3 Electronic Structure
3.11 Parameterization
the one-centre integrals, and are again parameterized as in eq. (3.88). The approxi- mations for the one-electron integrals in CNDO are the same as for INDO. The Pariser–Pople–Parr (PPP) method can be considered as a CNDO approximation where only π-electrons are treated.
The main difference between CNDO, INDO and NDDO is in the treatment of the two-electron integrals. While CNDO and INDO reduce these to just two parameters (gAA and gAB), all the one- and two-centre integrals are retained in the NDDO approximation. Within an sp-basis, however, there are only 27 different types of one- and two-centre integrals, while the number rises to over 500 for a basis contain- ing both s-,p- and d-functions.
The group centred around M. J. S. Dewar has used a combination of methods (2) and (3) for assigning parameter values, resulting in a class of commonly used methods.
The molecular data used for parameterization are geometries, heats of formation, dipole moments and ionization potentials. These methods are denoted “modified” as their parameters have been obtained by fitting.
3.11.1 Modified Intermediate Neglect of Differential Overlap (MINDO)
Three versions of Modified Intermediate Neglect of Differential Overlap (MINDO) models exist, MINDO/1, MINDO/2 and MINDO/3. The first two attempts at parame- terizing INDO gave quite poor results, but MINDO/3, introduced in 1975,45produced the first general purpose quantum chemical method that could successfully predict molecular properties at a relatively low computational cost. The parameterization of MINDO contains diatomicvariables in the two-centre one-electron term, thus the bAB
parameters must be derived for all pairsof bonded atoms. The Imparameters are ion- ization potentials.
(3.89) MINDO/3 has been parameterized for H, B, C, N, O, F, Si, P, S and Cl, although certain combinations of these elements have been omitted. MINDO/3 is rarely used in modern computational chemistry, having been succeeded in accuracy by the NDDO methods below. Since there are parameters in MINDO that depend on two atoms, the number of parameters rises as the square of the number of elements. It is unlikely that MINDO will be parameterized beyond the abovementioned in the future.
3.11.2 Modified NDDO models
The MNDO, AM1 and PM3 methods46are parameterizations of the NDDO model where the parameterization is in terms of atomicvariables, i.e. referring only to the nature of a single atom. MNDO, AM1 and PM3 are derived from the same basic approximations (NDDO), and differ only in the way in which the core–core repulsion is treated and in how the parameters are assigned. Each method considers only the valence s- and p-functions, which are taken as Slater type orbitals with corresponding exponents zsand zp.
The one-centre one-electron integrals have a value corresponding to the energy of a single electron experiencing the nuclear charge (Usor Up) plus terms from the poten- tial due to all the other nuclei in the system (eq. (3.83)). The latter is parameterized in terms of the (reduced) nuclear charges Z′and a two-electron integral.
(3.90)
h U Z
U
a a N
mn mn m
m
m n d m m n n
m n
= = − ′
= − ∇ −
∑
≠A B A A A A
A
A A A
nuclei
h
1 V
2 2
m n m n
b m n
mn m n
mn
A B A A B B
AB
A B
h = − ∇ −V −V
= ( + )
=
1 2
2
S I I
S
The two-centre one-electron integrals given by the second equation in eq. (3.83) are written as a product of the corresponding overlap integral multiplied by the average of two atomic “resonance” parameters,b.
(3.91) The overlap element Smn is calculated explicitly (note that this is not consistent with the ZDO approximation, and the inclusion is the origin of the “Modified”
label).
There are only five types of one-centre two-electron integralssurviving the NDDO approximation within a sp-basis (eq. (3.85)).
(3.92)
The G-type parameters are Coulomb terms, while the H parameter is an exchange integral. The Gp2 integral involves two different types of p-functions (i.e. px, py or pz).
There are a total of 22 different two-centre two-electron integralsarising from an sp- basis, and these are modelled as interactions between multipoles. Electron 1 in an
〈sm|sm〉type integral, for example, is modelled as a monopole, in an 〈sm|pm〉type inte- gral as a dipole and in a 〈pm|pm〉type integral as a quadrupole. The dipole and quadru- pole moments are generated as fractional charges located at specific points away from the nuclei, where the distance is determined by the orbital exponents zsand zp. The main reason for adapting a multipole expansion of these integrals was the limited com- putational resources available when these methods were developed initially. In the limit of the two nuclei being placed on top of each other, a two-centre two-electron integral becomes a one-centre two-electron integral, which puts boundary conditions on the functional form of the multipole interaction. The bottom line is that all two- centre two-electron integrals are written in terms of the orbital exponents and the one- centre two-electron parameters given in eq. (3.92).
The core–core repulsionis the repulsion between nuclear charges, properly reduced by the number of core electrons. The “exact” expression for this term is simply the product of the charges divided by the distance, Z′AZ′B/RAB. Due to the inherent approximations in the NDDO method, however, this term is not cancelled by electron–electron terms at long distances, resulting in a net repulsion between uncharged molecules or atoms even when their wave functions do not overlap. The core–core term must consequently be modified to generate the proper limiting behav- iour, which means that two-electron integrals must be involved. The specific functional form depends on the exact method, and is given below.
Each of the MNDO, AM1 and PM3 methods involves at least 12 parameters per atom, orbital exponents:zs/p; one-electron terms:Us/pand bs/p; two-electron terms:Gss, Gsp,Gpp,Gp2,Hsp; and parameters used in the core–core repulsion,a, and for the AM1 and PM3 methods also a,band cconstants, as described below.
ss ss G sp sp G ss pp H pp pp G pp pp G
ss sp sp pp p
=
=
=
=
′ ′ = 2
mAhnB = 12Smn(bm+bn)
3.11.3 Modified Neglect of Diatomic Overlap (MNDO)
The core–core repulsion of the Modified Neglect of Diatomic Overlap (MNDO) model47has the form given in eq. (3.93).
(3.93) The aexponents are taken as fitting parameters.
Interactions involving O—H and N—H bonds are treated differently.
(3.94) In addition, MNDO uses the approximation zs=zpfor some of the lighter elements.
The Gss,Gsp,Gpp,Gp2 and Hspparameters are taken from atomic spectra, while the others are fitted to molecular data. Although MNDO has been succeeded by the AM1 and PM3 methods, it is still used for some types of calculations where MNDO is known to give better results.
Some known limitations of the MNDO model are:
(1) Branched and sterically crowded hydrocarbons (such as neopentane) are pre- dicted to be too unstable, relative to their straight-chain analogues.
(2) Four-membered rings are too stable.
(3) Weak interactions are unreliable, for example MNDO does not predict hydrogen bonds.
(4) Hypervalent molecules, such as sulfoxides and sulfones, are too unstable.
(5) Activation energies for bond breaking/forming reactions are too high.
(6) Non-classical structures are predicted to be unstable relative to classical struc- tures (for example ethyl radical).
(7) Proton affinities are poorly predicted.
(8) Oxygen-containing substituents on aromatic rings are out-of-plane (for example nitrobenzene).
(9) Peroxide bonds are too short by ~0.17 Å
(10) The C—X—C angle in ethers and sulfides is too large by ~9°.
MNDOC48(C for correlation) has the same functional form as MNDO, however, elec- tron correlation is explicitly calculated by second-order perturbation theory. The derivation of the MNDOC parameters is done by fitting the correlated MNDOC results to experimental data. Electron correlation in MNDO is only included implic- itly via the parameters, from fitting to experimental results. Since the training set only includes ground state stable molecules, MNDO has problems treating systems where the importance of electron correlation is substantially different from “normal” mole- cules. MNDOC consequently performs significantly better for system where this is not the case, such as transition structures and excited states.
3.11.4 Austin Model 1 (AM1)
After some experience with MNDO, it became clear that there were certain system- atic errors. For example, the repulsion between two atoms that are 2–3 Å apart is too high. This has as a consequence that activation energies in general are too large. The source was traced to a too repulsive interaction in the core–core potential. In order to
VnnMNDO(A H, )= ′ ′Z ZA H s sA A s sH H
(
1+RAHe−aA AHR +e−aH AHR)
VnnMNDO(A B, )= ′ ′Z Z s sA B A A s sB B
(
1+e−aA ABR +e−aB ABR)
remedy this, the core–core function was modified by adding Gaussian functions, and the whole model was re-parameterized. The result was called Austin Model 1(AM1)49, in honour of Dewar’s move to the University of Austin at the time. The core–core repulsion of AM1 has the form given in eq. (3.95).
(3.95) Here kis between 2 and 4, depending on the atom. It should be noted that the Gauss- ian functions were added more or less as patches onto the underlying parameters, which explains why a different number of Gaussians is used for each atom. As for MNDO, the Gss,Gsp,Gpp,Gp2andHspparameters are taken from atomic spectra, while the rest, including the ak,bkand ckconstants, are fitted to molecular data.
Some known improvements and limitations of the AM1 model are:
(1) AM1 does predict hydrogen bonds with an approximately correct strength, but the geometry is often wrong.
(2) Activation energies are much improved over MNDO.
(3) Hypervalent molecules are improved over MNDO, but still have significantly larger errors than other types of compounds.
(4) Alkyl groups are systematically too stable by ~8 kJ/mol per CH2group.
(5) Nitro compounds are systematically too unstable.
(6) Peroxide bonds are too short by ~0.17 Å.
(7) Phosphor compounds have problems when atoms are ~3 Å apart, producing wrong geometries. P4O6for example is predicted to have P—P bonds differing by 0.4 Å, although experimentally they are identical.
(8) The gaucheconformation in ethanol is predicted to be more stable than the trans.
3.11.5 Modified Neglect of Diatomic Overlap, Parametric Method number 3 (PM3)
The parameterization of MNDO and AM1 had been done essentially by hand, taking the Gss,Gsp,Gpp,Gp2andHspparameters from atomic data and varying the rest until a satisfactory fit had been obtained. Since the optimization was done by hand, only relatively few reference compounds could be included. J. J. P. Stewart made the optimization process automatic by deriving and implementing formulas for the deriv- ative of a suitable error function with respect to the parameters.50All parameters could then be optimized simultaneously, including the two-electron terms, and a signifi- cantly larger training set with several hundred data could be employed. In this re- parameterization, the AM1 expression for the core–core repulsion (eq. (3.85)) was kept, except that only two Gaussians were assigned to each atom. These Gaussian parameters were included as an integral part of the model, and allowed to vary freely.
The resulting method was denotedModified Neglect of Diatomic Overlap,Parametric Method Number 3(MNDO-PM3 or PM3 for short), and is essentially AM1 with all the parameters fully optimized. In a sense, it is thebest set of parameters (or at least a good local minimum) for the given set of experimental data. The optimization process, however, still requires some human intervention in selecting the experimen- tal data and assigning appropriate weight factors to each set of data.
V V Z Z
R ak b R c a
k b R c
k
k k k k
nnAM
nnMNDO A B
AB
A B
A B A, B e A AB A e B AB B
1( , )= ( )+ ′ ′
∑ (
− ( − )2+ − ( − )2)
Some known limitations of the PM3 model are:
(1) Almost all sp3-nitrogens are predicted to be pyramidal, which is contrary to experi- mental data.
(2) Hydrogen bonds are too short by ~0.1 Å.
(3) The gaucheconformation in ethanol is predicted to be more stable than the trans.
(4) Bonds between Si and Cl, Br and I are underestimated, the Si—I bond in H3SiI, for example, is too short by ~0.4 Å.
(5) H2NNH2is predicted to have a C2hstructure, while the experimental result is C2, and ClF3is predicted to have a D3hstructure, while the experimental result is C2v. (6) The charge on nitrogen atoms is often of “incorrect” sign and “unrealistic”
magnitude.
Some common limitations of MNDO, AM1 and PM3 are:
(1) Rotational barriers for bonds that have partly double bond character are signifi- cantly too low. The barrier for rotation around the central bond in butadiene is calculated to be only 2–8 kJ/mol, in contrast to the experimental value of 25 kJ/mol.51Similarly, the rotational barrier around the C—N bond in amides is calculated to be 30–50 kJ/mol, which is roughly a factor of two smaller than the experimental value. A purely ad hocfix has been made by adding a force field rota- tional term to the C—N bond that raises the value to ~100 kJ/mol and brings it into better agreement with experimental data.
(2) Weak interactions, such as van der Waals complexes or hydrogen bonds, are poorly predicted. Either the interaction is too weak, or the minimum energy geometry is wrong.
(3) Conformational energies for peptides are poorly reproduced.52
(4) The bond length to nitrosyl groups is underestimated. The N—N bond in N2O3, for example, is ~0.7 Å too short.
(5) Although MNDO,AM1 and PM3 have parameters for some metals, these are often based on only a few experimental data. Calculations involving metals should thus be treated with care.
The MNDO, AM1 and PM3 methods have been parameterized for most of the main group elements,53 and parameters for many of the transition metals are also being developed under the name PM3(tm), which includes d-orbitals. The PM3(tm) set of parameters are determined exclusively from geometrical data (X-ray) since there are very few reliable energetic data available for transition metal compounds
3.11.6 Parametric Method number 5 (PM5) and PDDG/PM3 methods
Two approaches have appeared that try to further improve on the performance of the PM3 method. The PM5 (PM4 being an unpublished experimental version) method re- introduces diatomic parameters for the core–core repulsion, and the published results suggest that PM5 represent a slight improvement on the PM3 results.54No details of the methodology and parameterization have been published so far.
A similar approach using Pairwise Distance Directed Gaussian(PDDG) in connec- tion with the MNDO and PM3 methods has also been reported.55The idea is related to the concept used in AM1 and PM3 by introducing parameterized Gaussian func-
tions for describing the core–core repulsion, except that the modification is based on interatomic distances, although the parameters are still purely atomic. The latter pre- vents the exponential increase in parameters with the number of atoms. The available results suggest a slight improvement over the regular MNDO or PM3 methods, but it is difficult to assess whether the improvement is simply due to more fitting parameters or to fundamentally better modelling of the underlying physical problem.
3.11.7 The MNDO/d and AM1/d methods
With only s- and p-functions included, the MNDO/AM1/PM3 methods are unable to treat a large part of the periodic table. Furthermore, from ab initiocalculations it is known that d-orbitals significantly improve the results for compounds involving second row elements, especially hypervalent species. The main problem in extending the NDDO formalism to include d-orbitals is the significant increase in distinct two-elec- tron integrals that ultimately must be assigned suitable values. For an sp-basis there are only five one-centre two-electron integrals, while there are 17 in an spd-basis. Sim- ilarly, the number of two-centre two-electron integrals rises from 22 to 491 when d- functions are included.
Thiel and Voityuk have constructed a workable NDDO model that also includes d- orbitals for use in connection with MNDO, called MNDO/d.56With reference to the above description for MNDO/AM1/PM3, it is clear that there are immediately three new parameters:zd,Udand bd(eqs (3.90) and (3.91)). Of the 12 new one-centre two- electron integrals, only one (Gdd) is taken as a freely varied parameter. The other 11 are calculated analytically based on pseudo-orbital exponents, which are assigned such that the analytical formulas regenerate Gss,Gppand Gdd.
With only s- and p-functions present, the two-centre two-electron integrals can be modelled by multipoles up to order 4 (quadrupoles), however, with d-functions present multipoles up to order 16 must be included. In MNDO/d all multipoles beyond order 4 are neglected. The resulting MNDO/d method typically employs 15 parameters per atom, and it currently contains parameters for the following elements (beyond those already present in MNDO): Na, Mg, Al, Si, P, S, Cl, Br, I, Zn, Cd and Hg. Recently this technology has been used in connection with the AM1 model as well, which at least for phosphorous yields a further improvement.57
3.11.8 Semi Ab initio Method 1
The philosophy behind the Semi Ab Initio Method 1 (SAM1 and SAM1D) model is slightly different from the other “modified” methods.58It is again based on the NDDO approximation, but instead of replacing all integrals with parameters, the one- and two- centre two-electron integrals are calculated directly from the atomic orbitals. These integrals are then scaled by a function containing adjustable parameters to fit experi- mental data (RABbeing the interatomic distance).
(3.96) The advantage is that basis sets involving d-orbitals are readily included (defining the SAM1D method), making it possible to perform calculations on a larger fraction of the periodic table. The SAM1 method explicitly uses the minimum STO-3G basis set,
m nA B m nA B →f R( AB)m nA B m nA B
but it is in principle also possible to use extended basis sets with this model. The actual calculation of the integrals makes the SAM1 method somewhat slower than MNDO/AM1/PM3, but only by a factor of ~2. The SAM1/SAM1D methods have been parameterized for these elements: H, Li, C, N, O, F, Si, P, S, Cl, Fe, Cu, Br and I. Although the SAM1 method was proposed in 1993, no details of the functional form or para- meterization have been published, and there do not appear to have been any recent developments.