spin states in a variety of different coordin- ation environments.225PM3(tm) calculations can be performed usingHyperChem 7, Spartan, and Titan. PM3(tm) is not available by default in MOPAC but can be added via an external parameter file.
2.38.3.10 PM3d
PM3d240,241 is another extension of the PM3 method for TM species. Instead of the multipole model,228the Klopman–Ohno potential expansion into the real spherical harmonics272is used to calculate the TERIs. The method was parametrized to reproduce the heats of formation, ioniza- tion potentials, dipole moments, and equilibrium geometries. The parameters are available for C, N, O, H, and Cr.240,241No further parameter development is planned.289 Mean absolute errors for heats of formation and bond lengths for a reference molecule set, which includes 23 Cr compounds, are 14.8 kcal mol1 and 0.05 A˚ respectively.241 PM3d is available from the authors (http://quark.unn.runnet.ru/TCG_SOFTWARE.htm).
2.38.3.11 PM5
PM5 is a new re-parametrization of the MNDO method.244 Currently, PM5 parameters are available for main-group elements and 16 TM elements, however the parameters have not been published yet.290Average errors for heats of formation, ionization potentials, and bond lengths in the new parametrization are reduced quite considerably compared to those from MNDO/d, AM1/d, and PM3 calculations.244,290PM5 is available in MOPAC 2002.
2.38.3.12 NDDO/MC
This NDDO-based method is available for prediction of binding energies and molecular geom- etries.198–206 Its features include orthogonalization corrections to the core Hamiltonian matrix, the use of a modified formula for the resonance integral, and the model Coulomb hole function (Equation (29)) in the electronic repulsion potential in calculations of the TERIs.
Mean absolute errors for binding energies and bond lengths for a reference molecule set, which includes 36 Co and Ni compounds, are 5.0 kcal mol1 and 0.064 A˚ respectively.198 NDDO/MC has been also applied to calculate transition energies using the CIS approach (see Section 2.38.4.2).206 User-friendly software for performing NDDO/MC calculations has never been distributed.
2.38.3.13 SAM1
SAM1 is an expansion of the NDDO model with focus on thermochemical and structural predictions.207–216 It can also be useful in the calculation of vibrational frequencies, as long as certain deficiencies are recognized.210In SAM1, the TERIs are calculated using a minimal basis set of Gaussian functions to compute these integrals directly, limiting the actual number of integrals calculated by the NDDO approximation. The values of the computed integrals are then scaled.
SAM1 differs from AM1 in derivation of the one-center TERIs. At the first stage of the parametrization, Slater orbital exponent values are derived using atomic data. These orbital exponents are then used to calculate all the one-center two-electron integrals in the spd basis and these values are then fixed. The second stage is a ‘‘molecular’’ parametrization where the usual parameters from the NDDO models are augmented with the needed parameters for SAM1.
The method was parametrized more carefully and with more extensive searches of the para- meter hypersurface. SAM1 parameters are only available for 3 TM elements (Fe, Cu, and Zn). No further parameter development is planned.291SAM1 is available inAMPAC.