METHODS 1. Hu ¨ ckel Method

The most serious approximations are made in the Hu¨ckel MO (HMO) method developed for conjugated planar hydrocarbons [1]. In the original method, the ele-ments of the effective Fock matrix [Eq. (8)] are completely parameterized and no molecular integral has to be calculated. Only one 2pp AO per C atom

is considered in the LCAO [Eq. (6)]. For a system with NC carbon atoms, one gets

/^HMO_a ¼X^NC

l

claClð2ppÞ ð17Þ

The Hu¨ckel MOs /^HMO_a are normalized.

X^NC

l

c²_la¼ 1 ð18Þ

The diagonal elements of the HMO ‘‘Fock’’ matrix F_ll^HMO, the so-called Coulomb integrals, are assumed to have the same value a for every carbon atom in the molecule, and the only nonzero off-diagonal elements F_lr^HMO, the resonance integrals, are those between neighboring C atoms l and r. They are also set to the same value, b, irrespective of the molecular structure. This parameter can be obtained from a fitting to experimental optical spectra. Both parameters a and b have the dimension energy and are usually given in atomic units, a.u., or Hartree, Eh. In order to convert a.u. into the more familiar caloric or SI units, the following conversion factors have to be used [17]: 1 a.u.u 627.5 kcal/mol u 2625 kJ/mol.

The hydrogen atoms are not taken into account. For this reason, and since only nearest-neighbor interactions are considered, the Hu¨ckel matrices for cis- and trans-butadiene C4H6are identical (Fig. 1).

F^HMOðbutadieneÞ ¼

a b 0 0

b a b 0

0 b a b

0 0 b a

0 BB

@

1 CC

A ð19Þ

In general, the corresponding Hu¨ckel orbitals /_a^HMO and eigenvalues e_a can be obtained by standard diagonalization techniques for which a large number of pro-grammed routines are available [18–20], in correspondence to the ab initio procedure [Eq. (16)]. Due to the simple structure of the Hu¨ckel matrix, it is, however, possible to obtain analytical expressions for the eigenvalues. In the case of conjugated chains with NCcarbon atoms, solving the secular equationjF^HMO
ej=0 gives [21]

ea¼ a
xab xa¼ 2 cos ap

NCþ 1

 ð20Þ

Figure 1 Numbering of C atoms in (a) cis- and (b) trans-butadiene.

Since the HMO Fock matrix is not dependent on the orbital coefficients, no SCF procedure as for ab initio methods (Sec. 2) has to be performed. In the present example with NC= 4, the four orbital energies are x1= 1.618b, x2= 0.618b, x3=
0.618b, and x4=
1.618b. Since the parameter b is chosen to be negative, the first energy x1

has the lowest value, and the fourth x4has the highest value. This corresponds to the usual convention in quantum chemistry. From these values, the orbital diagram of butadiene can be drawn.

In Fig. 2, the two lowest MOs were occupied with two electrons of opposite spin according to the aufbau principle. Two results can be obtained from an HMO calculation: the excitation energies comparable to optical transitions in absorption spectroscopy [22] and the total energy E^el. The lowest HMO excitation energy of butadiene corresponding to the transition of one electron from the highest occupied MO (HOMO), No. 2, to the lowest unoccupied MO (LUMO), No. 3 (Fig. 2), is e3
e2

= 1.236jbj. In spectroscopy, excitation energies are usually given as reciprocal wavelength (wave numbers, unit cm
¹).

1

kðbutadieneÞ ¼ eLUMO
eHOMO

hc ¼ 1:236j
j ð21Þ

Here, h and c are Planck’s constant and the speed of light, respectively [23]. On the other hand, Eq. (21) can be used to determine the semiempirical parameter b of the HMO method if the reciprocal wavelength 1/k is taken from an accurate measurement.

In practice, b is chosen to approximate the excitation energies of a large variety of hydrocarbons [1] as close as possible. For excitation energies, the parameter a does not have to be specified at all and could be set to zero. It only plays a role for the calculation of the total energy. Since the HMO theory does not contain Coulomb or Exchange operators, the calculation of the total energy [Eq. (12)] simplifies to

E^el;HMO¼X^M

a

naea ð22Þ

where nais the occupation number of the ath MO. nais, in general, equal to 2 but is 1 for the HOMO of conjugated polyenes with an odd number of carbon atoms. For

Figure 2 Hu¨ckel MO diagram for butadiene derived from analytical expression (21).

butadiene, Eq. (22) gives E^el,HMO(butadiene) = 4a+4.472b. A famous example of the use of HMO total energies is the calculation of delocalization energies or resonance energies for aromatic compounds. For the prototype aromatic molecule benzene C6H6, the HMO energy is 6a+8b. If the molecule would consist of three separated p systems (or, alternatively, three C2H2molecules), the HMO energy would be 6a+6b.

Thus the stabilization energy due to delocalization of the three p orbitals is 2b (Fig. 3).

The resonance energy per p electron (REPE) has been used as a measure for the aromaticity of molecules [24]. For a recent and comprehensive overview of the aromaticity concept, see, for example, Refs. [25,26].

A generalization of the Hu¨ckel method to nonplanar systems comprised of carbon and heteroatoms is the Extended Hu¨ckel Theory (EHT) [27–30]. It takes explicitly into account all valence electrons, i.e., {1s} for H and {2s,2p} for C, N, O, and F. Similar to the HMO method, the ‘‘Fock’’ matrix in EHT F^EHTdoes not contain two-electron integrals. The diagonal elements FllEHT

are obtained from experimental ionization potentials (IPs) where the Koopmans theorem [31] has been used.

IPaðKoopmansÞ ¼
ea ð23Þ

The off-diagonal elements are approximated by the Wolfsberg–Helmholz formula [27]

F_lr^EHT¼1 2KSlr



F_ll^EHTþ F_rr^EHT

ð24Þ This expression takes into account the overlap Slr between two AOs vl and vr

centered at different atoms. The atomic basis functions v are represented by Slater functions [32],

v^Slater¼ ½2f^nþ1=2

½ð2nÞ!¹⁼²r^n
1_A e
^frAY^m₁ðh; uÞ ð25Þ where f is the orbital exponent and Y is a spherical harmonic. The two-center integrals Slrover Slater functions can be easily evaluated [32].

The overlap matrix S is also taken into account in the EHT version of the general eigenvalue equation (7) which can be solved by applying the mentioned orthogonal-ization transformation and matrix diagonalorthogonal-ization techniques. Due to the independ-ence of F^EHTfrom the orbital coefficients, no SCF procedure has to be performed. This is similar to HMO.

Since the overlap depends on the interatomic distance, the EHT distinguishes between molecular conformations and it is possible to calculate equilibrium structures

Figure 3 Resonance energy of benzene: transition from a fictitious system with three separated p bonds to the delocalized ground state.

using standard minimization techniques [33]. Thus the EHT represents the most simple all-valence electron semiempirical method. It has been successfully used by Hoffmann et al. in applications to a vast variety of systems, and it is still frequently used nowadays (see Sec. 6).

3.2. Pariser–Parr–Pople Method

The Hu¨ckel Theory in general gives reliable values only for the lowest excitation energies of the HOMO! LUMO transition of aromatic or conjugated hydrocarbons.

It cannot also distinguish between singlet and triplet excited states that are exper-imentally known to have different luminescence characteristics. The reason for this deficiency is the neglect of electron–electron interactions in the Hu¨ckel method. A semiempirical p electron method that explicitly takes into account electron–electron repulsion Vˆeein the effective Hamilton operator [Eq. (2)] is the Pariser–Parr–Pople (PPP) method [34,35]. It has been designed for the calculation of optical absorption spectra of aromatic hydrocarbons. Similar to the HMO theory, only C 2p p functions are taken into account and all other atoms in the molecule are ignored.

In the evaluation of the effective PPP Fock matrix elements, FlmPPP

, the zero differential overlap(ZDO) approximation is used.

v_lð1Þv_mð1Þdq₁¼ 0 for l p m ð26Þ

This is a fundamental assumption that is used more or less strictly in all semiempirical SCF MO methods. It has several important consequences.

1. The overlap matrix S becomes the unit matrix E (or, by integrating Eq. (26),R mA(1)m_r(1) dq1=Slm=dlmis obtained.)

2. From all two-electron integrals (lm|qr), only the ‘‘diagonal’’ terms of type (ll|qq) remain. This reduces the size dependence of semiempirical methods from formally N⁴(as is the case for HF methods) to only N². Here lies the main reason why semiempirical methods are able to treat very large systems with up to N=10 000 electrons. (It has to be kept in mind that for the PPP method, the number of electrons and the number of atomic basis functions are identical. This is, in general, not the case for all-valence semiempirical methods and certainly not for ab initio methods using extended basis sets.)

At first sight, these are rather drastic approximations. However, it has been proven that they are at least partly justified in an orthogonalized basis after a transformation according to Eq. (14). In the orthogonal basis, most of the integrals neglected in the ZDO approximation become smaller in absolute magnitude, and their relevance for the total energy and excitation energy is diminished [1,3]. Recently, the fundamental reasons for the PPP model Hamiltonian to qualitatively and semiquantitatively reproduce spectroscopic features of conjugated polyenes have been reexamined on the basis of high-level coupled cluster calculations [36].

The explicit form of the PPP Fock matrix [compared to the ab initio expression (9)] is

F_lr^PPP¼ H_lr^PPP
1

2PlrcAB l at atom A; r at atom B ð28Þ The integrals H^PPP_lr and cABare calculated from semiempirical formulas that contain adjustable parameters. These are optimized in order to reproduce properties of a given class of molecules in an optimal manner (see Sec. 4). The intra-atomic electron repulsion integral c_AAis estimated from experimentally measured ionization energies and electron affinities (see, for example, Ref. [23]).

The PPP method is the first semiempirical method presented here where the Fock matrix does depend on the MO coefficients C [via the density matrix elements P, see Eq. (11)]. Therefore, the Roothaan equations (by definition due to the ZDO approx-imation) in the orthogonal basis, Eq. (13), have to be solved in an iterative process until self-convergence is achieved [self-consistent field (SCF) procedure]. As starting coefficients C⁰, usually the orbitals of an HMO calculation are used.

Within the framework of PPP theory, it is possible to obtain excitation energies with the Configuration Interaction (CI) method. This method is described in greater detail in other chapters of this book. Here only a brief description is given. The PPP ground-state wave function is a single Slater determinant [Eq. (5)]. For closed-shell molecules with an even number N of electrons, M=N/2 MOs are doubly occupied and also M orbitals remain unoccupied. It is now possible to construct ‘‘excited’’

determinants by exchanging one or more occupied spin orbitals [i.e., either with a spin (/) or with b spin (/)] with unoccupied orbitals. For example, a determinant where the occupied a spin orbital a has been substituted by the unoccupied a spin orbital p is denoted as W^p_a. An improved ground-state wave function W^CIcan then be obtained by a linear combination of the original ground-state Slater determinant W0

and the modified determinants.

The optimal coefficients C of the CI ground state are obtained by a variational procedure. At the same time also, CI states of higher energy are obtained. In the simplest CI expansion, only single substitutions of the type aX b and a X b are taken into account in Eq. (29). This procedure is called single-excitation CI (SCI). Only singlet (multiplicity of 1) and triplet (multiplicity 3) states are obtained in this way, while the inclusion of multiple substitutions gives rise to states of multiplicity 5 and higher. The lowest singlet state¹C^CI₀ is the CI ground state, and the higher CI singlet states are denoted by¹C^CI₁ ;¹C^CI₂ ;: : :. The energy difference Eð¹C^CI₁ Þ
Eð¹C^CI₀ Þ is then comparable to the lowest excitation energy of an experimental absorption spectrum.

The PPP method is little used nowadays and has been largely superseded by more general semiempirical methods (see the following sections). However, it is still worth to mention since the approximations of the PPP approach are the prototype model of all methods that were developed later.

3.3. CNDO Method

The Complete Neglect of Differential Overlap method (CNDO) of Pople et al. [8,37–39]

makes use of the ZDO approximation [Eq. (26)] for all pairs of atomic basis functions.

It treats explicitly all valence electrons (e.g., C 2s, C 2p), but neglects completely the

effect of inner (core) electrons (e.g., C 1s). The CNDO Fock matrix elements therefore reduce to

F_ll^CNDO¼ H_ll^CNDOþX^A

m

Pmmc_AAþX

BpA

X^B

r

Prrc_AB

F_lm^CNDO¼
1

2Plmc_AA ð30Þ

F_l^CNDO¼ H_l^CNDO
1

2Plrc_AB ðl; m at atom A; r at atom BÞ ð31Þ For symmetry reasons, the intra-atomic one-electron integrals H^CNDO_lm vanish. The one-electron matrix elements Hll are subdivided into intra-atomic contributions U and interatomic contributions due to the electron attraction V^B_enwith other nuclei B.

H_ll^CNDO¼ Ull
X

Bp A

V_en^B

H_lr^CNDO¼ b_ABSlr ð32Þ

c_AB¼ ðsAsAjsBsBÞ

In fact, the nuclear attraction integrals include an effective core charge Z* which is the nuclear charge Z reduced by the number of core electrons. The intra-atomic integral Ullcontains the kinetic energy and the nuclear attraction with the nucleus where the AO l is centered. The two-electron repulsion integrals (ll|rr) are calculated over s-type functions invariably, and also when l or r are p-s-type atomic functions. These approximate integrals are called cAB. The replacement of p functions by angular-independent s functions is necessary in order to fulfill the requirement of rotational invariance as discussed in Refs. [37,40]. Calculated molecular properties must be independent from the orientation in a global coordinate system (in the absence of an external field). For the same reason, the nuclear attraction integrals V^B_enare calculated using s-type AOs only.

All two electron one-center integrals (ll|mm) are reduced to a single cAA. The two-center one-electron integrals H^CNDO_lr are set proportional to the overlap integral Slr

which is calculated over Slater-type functions (and only here not neglected according to the ZDO assumption). The coefficient bAB is calculated either from atomic parameters bAor orbital-dependent parameters b_l.

b_AB¼¹₂ðb_Aþ b_BÞ or

b_lAmB ¼¹₂ðb_lAþ b_mBÞ

ð33Þ

According to the Wolfsberg–Helmholz formula (24), the b parameters can be regarded as diagonal terms of the CNDO Fock matrix.

There are two versions of CNDO, CNDO/1 and CNDO/2, which differ in the evaluation of the intra-atomic integral Ull. In CNDO/1, it is taken from experimental IP of the free atom [8]. In CNDO/2, a modified procedure is used [41]. In order to compensate for deficiencies of CNDO/1 in describing long-range intermolecular

electrostatic interactions, the evaluation of V^B_en was modified in CNDO/2 (see Ref.

The CNDO method is used to a lesser extent than EHT or less approximate semi-empirical methods (see Sec. 6). The accuracy of energetic and electronic properties obtained with CNDO is, in general, inferior to that of the methods described in the next sections, while the computational effort of the SCF calculation is comparable.

3.4. INDO Method

The Intermediate Neglect of Differential Overlap (INDO) method, originally devel-oped by Pople and Beveridge [8] and Pople et al. [44], uses the ZDO approximation [Eq. (26)] only for two-center integrals. The elements of the INDO Fock operator are therefore modified with respect to CNDO mainly by the inclusion of one-center exchange-type integrals (lm|lm). The one-center two-electron integrals (ll|mm) and (lm|lm) are partly calculated analyti-cally and partly derived from atomic spectra [8]. The original INDO method gives unsatisfying results for geometries and dissociation energies and was soon replaced by several improved versions which are still in use nowadays, namely, MINDO/3, INDO/

S, and SINDO1. The first of these newer INDO methods is MINDO/3 (third version of the modified INDO) by Bingham et al. [45], the successor of MINDO/1 [46] and MINDO/2 [47]. The new idea of MINDO was to replace the time-consuming analytical calculation of two-electron integrals cAB(e.g., using the Harris algorithm [48]; recent algorithms for two-electron integrals over Slater orbitals can be found in Ref. [49]) by a simple multipole expansion, suggested by Ohno [50] and Klopman [51].

c_ABðMINDOÞ ¼ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R²_ABþ¹₄ðcAAþ cBBÞ²

q ð39Þ

The calculation of two-center one-electron integrals has been modified compared to INDO.

H^MINDO_lr ¼ bABS_lrðH_ll^CNDOþ H_rr^CNDOÞ ð40Þ

Here bABare empirical interatomic parameters that have been optimized to minimize the errors in heats of formation with respect to experiment of some reference

com-pounds (see Sec. 4). The Pauli repulsion, which is not included in the CNDO or INDO method, has been incorporated into the core–core potential Vnnin the total energy calculation [Eq. (3)]. In order to obtain a balance between attractive and repulsive terms, the analytical 1/RABdependence has been replaced by cABplus a correction term. This also in part accounts for the neglected interactions of inner orbitals.

V_nn^MINDO¼X

with the additional bond parameter aAB. The one-center integrals are calculated by a method due to Oleari et al. [52]. The MINDO/3 version improved in general geom-etries and dissociation energies compared to the original INDO methods, but also had several failures for specific compounds.

The most frequently used INDO-type method nowadays, known as INDO/S or ZINDO, has been developed by Zerner and Ridley [54] and Bacon and Zerner [55].

From the very beginning, it has been designed for the calculation of molecular spectra of organic molecules and complexes containing transition metals, while its results for structural properties are less accurate. INDO/S starts from the original INDO Fock matrix terms [Eq. (38)] together with the analytic expression for the internuclear repulsion Vnn. Special attention has been given to the calculation of one-center two-electron integrals from Slater–Condon factors F and G and the evaluation of Ullfrom experimental ionization energies [55]. For example, the ionization process that removes an s electron of an atom with a {s^lp^mdⁿ} electronic configuration can be expressed in terms of Ull, F, and G.

The one-center Coulomb integrals (ll|mm)=F_lm⁰ are calculated analytically while the G integrals are taken as parameters. The IP are taken from atomic spectra. After rearranging Eq. (43), the Ussare obtained. A special feature of INDO/S is the use of distance-dependent Slater exponents for the calculation of two-center integrals [55].

fðRÞ ¼ a þ b=R for fðRÞ < fð0Þ ð44Þ

fðRÞ ¼ fð0Þ elsewhere ð45Þ

This procedure to some extent mimics the use of multiple-zeta basis sets in high-quality ab initio calculations. Later, also charge-dependent orbital exponents that are more flexible with respect to the chemical environment have been implemented in INDO/S [56]. For transition metal atoms, one-center integrals of the general form (lm|qk) are taken into account which do not appear in the original INDO method.

For the two-center one-electron integrals, a conventional INDO formalism is applied. There are two sets of optimized parameter sets for INDO/S. One has been optimized for electronic spectra and the other for molecular geometries. Therefore, in

principle, two successive INDO/S calculations would be necessary: first, a geometry optimization using the second parameter set, and then a calculation of spectroscopic properties at fixed geometry. Since it is known that the INDO/S geometries are not very accurate, usually another semiempirical method is used for the structure optimization in practice. Recently, a modification of the original INDO/S method has been reported [57].

One year after INDO/S, the method SINDO1 (symmetrically orthogonalized INDO/1) by Nanda and Jug [58] was introduced. Originally developed for organic compounds of first-row elements, it was later extended to elements of the second and third row [59,60]. This method has several distinct features. The most important is that the orthogonalization transformation [Eq. (14)] is taken into account by a Taylor expansion. The matrix S
^1/2is approximated as

S¼ E þ s S
¹⁼²¼ E
1

2s þ3 8s²
5

16s³þ: : : ð46Þ

and expansion (46) is truncated after the second order. Only the one-electron integral matrix H is transformed (for a discussion of the consequences of finite-order expansion on molecular integrals, see, e.g., Ref. [61]). Another special feature of SINDO1 is the explicit treatment of inner orbitals by a pseudo-potential proposed by Zerner [62]. The calculation of center integrals is similar to that in INDO/S. Two-center one-electron integrals Hlrare calculated by the following empirical formula:

H^SINDO1_lr ¼ Llrþ DHlr l at atom A; r atatomB ð47Þ

Here L is a correction of the Mulliken approximation for the kinetic energy and DH is entirely empirical and contains adjustable bond parameters. These are optimized in order to minimize the deviation from experiment for a set of reference compounds. In a way similar to INDO/S [Eq. (45)], two sets of Slater orbital exponents are used: one

Here L is a correction of the Mulliken approximation for the kinetic energy and DH is entirely empirical and contains adjustable bond parameters. These are optimized in order to minimize the deviation from experiment for a set of reference compounds. In a way similar to INDO/S [Eq. (45)], two sets of Slater orbital exponents are used: one

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