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MODEL HAMILTONIANS FOR THE CALCULATION OF ATOMIC AND MOLECULAR SPECTROSCOPY By JOHN DAVID BAKER A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1991 ACKNOWLEDGMENTS I would like to thank first and foremost my wife, Rebecca, for her under standing and support that was crucial to the success of this work. Secondly, I would like to thank Tennessee Eastman for the financial support that accelerated this project. A special thanks goes to my committee who have struggled with me to get many recommendations and resumes in the mail. I thank my advisor, Dr. Zerner, for allowing me the freedom I desire to explore my own ideas while supporting me in the process. This, I feel, is truly unique in the academic business and I certainly realize the personal sacrifice he has made to allow his guys this privilege. And finally, I would like to thank the guys in the clubhouse who put up with my songs and occasional bouts of sarcasm. TABLE OF CONTENTS ACKNOWLEDGMENTS .. ..... ...................... ii ABSTRACT.................................... .. v CHAPTERS 1. INTRODUCTION ................... .............. 1 Motivation .... .. .............. ............... 1 Time Dependent Perturbation Theory ........... ....... 5 Molecular Response to Radiation ...................... 7 The Reporting of Theoretically Calculated Spectral Probabilities .. 13 2. GENERAL THEORY ...................... ... 20 The HartreeFock Equations . . ... 20 The Roothaan Equations . .. . 27 The Calculation of Electronic Spectra ...... .......... 31 Singlet Excitations with a Finite Basis ............... 41 3. THE INDO/RPA METHOD ......................... .44 Theory ..... ...... ..... .............. .. .. 44 Benzene and Pyridine . .. . .. 51 Naphthalene and Quinoline ....... ............ ... .. 58 The Diazines .................. ................ 61 The RPA for Extended Systems ... ..... ........... 65 4. THE UVVIS SPECTRA OF FREE BASE AND MAGNESIUM PORPHIN .................... ............... 70 Motivation ..... ............ ................ 70 Results ....... .. ....... .................. 73 Discussion .. ..... ........................ ... 76 iii 5. THE INDO/RPA CALCULATION OF NMR CHEMICAL SHIELDING The Concept of Chemical Shift ...................... Theory . . . . . Localization and Integral Evaluation ................... Reduced Linear Equations ........................ Diamagnetic Shielding in Molecules ................ .... The INDO/RPA Method for Chemical Shift .............. 6. A CHARGE DEPENDENT HAMILTONIAN FOR ATOMIC PROPERTIES ............................ ..... Im petus ... .. ..... .... . Atomic Parameterization ............... A New Atomic Parameterization .......... A analysis ... .. .. ... . 7. A DENSITY PARTITIONED HAMILTONIAN FOR ELECTRONIC SPECTRA ............... Density Partitioning .................. Atomic Hamiltonian ................. Molecular Hamiltonian ................ Modified SCF Procedure ............... Results .. .. . . 8. CONCLUSIONS...................... Spectroscopy............ .......... Development of a Better Model .... ...... BIBLIOGRAPHY ..................... BIOGRAPHICAL SKETCH ............... ........... ........... ........... MOLECULAR . . . ........... ........... ........... ........... ........... . . 161 . .. .. .. 169 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MODEL HAMILTONIANS FOR THE CALCULATION OF ATOMIC AND MOLECULAR SPECTROSCOPY By JOHN DAVID BAKER May, 1991 Chairman: Michael C. Zerner Major Department: Chemistry Semiempirical models appropriate for the calculation of atomic and molecular spectroscopy for large systems were developed. Through equation of motion (EOM) constraints, equivalent to the random phase approximation (RPA), elec tronic transitions and oscillator strengths are calculated directly. The RPA formalism is demonstrated to give accurate transition energies for electronic excitations. The EOM constraints guarantee equivalence between calculated oscillator strengths in differing formalisms. Although this provision is only true for a complete basis, considerable improvement in predicting oscillator strengths that compare with experiment, and differing formalisms, is demonstrated for a considerably smaller basis. The computational tools associated with the RPA were applied to the cal culation of the electronic spectra of free base and magnesium porphin. The ability of the RPA formalism to predict accurate oscillator strengths and selectively correlate transition energies has allowed direct corroboration of the experimental spectra associated with these molecules. The RPA formalism is also used for the calculation of isotropic nuclear magnetic resonance (NMR) chemical shift. The method is capable of demonstrating the inductive effects associated with electronwithdrawing substituents and is capable of differentiating types of bonding environments. The prediction of paramagnetic substituent effects, however, was found to be deficient due to the small size of the basis utilized. Minimal basis set (MBS) formalisms are restricted in their ability to predict accurate results for properties. In order to maximize the capability of model Hamiltonians and maintain the efficiency of such MBS formalisms, a complete chargedependent intermediate neglect of differential overlap (INDO) Hamiltonian appropriate for the calculation of molecular spectra was developed and examined. The new model is capable of accurately describing the local chemical potential of atoms in molecules, and surpasses the current INDO/S model in the prediction of carbonyl excitation energies. A procedure for the acquisition of atomic and molecular parameters was developed that makes use of readily available exper imental data. Simplified parameter functions were developed which allow more information to be incorporated into the model with less effort than previously. CHAPTER 1 INTRODUCTION Motivation Chemistry is the study of interactions at the atomic and molecular level. These interactions can be between the atoms that constitute a molecule, between molecules or between a molecule and its environment. Quantum chemistry dedi cates itself to the task of predicting observables associated with these interactions. One specific area of study is the interaction of matter with long wavelength radi ation. Molecules react to shortlived pulses of radiation in many ways depending on the frequency of the radiation. Highenergy radiation may result in dissocia tion or ionization. Lowenergy radiation may result in orientational changes of a molecule. In all cases, however, specific molecules respond to radiation of dif fering wavelength very uniquely and can be characterized by their specific ability to interact with radiation of a given frequency. The range of wavelength associated with visible and ultraviolet light usually corresponds to electronic transitions in molecules. Light of a set wavelength may be absorbed by the material resulting in a new electronic state. This absorption will take place with a given probability which is again a function of the molecule of interest. In addition, the probability for which a molecule absorbs 2 radiation of fixed wavelength but opposite polarization may differ. The qualitative and quantitative prediction of these types of interactions has been prominent in quantum chemistry from its inception. While quantum mechanics can in principle yield exact results for all molecular interactions, the difficulty in obtaining these solutions has led workers to seek approximate solutions. Early Hiickel theory models are successful in predicting the qualitative nature of molecular absorptions of visible and ultraviolet light for planar conjugated systems [1]. More exact models such as that of Pariser, Parr and Pople (PPP) for conjugated systems were developed and the agreement between theory and experiment was extended to the accurate prediction of transition energies [24]. Hiickel theory was also extended to nonplanar systems [5]. Modern approximate methods usually involve the use of wellcharacterized experimental data for small systems to fit quantum mechanical models that can be extrapolated to much larger systems. The spectroscopic parameterization of the Intermediate Neglect of Differential Overlap (INDO/S) model Hamiltonian has been very successful in the prediction of properties associated with electronic tran sitions [6, 7]. The use of the INDO/S Hamiltonian for the prediction of electronic transition moments and properties has been primarily a stateoriented approach. In other words, electronic states are calculated directly yielding transition energies as differences in state energies and transition probabilities as transition amplitudes between states. 3 In this work a transitionoriented approach is developed with the INDO/S model Hamiltonian and compared to the more traditional stateoriented approach. Through the use of polarization propagator (PP) techniques [8], transition energies and probability amplitudes can be calculated directly without the need to describe the individual states. If exact state functions can be found then both the state and transitionoriented approaches are equivalent. For almost all cases, however, approximate state solutions are employed and methods that are capable of giving the most consistent solution to a given property must be exploited. A consistent approach to the calculation of molecular response to electromagnetic radiation will involve an equitable treatment of transition moments as calculated in both the dipole length and dipole velocity formalisms. This consistency requirement will be sufficient to derive the equations associated with the Random Phase Approximation (RPA) [9, 10]. In this way a balanced treatment of transition amplitudes can be achieved. Many properties are computationally related to the transition energies and moments associated with electronic excitations. Such properties are usually associated with the response of the total energy of a molecule to an external perturbation such as a constant magnetic field [11, 12]. These properties are often expressed as a perturbative expansion of the total energy and truncated at some order. For a small perturbation the expansion can be truncated at second order which usually involves a formal sum over all transition moments and associated 4 energies. Computational tools used must not become prohibitively expensive as the size of the system becomes larger regardless of the Hamiltonian used. For properties that involve a formal sum over all the states of a system, the techniques necessary to calculate individual state transition moments become too difficult. The method of reduced linear equations [13] as recently employed in the calculation of molecular polarizabilities [14] meets the criteria of expediency for sum over state properties. This technique is used in the calculation of Nuclear Magnetic Resonance (NMR) shielding tensors at the RPA level of theory for large compounds which would not be possible otherwise [15]. Accurate ab initio calculations with the RPA for small systems has yielded excellent results but ab initio techniques are often too computationally intensive for large molecules. If quantum chemistry is to be predictive for the majority of chemical systems model Hamiltonians will continue to be a necessary tool. It is therefore prudent to develop and characterize the basic structure of model Hamiltonians and the content of the parameters in such models. To this end the basic form of the atomic parameters in the INDO model are examined. In conjunction with recent developments in density functional theory [16] this examination suggests generalization and improvement of the current Hamiltonian used for electronic spectra. 5 Time Dependent Perturbation Theory If a molecule is subjected to a short duration of long wavelength radiation the response of the molecule to such radiation can be treated as a time dependent perturbation. Stationary state solutions (Ji) to the time dependent Schr6dinger equation Hgo (11) 1 at for time independent Hamiltonians (Ho) can be expressed as, I o(q, t) = exp(iE t/h) i(q) (12) where Ef is the total energy of stationary state i. The Oi are orthonormal functions of space and spin coordinates (q) only and are eigenfunctions of Ho. For a time dependent potential energy perturbation (H') Equation (11) becomes (H + H') =  (13) i at For a fixed point in time, solutions to Equation (13) can be expressed as a linear combination of %0 since they are eigenfunctions of the Hermitian operator HI and thus form a complete set. For different points in time the coefficients of the expansion will in general be different; therefore, solutions for Equation (13) of the form T = aj(t)1(q, t)= a(t) exp(iEt/h)jJ(q) (14) j j 6 must be sought. Substitution of Equation (14) into Equation (13) yields, [E aj(t) + H'aj(t)] exp(iE t/h)Vkj(q) = j h daj(t) (15) Eaj(t) d ] exp(iEjt/h)n (q) ( J Utilizing the orthonormality of Oi one obtains, aj(t) exp(iEt/h)(kIi'J h dak(t) exp(iE t/h) (16) j so that Equation (13) becomes a series of differential equations in time for the coefficients in Equation (14) without loss of generality [17]. dak(t) i dt 1 aj(t)exp[i(E Ef)t/lh](kH' j) (17) j For a system in state m in the absence of a time dependent perturbation we can choose initial conditions such that am ; 1 and anm w 0. For a perturbation applied at time 0 of sufficiently short duration that the coefficients are close to their initial values we can approximate Equation (17) as dak(t) i dat) exp[i(E' Em)t/h](klIH' Im) (18) dt h and integrate from t=0 to t' to obtain an approximate expression for the coef ficients. t' to Jdak(t) /exp[i(E E'm)t/h](k IH'Ilm)dt 0 0 (19) ak(t) ak(O) exp[i(E Em)t/h](klH'lm)dt 0 7 If the perturbation ceases after t' then the final state function can be expressed as Equation (14) with constant ak(t). This result is a superposition of eigenstates of the unperturbed Hamiltonian. The probability of observing the system in state n with energy E after the perturbation is then [18], Pn = an (t') exp(iEot/h)2 = an (t')12 (110) The effect of the perturbation is a transition from the initial state m to final state n with probability Pn. Molecular Response to Radiation Electromagnetic radiation consists of both electric (E) and magnetic (B) fields that can interact with the electrons within a molecule. The classical potential energy of interaction between these fields and a system of n identically charged particles (q) can be written as [19], Ep= q Et.r+B_ E.d ./ (111) i \ 2mic where the position, momentum and mass are represented by r, p and m, respec tively, and c is the speed of light. The second equality expresses the interaction energy in terms of the system dipole and magnetic moments, respectively. If a classical description of the electromagnetic fields is maintained and we replace 8 the dipole and magnetic moments with their quantum mechanical operators the interaction energy becomes for electrons of charge e ielir x Vi r eL 2mc i=1 2mrnc (112) In the above L is the angular momentum operator. The space and time variations of the electromagnetic fields of transverse plane waves in a vacuum can be expressed in a righthanded coordinate system (e'1, e2, e3) with propagation along ^3 as E(r, t) = (Ele1 + E2e2) exp(ike3 r iwt) (113) B(r, t) = e^3 x E The magnitude of the wave vector is denoted k and has dimensions of reciprocal wavelength. The variation in coordinate for an electron in a molecule is of the order of Angstoms while the wavelength of radiation in the ultraviolet region is of the order of thousands of Angstroms; therefore, the spatial variation of the electromagnetic field can be neglected to obtain [20] E(, t)= (Elei + E2e2)exp(iwt) = Er exp(iwt) (114) B(, t) = (Ele2 E2e6i)exp(iwt) =Br exp(iwt) . The terms El and E2 are, in general, complex. Two special cases are of interest, that of linear and circularly polarized radiation. Let El be real and E2 purely imaginary and of equal magnitude as 9 E1. Through the use of the identity exp(iO) = cos 0 + i sin 0 the real component of the fields can be expressed EL (, t) = El(e1 cos wt e2 sinwt) (115) BL (, t) = E(e&2 cos wt F ei1 sinwt) R The upper sign designates left circularly polarized light as the rotation of the polarization vector is counterclockwise as observed towards the origin from the positive axis of propagation. The lower sign designates right circular polariza tion. The electric and magnetic vectors maintain mutual orthogonality. Linearly polarized radiation can be expressed as an average of pure left and right circular polarizations. Combining Equations (111) and (115) the interaction Hamiltonian of in terest is obtained. H'L = E [(ei d + e2* fl)coswt (e2 d d ) sin wt] S(116) H'L = EI(HI1 cos wt H2 sin t) R 10 where HI and H2 are independent of time. Defining wkm (E' E )/h Equa tion (19) becomes t/ akL (t) =k() + i exp(iwkmit) cos t( klilI m)dt 0 t' iE J exp(iwkmt) sin wt(bklI2 m)dt 0 t(117) akL (t') ak(0) + (bkHli ti m) i exp(iwkmt)cos wtdt 0 t' (bk IH21im) / exp(iikmt) sin wtdt 0 The sine and cosine terms can be written as exponentials cos wt = (exp(iwt) + exp(iwt)) 2(118) sin wt = (exp(iwt) exp(iwt)) 2 and the explicit integration over time performed [19]. ak(t) = ak(0) R SliEl exp[it'(wkm + _)] 1 exp([it'(wkm )]  2+( kkm + O km + (119) T fi2m) _El [exp[it'(wkm w)] 1 exp[it'(wkm + w)] 1 T (0kH2 m)^  ^  2h wkm O Wkm + W Equation (119) was derived on the premise that t' is small; therefore, the only cases for which ak becomes of appreciable magnitude is if w approaches Wkm. Since exp(itIO) 1 iexp(it = it' (120) a0o 0 11 the expression does not become infinite [18]. If E )E0 then the case of w=wk corresponds to an absorption of radiation to go from state m to state k. For the opposite case we obtain the probability of stimulated emission from a higher state to a lower state. The probability for absorption for k 5 m is then given by Equation (110) recalling the initial conditions on ak. Pk, = akL (t')ak (t') = k (H1,H I2 x Pk(E, t') = R R R R {l(klHil lm)12 + 1(OkljA2[m)j2 (121) Ti[(kHlCm)( kH2m) (kklHllm)*(k IH2ilm)]} E 2 exp[it'(wkm w)] exp[it'(wkm w)] 412 km )2 The time dependent portion of the probability can be simplified to yield ( ) E sin [.5(wkm  (1)t'] Pk L = Pki Hk) ,I x [Si2[. m  W)2 (122) R R Ih (km W)2 For a source of radiation that has equal portions of left and right circularly polarized light we can define two different types of absorption processes. The average of left and right polarizations (Pklin = Pk = .5(PkL + PkR)) yields linear polarization and the absorption process is the total response of the material to the perturbation. The remaining component corresponds to the difference in absorp tion between left and right polarizations (Pkdif PkCD = .5(PkL PkR)). This 12 is known as circular dichroism [21]. The spatial components of the probability of absorptions becomes, Pk(Hil,H2) = I(kHllim) 12 = 1(kklei dl m)2 + I(^kl2 *l m)12 (123) = dle dkm2 + e2 Fkml2 and Pkco (Hi, IH2) = [(ObklI3 2 1m)(k)kI2jlmbmm* (1kHllm) *(iPkI72zlm)] (124) 2 = i[(1i dkm)(e'l km) + (e2 dkm)(e2 k Am)] Assuming that the energy for the absorption process is supplied by the external radiation of frequency v then Wkm W = (E' Em hv)/h = 27r(E~ Eo hv)/h (125) = 27(vkm v) Normal radiation is not composed of a single frequency but is better described by a frequency distribution. The magnitude of the field E1 is related to the radiation per unit volume (U1) via UI = (E1)2/8r which is also in general a function of the frequency [22]. Let ul(v)dv be the electromagnetic radiation per unit volume for frequency v in the differential range v to v + dv. In terms of probability per unit frequency the field dependent portion of the transition probability becomes [18] 8x sin2[ (km u)t'] d  Pk(El,t',dv) = ul() (m dv (126) h (Vkm V)2 13 Only for v close to vUk does the frequency dependence of the probability have appreciable magnitude. The frequency of visible light is on the order of 1015 cycles per second or Hertz (Hz). We can therefore approximate ul(v) to be constant in the region of Vkm and integrate over all frequencies to obtain, via f dx sin2(ax)/x2 = ar, Pk(E, t') = Pk(Vkm, t') = 27r1(km)t (127) The probability per unit time is obtained by dividing by t'. Thus far we have only been dealing with radiation that is propagating along a specific axis. For isotropic radiation (ul = u2 = u3 = u/3) we have to sum the probabilities over all orientations to obtain the total probability of absorption. The final expressions for the probability of absorption of a single molecule to go from state m to state k per unit time is Pk = [ldkmi2 + kml2] (Vkm) 2r 2w (128) PkcD = 2i[dkm /km] 2U(V(km) 3h The Reporting of Theoretically Calculated Spectral Probabilities The total probability per unit time for the absorption of a system of N molecules is simply N times the single molecule probability. There are two parts of the transition probability that are characteristic of the molecule and not a 14 function of time or the radiation field in this treatment. The first is the frequency or energy of the transition. The second is the proportionality constant that is a function of the transition matrix elements of the dipole and magnetic moment operators. The probability for absorption can be formulated as a proportionality constant times the density of the radiation times the number of molecules. The proportionality constant then becomes [18] 21 Bmk [dkml + kmI 2 i2 3 (129) 2z BmkcD = 2i[dkm (km]2 The problem that must be addressed theoretically, therefore, is the attainment of the transition energies and the transition moments themselves. The transition moments have been expressed as matrix elements between wave functions of the total time independent Hamiltonian operator. NT 2 NT Si + E q (130) 2m; ij i i,j(i These state functions therefore include both the nuclear as well as the electronic components. In this treatment the electronic portion of these transition moments is of interest. The mass of the proton is approximately 1836 times the mass of the electron; therefore it is reasonable to assume that the electronic motion can be treated separately from the nuclear motion. In other words, since the electronic motion should be very fast with respect to the nuclear motion the nuclei can be treated as fixed with respect to the electronic motion and treated as a potential 15 field. This is commonly known as the BornOppenheimer approximation [17, 23]. For n electrons and M nuclei the Hamiltonian for fixed nuclei becomes S2 "2 ( 1 M M 2 Hei= m + 2 ZE Z Z (131) i mi j(i rij i ,l a r# Solutions to Equation (131) for arbitrary nuclear positions then defines a potential energy function (V(ri)) from which the nuclear motion Hamiltonian (HT) can be defined. M 2 Hoi __ V2 2M (132) l= 2m V + V(ri) Under the BornOppenheimer approximation the Hamiltonian operator is par titioned between the electronic and nuclear motion. Therefore the state functions for the total Hamiltonian can be expressed as a product of nuclear and electronic functions in which the electronic wave function depends parametrically on the coordinates of the nuclei. Ok = 0kel (ri, {r.}) ONk (r.) (133) Transition elements for perturbation operators such as dipole then become (OklbIdm) = (~ elk/Nk mIdl mEel Nm,) (= ('Nk, (elk Ide,)em Nm,) + (melkI(Nk, kdN Nmi)'eim) (134) = (bNk, (Welk delCJel mbN,,), k $ m 16 The nuclear contribution to the dipole operator vanishes for transitions between electronic states and will not be considered further. If for given initial nuclear and electronic state m and specific electronic state k, (ielk Id.elm,) is treated as an average electronic moment for all excited nuclear states, k', the electronic integral can be removed and the squared matrix element written Idkm2 = 2 I elk dl elm)l2 X (Nk INm,)( I'N,) (135) For a given electronic transition from state m to k many possible nuclear transitions are possible. The squared overlap between nuclear state functions is known as the FranckCondon factor for the electronic transition. The nuclear states correspond to various vibrational and rotational states for the molecule. The FrankCondon factors then give the relative intensities of electronic transitions as divided among the various final nuclear states [18]. Considering that the ONm, are eigenfunctions of a Hermitian operator corresponding to electronic state m, they form a complete set of orthonormal functions and therefore the total electronic transition probability over all final nuclear states is ldkml2 = I()elk dl el m)2 X ZiN NIN )(NmIlNk,) m' m' (136) = I(elk el X N,,)N,, = I(< elkl dI l)I2 Therefore the total probability of an electronic transition under the Born Oppenheimer approximation can be expressed at the average electronic state 17 probability. This average is usually further approximated as the value obtained between electronic states at a fixed geometry. With the approximations to the transition moments presented above, the theoretical problem is reduced to the determination of the state functions and energies of the electronic Hamiltonian for a molecule. The frequency of a transition is obtained from the difference in state energies and the probabilities of interest are found as straightforward integration of the perturbation operators over the electronic state functions. The total probability for electronic transitions is proportional to the sum of both dipole and magnetic moments. For dipole allowed transitions the magnetic moments are usually 104 those of the dipole transition moments and are usually neglected. Consider a three dimensional harmonic oscillator of the same mass and charge of an electron initially in its ground vibrational state with vibrational frequency the same as that of the electronic transition in question. The proportionality constant for a change of one vibrational quantum number from the ground state is given by [18] 7re2 we2 B01 = (137) Vkmmh Ekmm The convention for reporting total electronic absorption probabilities is in the form of oscillator strengths which is defined as the ratio of the electronic proportionality 18 constant to that of the pure harmonic oscillator described above. The result is a dimensionless constant usually designated f,. S47rVkmm 2 2Ekmm 2 f = IK2 Ikm = e2Idkm 3he2 3h2e2 2Eknm 12 (138) The "r" indicates the use of the dipole length operator to calculate the transition matrix elements [24]. For circular dichroism the proportionality constant can be rewritten 27r 44r BmkcD = 2i[dkm km]3h = Rmk 2 mk = ikdbm}) (bklIkm) (139) n 1 2mc1l k m i x i,k) i i Rrk is known as the rotary strength of the electronic transition from state m to k. Rotary strength has both sign and magnitude since it represents the difference between the probabilities of absorption for left and right polarized light [21, 25]. For equal absorption of left and right circularly polarized radiation the rotary strength would be zero even though the total oscillator strength could be substantial. As with oscillator strength rotary strength is proportional to the total electronic probability averaged over all final nuclear states. Oscillator strength is a dimensionless quantity but some system of units must be used in order to account for the magnitude of rotary strength [26]. The most 19 commonly used system is cgs units. Rotary strength in cgs units becomes n n Rnk(cgs) = 4.45458 x 1030o(mI I i I'k)(bm i X x i ik) (140) i i It is customary in quantum mechanical calculations to use atomic units to calculate transition energies and moments. In atomic units h = e = m = 1. The atomic unit of energy is the Hartree and the atomic unit of length is the Bohr. Using atomic units to calculate transition moments and energies, oscillator strength and rotary strength becomes S 2Ekm(au) nm 2 fr = 3 I(lmI Srik)l2 n n (141) Rmk(cgs) = 235.7262 x 1040(m I rk)( m i X Vi 'k) i i Common units for reporting transition energies are electron volts and reciprocal centimeters (1 au = 27.212 eV = 219474.6 cm1). The conversion constant for rotary strength from cgs to SI inits is 1.19x106. CHAPTER 2 GENERAL THEORY The HartreeFock Equations In order to determine the electronic transition moments and energies for a molecule it is necessary to determine in some fashion the eigenfunctions and energies of the electronic Hamiltonian. In atomic units the electronic Hamiltonian becomes n 2 n 1 M o +  E + Vnuc. (21) ii j(i a The first term is the electron kinetic energy operator. The second sum contains the potential energy of repulsion of the electrons and the potential energy of attraction between the electrons and nuclei. The final term is the nuclear potential energy and is a constant for fixed geometry regardless of electronic state. Since we are interested in differences in electronic states this term is dropped. We seek solutions to Equation (21) of the form H j) = Ejljj) (22) The eigenstates Oj are functions of the electronic coordinates and spin. The nonrelativistic Hamiltonian used in this treatment is not a function of the spin 21 coordinates of the electrons. Since electrons are spin 1/2 and therefore fermions the eigenstates for a system of electrons must be antisymmetric with respect to the interchange of the coordinates of any two particles [17]. It is necessary therefore to seek solutions that have the property j(ql, q2," " ) = j(q2, q1, ) (23) where ql is the space and spin coordinate for electron one. According to the variation principle, for an arbitrary well behaved normalized wave function, 14), the expectation value for the Hamiltonian of a system is an upper bound to the exact ground state energy. In other words Eo < (IHIO4) (24) If 1() contains adjustable parameters they can be varied so that E =(IHoj) is a minimum and thus 14) becomes the best possible approximation to the ground state wave function and E the best possible approximation to the exact ground state energy. The calculated energy (E) only becomes the exact if 1j)) is the exact wave function [19]. In general it is not possible to solve Equation (22) for more than two electrons. It becomes necessary to seek expedient approximations that allow generalized methods of solution. The most common such approximation is the molecular orbital or HartreeFock (HF) approximation [23]. In the HF 22 approximation Oj is expressed as the best possible determinant of oneelectron spin orbitals. Since a determinant changes sign when two rows or columns are transposed, which in this case represents a single electron function, the fermion property of antisymmetry is attained whereas a simple product of spin orbitals would not have this property. For n electrons j) 4 ll(qi)02( (q2) m(qm)... n(qn)) n! (25) = (1)"Pij 1(ql1)2(q) m(qm) n(qn) (25) i=1 and is known as a Slater determinant. The symbol Pi is the permutation operator for the electron labels and the sum is over all n! permutations. The number of single label interchanges necessary to restore the i'th permutation to the initial order is denoted pi. The 1/vn! factor is the normalization constant so that ( j lIj) = 1. Utilizing the variational principle with a Slater determinant of orthonormal spin orbitals yields Eo < E = (o0H110o) (26) The calculated energy becomes a function of the molecular orbitals that make up the determinant. The remaining discussion is restricted to closed shell deter minants. Insertion of Equation (21) into (26) and integrating over space and spin yields E = O q*() Zi()dq i=l 27 L i ql)dq =1 =1 + f / O(ql)Oj(q2) [i(qi)Oj(q2) Oj(qi)Oi(q2)]dqidq2 n n (27) E Oilfo) + 2 [(E jlij) (Oij0j0\ i=l i,j=l n n (iIlhi) + (iij i=1 i,j= The operator h is a oneelectron operator consisting of kinetic energy and nuclear electronic attraction. The i'th term is usually denoted hii. The two terms relating to the electron repulsion are known as the coulomb (Jij) and exchange (Kij) integrals respectively. The exchange integral is the direct result of using a determinantal form for the wave function and would be absent if the wave function were a simple product. In simplified notation Equation (27) becomes n n E = E(ili) + 1 (ijllij) i=l i.j=1 n n (28) = hii + 2 [Jij Kij] i=i i.j=1 In order to obtain the best possible solution to Equation (28) the spin orbitals are varied under the constraint of maintaining orthonormality. Let eij be a set of 24 Lagrange multipliers so that the Lagrange variational expression becomes L = E eji[(iIj) 6ij] ij (29) 6L = 6{E eji[(ilj) bij]} = 0 ij Differentiating the orbitals and rearranging terms yields J dqis l(1){h(1) i(1) + [dq2O(2) ji(2) ]i(1) i=1 j=1 dq2o (2) Pi2j(2) i(l) + complex conjugate (210) j=lV Jj J J n n = i dql6 (1) e jiij(1) + complex conjugate i=1 j= The terms in brackets are known as the coulomb and exchange operators respec tively. The variation of the spin orbital is arbitrary so Equation (210) must hold term by term or 1h(1) + E [ j] }ki(1) = f(1)i 1) = Esjij() (211) j=1 j=1 where the first equality defines the oneelectron Fock operator. Multiplying by Ok and integrating one obtains (k [fli) = Iki. The Lagrange multipliers are matrix elements of the Fock operator in the basis of the spin orbitals. Since the Fock operator is invariant to a unitary transformation and the matrix of Lagrange multipliers is Hermitian it is possible to choose the Oi so that the matrix of Lagrange multipliers is diagonal. That set of Oi and ei are known as the 25 canonical HartreeFock equations. The solution of the HF equations reduces to the determination of the eigenvalues and eigenfunctions of the Fock operator. f i) = ii) (212) with the Fock operator defined in Equation (211). Since the Fock operator itself is a function of the spin orbitals the methods for the solution of Equation (212) are in general iterative in nature. The energy minimization that led to Equation (212) pertained only to the n "occupied" molecular orbitals directly associated with the electrons. Once these orbitals are determined, the Fock operator becomes a welldefined Hermitian operator and thus possesses a complete set of solutions. Those orbitals beyond the active occupied orbitals are known as virtual orbitals. The eigenvalue for an arbitrary canonical orbital can be expressed n Ei = (iIf Ii) = hii + [Jij Kij] (213) j=1 Equation (28) represents the total electronic energy expression for n elec trons. As a reasonable first guess, the orbital solutions for n electrons can be used to estimate the energy for a system of (n+1) or (n1) electrons. The difference in energy between an n electron state and an (n1) electron state is known as an ionization potential (IP). The difference with an n and (n+1) state is known as an 26 electron affinity (EA). Using the orbitals obtained for an n electron system and Equation (28), one obtains a molecular ionization potential for arbitrary occupied orbital a and an electron affinity for arbitrary virtual orbital b. E,(n 1) E =IP = e (214) E Eb(n + 1) =EAb = Eb Equation (214) is an expression of Koopmans' theorem which interprets the eigenvalues of the occupied orbitals as the negative of the IP's for those orbitals and the eigenvalues of the virtual orbitals as the negative of the EA's for those orbitals [23]. The use of the molecular orbital approximation yields an approximate ground state for a molecular system. Since the solutions for the Fock operator form a complete set the exact wave function for a system can be expressed in terms of an expansion of the these solutions. From the HF ground state the occupied orbitals can be successively replaced by the virtual orbitals and classed in terms of single, double and higher replacements [27]. o) = [Cio0) + CI0) + c a C ab) +" ] (215) aa aafab The determinant generated by the replacement of occupied orbital a with virtual orbital a is denoted Ia) with corresponding expansion coefficient Ca. The coef ficients of the determinants become the parameters for the use of the variational principle. Matrix elements of the Hamiltonian in the basis of determinants must 27 be formed to solve the problem. The matrix elements between the HF ground state and a single replacement can be expressed n (oI ) =haa + (ailai) = i=1 (216) (0aIf #a) = Ea(0,a1a) = 0 The determinants formed from a single replacements do not directly interact with the HF solution for the ground state. This is an expression of Brillouin's theorem. The lowest order correction to the HF ground state is through double replacements only. The eigenvalue problem in the complete basis of determinants yields not only the exact ground state energy but also the wave functions and eigenvalues of the excited states of the system [28]. The Roothaan Equations The HartreeFock equations in the previous section do not suggest a method by which to obtain the molecular orbitals that define the Fock operator. The most common way is to assume a basis set that is pertinent to the system of interest. For molecules this usually involves an atomiclike basis. Each atom in a molecule contributes a set of functions to the total basis for the solution of the HF equations. The number and type of functions that an atom can contribute is limited only to the computational resources of the researcher. The smallest basis that an atom can contribute is called the minimal basis set (MBS) and is usually that basis associated with the shell structure of an atom. Each principle quantum 28 number on an atom contributes a fixed number of basis functions of specific angular momentum. The molecular orbitals associated with the Fock equations become linear combinations of atomic orbitals (LCAOMO). N i = I Cix (217) p=1 The expansion coefficients become the variational parameters. The xy are the atomic basis functions and in general are a function of the electronic coordinates only. The total number of basis functions is N. The spin of a molecular orbital is denoted by the existence a bar over the symbol for orbitals that are spin 1/2. No bar is present for spin 1/2. Alternatively, spin 1/2 is often denoted as / and spin 1/2 as a. Orbitals of differing spin are orthogonal. Placing Equation (217) into Equation (212) and multiplying by the atomic basis functions and integrating yields N N (Xl1 // CEiIXu)= eiCCiIX}, v=l v=l N N SCi(xlf xv) = i E Ci(X Xv) (218) v=l v=l N N SF rVi = C i Y St Cvri V=1 V=1 where the elements of the Fock (F) and overlap (S) matrix are defined. In terms of matrices FC = SCe (219) 29 is known as the Roothaan equation [29] and is the most used formalism for the solution of the HF system of equations. The columns of the coefficient matrix are the expansion coefficients for the molecular orbitals and e is a diagonal matrix of orbital eigenvalues. For N basis functions one obtains N molecular spin orbitals, n of which are occupied and (Nn) are virtual. A determinant in which each spatial orbital is associated with both spin a and p functions and both are occupied is known as restricted HartreeFock (RHF) and is the most common type of calculation performed. Introduction of Equation (217) into the expression for the Fock operator and calculating the matrix elements of the Fock operator in Equation (218) for a RHF determinant yields n/2 N F, = h, + [ CC L [2(r a) (Irjov)] (220) The iterative nature of the Fock equations is demonstrated in that the coefficients of the occupied orbitals must be known in order to define the Fock operator but these are the same coefficients that are sought in the calculation. The intermediate sum over the occupied orbitals and coefficients in (220) defines the so called density matrix P = CnCt. Here n represents the diagonal matrix of orbital occupancies the trace of which is the number of electrons. For an RHF determinant the spatial orbital occupancies are either 2 or 0. n/2 Par = 2CiCri (221) 1 30 The elements of the Fock matrix become SN FV = h, + Pr[2(rTjv) (srl 7r)] (222) The matrix elements of the oneelectron operator in the atomic basis do not depend on the density matrix. For real spatial atomic orbitals the overlap matrix is symmetric and can be diagonalized and real roots taken of the matrix. Let X be the transformation matrix that diagonalizes S so that XtSX = s. S = XsXt = Xsl/2s'/2Xt = [XS1/2Xt][Xsl/2Xt] (223) If there are not any linear dependencies in S then 1 = [Xs1/2Xt][Xsl/2Xt] = U[Xs/2Xt] (224) defining the matrix U = S1/2. Consider the transformation of Equation (219). Ut // FIC = FU[Xsl/2Xt]C = [Xsl/2Xt][Xs1/2Xf]Ce = SCe [UtFU]C' = {Ut[Xsl/2Xt]}C'e (225) F'C' = C' This procedure by which the atomic basis is orthogonalized is known as symmetric or L6wdin orthogonalization [30] and the Fock procedure is reduced to the matrix eigenvalue problem C'tF'C' = C'tC'e = e. Once the atomic basis is chosen for a problem the solution usually follows well established lines. The general algorithm is to first evaluate all the integrals in 31 the atomic basis. The overlap matrix is diagonalized and the transformation matrix U is formed. An initial guess for the density matrix is formed and the Fock matrix is built, transformed and diagonalized. The coefficient matrix in the atomic basis is formed by the back transformation C = UC'. A new density matrix can then be formed yielding a new Fock matrix and the pattern repeated. The procedure is considered converged when differences in successive density matrices are within some set threshold. This technique is known as the selfconsistentfield (SCF) procedure. The Calculation of Electronic Spectra The HartreeFock procedure for a given basis provides a convenient set of orbitals from which Equation (215) may be variationally solved for all the electronic states of a system. The full expansion provided by all orbital replacements is known as a full configuration interaction (CI). For the special case of a complete basis set, the full CI is said to be "complete" and represents an exact solution of the problem represented by the assumed Hamiltonian. For other than very small systems, however, full CI is not feasible and some truncation is necessary at some order in the expansion. From Brillouin's theorem the lowest order correction to the ground state comes at the inclusion of double replacements only. Since the single replacements do not mix directly with the ground state they become the basis for the lowest order approximation to singly excited states. It is 32 the transition moments and energies between such excited states and the ground state that are of interest here. The expressions for oscillator and rotary strength in Chapter 1 involve matrix elements between states for the electron position and angular momentum oper ators. These expressions are not unique however. The commutation relation between the Hamiltonian for the system and the position operator [31] is [ io = f i (226) i=1 i=1 where Vi is the velocity operator for electron i. The evaluation of matrix elements between the ground and excited states of the Hamiltonian operator yields n n n (001 NolO) = (iol1 rlH0 H iik) i=l i=l i=l n = (Ek E)(o  i k) (227) i=1 n n (o I Vilk) = EkO(O 1 E i 4l k) i=1 i=1 Equation (227) indicates that the matrix elements for the position operator can be replaced by those of the velocity operator divided by the transition energy [32, 33]. Equations (138) and (139) for oscillator and rotary strength in terms of the velocity operator become 2Ek0 n , i=l (228) f_ 2 n, i=1 and n n R = CR( O ri kii 0k)(oI >i x Vib) = i=1 i=1 (229) SCR EkO (0 dik)(0 i XVi k) i= i=1 For exact eigenfunctions of the Hamiltonian, transition moments in both length and velocity formalisms are equivalent. Since approximate solutions to ground and excited states do not necessarily obey Equation (227), the equivalence of oscillator and rotary strength in Equations (228) and (229) can not be assumed. Equation (227) constitutes a formal constraint that must be incorporated into the calculation of approximate excited and ground states if equivalence is to be achieved. For practical purposes the HF approximation and the HF plus doubles approx imation will be considered for the ground state and singles only for the excited states [10]. Truncations at higher order are not feasible for use with large sys tems. IYo = YKo[I'HF)+{y I C b a# }a#) aofab (230) yc Transition moments for a general oneelectron operator 0 is expressed in terms of the approximate states in Equation (230) as 1 k Cab*C( o I (231) aflbabc 34 Matrix elements for oneelectron operators are identically zero if the two deter minants differ by more that one orbital replacement. ( 1,o,^.) = (fl^ c) [6,. + ,, + 6 c + ~ ] (232) The indices of summation in Equation (231) are arbitrary so that (0o I0lk) = ^ KiC ,(rHF IOI) 7c C; I (233) aac but (Oa6)aOI) (,l (IO HF) yielding (0Ck)= KC(HF= Klt ) 7c (234) + { KiC Cc (,16OrHF)} aa 0 C For excitation k the following notation is adopted. xk = K Cc(k), yk, = ZK aC (k)CC(k) (235) aa This yields, (0i^O14k) = [xcHFI=OC) yc (236) +{ YI C(O/6B1HF)}] The commutation relation between the general operator 6 and the Hamiltonian operator with the exact state functions [9] is according to Equation (227) (4'oIN ) k)Eko = (Ool[0,HO ]01k) (237) 35 If the commutation relation itself yields a oneelectron operator as in Equation (226) the right hand side of Equation (237) becomes for the approximate states defined by Equation (230) RHS = E [Xc((HF6HIO ) ( HF 061C)) ye (238) +{Y (( 6f2o HF) (OcH6HF))} Since the eigenfunctions of the Fock operator form a complete set we can resolve the identity 1 = IHF)() H + { )( }+ {I )( 1} I )(a}+ (239) between the Hamiltonian and 0. Utilizing Brillouin's theorem and the fact that matrix elements of the Hamiltonian between two determinants that differ by more than two orbitals is zero the insertion of Equation (239) into (238) gives RHS = X X( (HF I Sd)( i C) 7c bd S( ~HI HF HF HF If6d)(o (IOI)) 6Sde SHF H(240) +YC((6 HF)HFHF) + 4 I11OHF) S'6de Z(4 IHI ) IO HF))}] bd 6 Q H) Defining Ad,,c (dIHt ( HFiHRHF)(1dc (241) B6d,,e = (oH^HF) 36 and simplifying according to Equation (232) Equation (237) takes the form Eko E d[X HF ) {Y bd 1OHF)}] bd 7c L 6d +{Y ((HF 6 ^d )Bbd,7c (46OizHF)A de,6d) 6d The operator 0 can in general be Hermitian as is the position operator or anti Hermitian as is the velocity and angular momentum operators. Most calculations are performed in a basis of real orbitals in which the A and B matrices are symmetric. For Hermitian operators and real orbitals Equation (242) becomes ( lnHVH )[(Xk y,)(A6d, Bsd,7c) 7c 6d Eko(Xd + Ykd)76cd] = 0 (243) (Xc  Yc)(Asd,7C Bsd,7c) = Eko(X d Yd) yc and for antiHermitian operators (Xkc + Ykc)(Abd,7c + B6d,yc) = Eko(Xk kd) (244) 7c Equations (243) and (244) thus define a set of simultaneous linear equations that must be solved in order to obtain the coefficients necessary for the calculation of transition moments for both Hermitian and antiHermitian operators. In matrix form they become (A B)(X Y) = (X + Y)EK (245) for Hermitian operators and (A + B)(X + Y) = (X Y)EK (246) for antiHermitian operators. The columns of X and Y are the expansion co efficients for the excitations and EK is a diagonal matrix of transition energies. Equations (245) and (256) are the RPA Equations [8, 11]. The first case to be considered is for excitations from the HartreeFock approximation to the ground state. This is obtained by setting the matrix Y identically to zero obtaining (A B)X = XEK (247) and (A + B)X = XEK (248) which are two eigenvalue problems for X and EK. Since A and B do not commute they do not posses a common set of solutions. Therefore transition moments cal culated in different formalisms will not in general be equal. Equation (237) can be solved for Hermitian or antiHermitian operators separately, however, by the diagonalization of (AB) and (A+B) respectively. If the matrix B is neglected, Equations (247) and (248) reduce to the diagonalization of A which is exactly the matrix obtained from the variational solution of the singles only configura tion interaction (CIS) or also known as the TammDancoff approximation (TDA) 38 [34]. The TDA is the most commonly used approximation for the calculation of transition moments and energies but as seen here cannot guarantee equivalence between transition moments as calculated in different formalisms since Equations (227) and (237) do not in general hold. Only in the limit of a full CI can a variational procedure guarantee this equivalence. The second case is the direct solution of Equations (245) and (246) which were derived on the basis of a ground state consisting HartreeFock and double replacements. Taking the linear sum and difference of Equations (245) and (246) yields AX + BY = XEK (249) BX + AY = YEK which is the linear form of the matrix equation [A B ] =[ X EK (250) Since we have assumed real orbitals the matrix of A and B is real and symmetric and can be solved as [27], [Xt Yt] [A B]j ] =[Xt ytj X ]EK = [XX YtY]EK = EK (251) with the special normalization constraint [XtX YtY] = 1. Direct solution of Equation (250) would involve the diagonalization of a matrix of twice the di mension of the number of single replacements in order to obtain the coefficients 39 and transition energies. The problem can be reduced to two successive diago nalizations of the same dimension as the number of single replacements. Since diagonalization scales as the cube of the dimension of the problem, two diagonal izations of dimension N are preferable to one diagonalization of dimension 2N. According to Ullah and Rowe [35] let M = A + B, N = A B, U = X + Y and V = X Y so that Equations (245) and (246) become NV = UEK, MU = VEK (252) The normalization constraint becomes VtU = UtV = 1 (253) since with real orbitals and real coefficients vtU = (Xt Yt)(X + Y) = XtX yty + Xty YtX = xtx yty + [XtY (Xty)t] = XtX yy = 1 (254) Xty = 4[UU VtV (UtV V'U)] = [UtU VtV] 4 If N is positive definite then V = N'UEK and NMU = NVEK = UEK~ (255) 40 Equation (255) is similar to the Roothaan equation and an analogous treat ment is sought. Let T be a unitary matrix that diagonalizes M so that TtMT = m, M = Tml/2m1/2Tt. Equation (255) can be transformed ml/2Tt // NTml/2m/2TtU = UEK [ml/2TtNTml/2][m/2TIU] = [ml/2TtU]E (256) N'U' = U'EK into a Hermitian eigenvalue problem. To obtain U the reverse transformation U = Tm1/2U' is performed. V can be obtained form Equation (252). TTt // MU = MTm1/2U' = VEK Tml/2U' = VEK (257) V = Tm/2U'Ei E In order to obtain X and Y the normalization constraint must be taken into account. UtV = [U'tm1/2Tt][TmI/2U'EK] = 1 (258) UV = U'tU'E1 = 1 The normalization condition is obeyed provided the vectors in U' are scaled so that U'tU = EK (259) With U' properly normalized and U and V obtained as above, X and Y are simply obtained from X= (U +V), Y = (UV) (260) 2 2 The use of HF plus doubles as a reference for the ground state and singles only for the excited states is successful in the maintenance of Equation (227) as a constraint. The resulting solutions are equal to those of the random phase approximation (RPA) or polarization propagator (PP). The RPA in a complete HF basis is the lowest level of theory in which a formal equivalence according to Equations (228) and (229) can be expected. With the RPA the transition moments and energies are obtained directly. No formal correlated ground state or excited states are explicitly formed. Singlet Excitations with a Finite Basis The RPA yields formally equivalent expressions for the calculation of optical properties for a complete HF basis. In practice, however, a finite basis must be chosen in which Equation (239) does not rigorously hold. Calculated optical properties in differing formalisms become a test for the completeness of the basis used. Often a full singles space calculation is prohibitive even in a finite basis and some truncation of the active orbitals is used to calculate transition moments and energies. If electronic excitation spectra is of primary interest usually only the lowest excited states are needed. It therefore becomes necessary to study the stability [36] and equivalence of the calculated transition energies and moments 42 for the lowest energy transitions as a function of the size and nature of the active space. For this study singlet excitations from a closed shell ground state will be considered. The RPA gives two sets of transition vectors [37]. For Hermitian operators (l016KHIk) = (KHF OH )[Xkc + Yk] (261) 7c and for antiHermitian operators (0olOAlIk) = E(]HFIOA c)[Xkc yk ] (262) 7c The singlet spin adapted excitation 1Cryc) in terms of spatial orbitals is 1l07c) = 1[114c) 7) I+lcE)] (263) The transition moments in Equations (261) and (262) become matrix elements of the oneelectron operator in the MO basis. ('lHFlOIlrtc) = [('ObHF6lOl1rc) + (1CHF1IO61',)] 1f2 = +[(< 610c) + ( 61c] (264) = 2 (161c) The operators are usually evaluated in terms of the atomic basis and, after the SCF coefficients are obtained, transformed into the molecular basis. N (6101c) = Ci'r(xi Ixj)Cjc Oj (265) Oc(MO) = [CtO(AO)C], 43 The singlet spin adapted matrix elements of the TDA matrix and B matrix defined by Equation (241) become Abd,7c = (r6 ec)6766cd + 2(ybScd) (yc Sd) (266) B6d,7c = 2(761cd) (76dc) where the twoelectron integrals have been suitably transformed from the atomic basis to the molecular orbital basis and the ei are the SCF orbital eigenvalues. CHAPTER 3 THE INDO/RPA METHOD Theory Use of the LCAOMO approximation without further simplifying assump tions creates a computational barrier. The obstacle is the evaluation of approxi mately N4/8 twoelectron integrals over the atomic basis set. N is the number of atomic basis functions. An MBS for benzene would consist of 42 basis functions and require approximately 400,000 integrals to be evaluated. Accurate calcula tions of molecular properties with ab initio techniques, however, often require extended basis sets. A second set of p functions on benzene increases the num ber of integrals to 2,400,000. Commonly used semiempirical Hamiltonians for electronic structure owe their efficiencies to reducing the number of integrals pro cessed to N2. In so doing matrix multiplications and diagonalizations, both N3 steps, dominate, at least at the HF level. In addition most semiempirical methods utilize a MBS of valence type orbitals only, further reducing N in the more com plex N3 steps. These models compensate for the reduction of integrals and small basis sets by parameterizing directly on atomic phenomena and molecular prop erties, and are thus often able to reproduce experiment in a more accurate fashion 45 than the MBS ab initio calculations on which they are modeled. A valence MBS for benzene consists of 30 basis functions and only 900 integrals. The zero differential overlap (ZDO) approximation at the intermediate neglect of differential overlap (INDO) level is adopted for this work [6, 3842]. The ZDO approximation assumes the differential element X,(1l)Xv(1)drl to be zero unless p=v where it occurs in integral evaluations. A complete description of the approximations for INDO and other levels of ZDO can be found in Sadlej [43]. The basic equations are summarized below. For INDO the ZDO approximation is applied to all twoelectron integrals in which two or more centers, denoted by capital letters, are involved and all oneelectron integrals associated with the SCF procedure. Certain twocenter integrals will be spherically averaged in order to maintain rotational invariance. The overlap matrix becomes the unit matrix. Sp,' = S6p, (31) All onecenter twoelectron integrals remain and the twocenter integrals are spherically averaged and symbolized by 7AB. (IAvBIACO'D) = (PAvBIlAvB)SAC6BD6SA6Ta = 7ABSACSBD6gASvo A 0 B (32) (/PAVBIAD) = (AVAAIAAAA)SAC6BD A = B Several cases exist for the oneelectron matrix elements. Kinetic energy integrals do not contain the differential element for which the ZDO approximation 46 is applied so these integrals remain. If both p and v are on center A the matrix element is partitioned into an atomic part and a twocenter part. As with the two electron integrals all onecenter integrals are kept, but with the use of a Slater basis the oneelectron onecenter integrals are diagonal by orbital symmetry. The core orbitals on one center have a negligible effect on the valence orbitals of a second, therefore, only the valence electrons are included in a calculation. The interaction between a core and valence electrons on a single center, however, cannot be neglected [44]. This energy of interaction scales linearly with the number of valence electrons and can be accounted for by a corevalence potential function, EA, that is added to the atomic portion of the oneelectron matrix elements. 1 1 ZA hPAVA = (PA E + EAIA) = AI Y + EAIVAn)6P 2 C= rlC 2 rlA ZC(YAIVCIVA)b,, COA (33) The symbol Vc is the potential energy of attraction for an electron to center C. The atomic oneelectron onecenter matrix elements are symbolized by UU/p and are called the core integrals. Since differential overlap has been neglected between centers, the attraction of an electron on one center to the core of a second should be similar to a like repulsion. This is accomplished by setting the magnitude of the twocenter nuclear attraction integral to 7AC [38]. hAVA =[Up ZCTAC]&, (34) CCA Twocenter oneelectron matrix elements are not zero due to the presence of the kinetic energy operator. The twocenter nuclear attraction integrals are of different type than those that occur on the diagonal of the oneelectron Hamiltonian and can be neglected by the ZDO approximation. This matrix element is called a resonance integral and is symbolized by /3p,. hAVB= flAVB, A B (35) Use of the ZDO approximation reduces the Roothaan equation to FC = Ce (36) which is the exact same form as the symmetrically orthogonalized equation without the ZDO approximation. Symmetric orthogonalization can be derived by the orthogonalization of a basis that maintains maximum overlap between the orthogonalized basis and the original basis [30]. The matrix elements in INDO can be viewed as if they had been evaluated directly in the orthogonal basis. In this way the coefficients determined from the INDO Fock matrix can be back transformed to the original MBS from which it is modeled. These coefficients, eigenvalues and atomic basis functions can be used to calculate properties such as transition moments and spectra. With this point of view, the 48 ZDO approximation then applies only to the SCF procedure and not properties that are to be calculated that rely on the SCF results. The matrix elements that remain in the INDO Hamiltonian can be evaluated directly in a basis but this approach results in a severe approximation to the full MBS calculation which is usually inadequate for the prediction of properties. Through the judicious use of experimental information the integrals are parameterized so that calculation of the same properties yields accurate results. A successful parameterization is capable of extrapolation to systems and properties for which it was not fit. Integral approximation schemes appropriate for the calculation of electronic spectra are of interest here. Two types of parameters need to be determined. The first are the parameters that are purely atomic in nature. These are the core integrals and onecenter two electron integrals. Since all atomic integrals are kept in INDO, atomic parameters are chosen to reproduce atomic phenomena explicitly. The total energy for an atom can be written in the same form as Equation (28) [45] ny 1 ny E=Ec + Uii [Jij Kij] (37) i=1 i,j=1 Ec is the constant energy of the core electrons and nv is the number of valence electrons. The exchange integrals, expressed in terms of CondonShortly integrals [46], are determined from differences in atomic state energies. The coulomb integrals are a linear combination of the spherically symmetric coulomb integral, 49 denoted Fo or 7AA, and the same CondonShortly integrals that make up the exchange integrals. According to the analysis by Pariser, Fo is set to the ionization potential of the atom less its electron affinity [47]. /7AA = F = (ssss) = IP EA (38) The core terms remain and are determined as that value for which Koopmans' theorem would predict the correct orbital ionization potential. There are two terms that must be addressed in the parameterization of two center quantities at the INDO level of approximation. These are the twocenter repulsion integrals and the oneelectron bond parameters or resonance integrals. The generalized form of many twocenter repulsion integrals used in semiempirical theory can be expressed as 7AB(RAB) = (39) [(A ) m +RAB] I/m K m In this equation fg is known as the Weiss factor [7] and RAB is the internuclear separation. AB(0) is the limit of AB for which RAB=O and is usually expressed as an average of the two onecenter Fo's. The Weiss factor is set to 1.2 and m is fixed at 1 according to the prescription of MatagaNishimoto [48]. With fg and m fixed no further parameters are necessary for ^AB. The INDO twocenter oneelectron matrix element denoted flv is usually expressed as the average of onecenter bond parameters weighted by the overlap 50 between two Slater orbitals and an appropriate fixed interaction factor, fi, for the type of orbitals involved. For example, S.5(3A + B)S, Y,V= s PAVB = .5(15A + B)S, fs = s, V = p (310) S5(PA + B)[fa(gaS,, + firgrSr,], /, V = p The interaction factors for spectroscopic INDO [7] are fs, = 1.0, f, = 1.267, f, = 0.585 (311) The pp interactions are divided between pure sigma and pure pi contributions. The Euler rotation factors are denoted by g. Properties such as spectral transition moments involve the calculation of integrals that are not involved in the SCF procedure and have not themselves been parameterized. In order to evaluate property integrals in the assumed molecular basis, it becomes necessary to view the parameterized SCF procedure as connected to an atomic basis set through orthogonalization. The majority of INDO/S calculations performed to date involve the calculation of oscillator strength in the dipole length formalism only. The dipole integrals are restricted to onecenter contributions and the SCF coefficients are maintained in the assumed orthogonalized basis. For this work all property integrals will be calculated over a Slater basis [49] and subsequently orthogonalized. This will lend to a more systematic comparison between oscillator strength as calculated in differing formalisms. Other properties such as NMR shielding involve many different 51 types of property integrals. Since there is currently no systematic way of parameterizing or truncating these integrals in a balanced fashion, working in a fully orthogonalized basis is the most logical point of reference to explore. Benzene and Pyridine The INDO/S method was originally parameterized to reproduce excitation energies for benzene and pyridine with a truncated CIS procedure. In this section the CIS and RPA methods are compared for these molecules. Oscillator strengths in both length and velocity formalisms are calculated. The effect the size of the active space has on oscillator strength and transition energies is also explored. Benzene is of point group D6h of which the highest occupied molecular orbital (HOMO) is a doubly degenerate pi orbital of Eig symmetry. The lowest unoccupied molecular orbital (LUMO) is also a doubly degenerate pi orbital of E2u symmetry. Single excitations from the HOMO to the LUMO generate excited states that are of Blu, B2u and Elu symmetry. The only dipole allowed transition from the ground state is to the degenerate E1u state. The experimental excitation spectra for benzene consists of three bands. The two lowest bands indicate low probability of absorption and the third has a large absorption confirming the simple analysis above. The presence of the nitrogen in pyridine reduces the symmetry of the molecule to C2v. The excitations from the upper two occupied pi orbitals into the lower two unoccupied pi orbitals are all now symmetry allowed. The 52 split degeneracies from that of D6h are small however such that there remains the same three bands as with benzene. In addition, excitations from the highest occupied nonbonding orbital on the nitrogen to the lowest unoccupied pi orbitals are also predicted. The lowest of these is allowed and the second is not. The SCF orbital eigenvalues for the outer valence region for benzene and pyridine are presented in Tables 31 and 32 and compared to the experimental orbital ionization energies. The agreement with experimental ionization potentials is reasonable with one noted exception. The nonbonded orbital is predicted to lie below the pi orbitals but is experimentally found to be nearly degenerate with the highest pi orbital. This represents the failure of Koopmans' theorem for such orbitals and not the model Hamiltonian itself. Koopmans' theorem does not take into account orbital relaxation which can be substantial for such orbitals. The model Hamiltonian correctly accounts for the experimental ordering when such effects are considered. The calculated spectra for all configurations below 70,000cm1 is presented in Tables 33 and 34. Both RPA and CIS give good estimates for the excitation energies. The RPA excitation energies are consistently lower than the CIS energies. It is well known that CIS dipole length oscillator strengths are usually too large and dipole velocity oscillator strengths are too small. From the form of Equation (228) a smaller excitation energy is necessary in order to reduce length oscillator strength in favor of velocity oscillator strength maintaining constant 53 Table 31 SCF orbital energies for benzene. MO Energy(AU) EXPa 12 0.4581 0.4222 13 0.4581 0.4222 14 0.3291 0.3399 15 0.3291 0.3399 16 0.0304  17 0.0304  a Ref. [50] Table 32 SCF orbital energies for pyridine. MO Energy(AU) EXpa 12 0.4784 0.4575 13 0.3709 0.3554 14 0.3633 0.3859 15 0.3328 0.3601 16 0.0151  17 0.0290  a Ref. [51] Symmetry(D6h) (a) (a) Eig (Tr) Elg (7r) E2u (r*) E2u (r*) Symmetry(C2v) (a) A1 (n) B2 (7r) A2 (r) B2 (7T*) A2 (7r*) transition moments. The RPA method only guarantees equivalence in a complete basis. It is demonstrated here however, that the RPA method does a good job in balancing the two formalisms with a finite basis.  54 Table 33 Electronic excitation spectra for D6h benzene. Sym. CIS ft fy RPA fr fv EXPa (cm1) (osc) 'B2u 37797 37306 38090 (0.001) 1B1u 48806 48305 48972 (0.10) Elu 54644 1.020 0.222 51566 0.678 0.541 55900 'Eln 54644 1.020 0.222 51566 0.678 0.541 (0.69) a Ref. [52] The nature of the length operator is to emphasize the long range region of the basis functions whereas the velocity operator emphasizes the region closer to the nuclei where the functions change the most. The dipole velocity oscillator strength for the lower n7r* transition for pyridine is larger than the length oscillator strength for both CIS and RPA. The nonbonding orbital is essentially located on the nitrogen which is more electronegative than carbon. The vicinity of the nitrogen atom is expected to gain charge and thus the electron density in the region will relax. If the assumed atomic basis took this fact into account the atomic velocity matrix elements would be diminished in favor of the length matrix elements for the pyridine nonbonding orbital. The INDO method however is based on a minimal basis formalism which does not allow atoms to become polarized in an asymmetric Table 34 Electronic excitation spectra for C2v pyridine. Sym. CIS fr fv RPA fr f EXPa (cm1) (osc) 'B2(n) 35981 0.009 0.210 35804 0.008 0.230 34771 (0.003) 'A2(n) 44158 44125  'B1 38751 0.061 0.010 38120 0.055 0.042 38350 (0.03) 'Al 49991 0.067 0.020 49230 0.099 0.083 49750 (0.2) 1A1 56282 0.732 0.037 53970 0.510 0.220 "55000 1B1 56682 0.887 0.210 54045 0.594 0.449 (1.30) a Ref. [52, 53] fashion. The relaxation of all the orbitals on an atom can be accomplished however with some success and will be the subject of other chapters. In order to do calculations for large systems some truncation of the active space must be accomplished. The effect the size of the active space has on the spectra of small systems can be studied. For benzene and pyridine no qualitative difference in the ordering of the excitation energies is noted in going from a very small space of all transitions below 55,000cm1 to a full singles space consisting of transitions above 400,000cm1. The very lowest transitions change very little 56 Table 35 Benzene 'Elu transition as a function of the active space. Active CIS fI fV RPA fr fV Space (cm1) (cm1) 55444 1.249 0.522 52208 0.829 0.786 55000 54644 1.020 0.222 51566 0.678 0.541 65000 53060 1.041 0.259 51026 0.708 0.567 80000 53850 0.994 0.198 50854 0.677 0.512 90000 53394 0.988 0.199 50407 0.681 0.509 100000 51054 0.770 0.043 48077 0.514 0.375 130000 50848 0.776 0.051 47863 0.520 0.382 200000 50139 0.768 0.056 47097 0.523 0.394 300000 50000 .771 0.061 46929 0.528 0.402 400000a aFull active space after all configurations below 65,000cm1 are included. The quantitative nature of the other excitation energies, however, systematically diminishes with increasing active space for both RPA and CIS. This is perhaps caused by excitations to the higher lying virtual orbitals calculated too low in energy when the overlap is neglected and mix too readily with the lower lying transitions. The relative magnitudes of the length versus velocity oscillator strengths remain the same for both CIS and RPA. Their absolute magnitude becomes smaller with the size of the space and reduces more rapidly for CIS than RPA. The data for the 'Elu Table 36 Pyridine 'Al transition as a function of the active space. Active CIS fr fv RPA fr fv Space (cm1) (cm1) 57477 1.082 0.452 54674 0.717 0.646 55000 56282 0.732 0.037 53970 0.510 0.220 70000 55562 0.703 0.023 53247 0.493 0.198 85000 54426 0.645 0.001 52111 0.471 0.140 100000 52887 0.546 0.013 50392 0.382 0.122 150000 52685 0.546 0.010 50171 0.383 0.125 170000 52327 0.551 0.010 49743 0.389 0.138 240000 52074 0.564 0.005 49410 0.404 0.156 310000 51971 0.571 0.003 49267 0.411 0.164 430000a aFull active space excitation for benzene and the 1A1 transition for pyridine are presented in Tables 35 and 36 for illustration The other excitations follow the same trend. On the basis of the above analysis the active space for the current model should be truncated in the region 6580000cm1 in order to maintain quantitative transition energies. All subsequent calculations will be truncated in this range unless otherwise noted. 58 Table 37 SCF orbital energies for Naphthalene and Quinoline. MO Energy(AU) Energy(AU) Naph. Quin. 20 0.44105 r 21 0.42976 x 0.38831 r 22 0.36977 7r 0.36693 n 23 0.31545 7r 0.31520 r 24 HOMO 0.28856 7r 0.30397 7r 25 0.00043 7r* 0.00892 r* 26 0.02225 7r* 0.01954 7r* 27 0.05681 7r* 0.04968 7r* 28 0.08862 7*r 0.08374 r* Naphthalene and Quinoline One success of the INDO/CIS method is that it extends to molecules larger than for which it was parameterized. For a model Hamiltonian this extension is not guaranteed and becomes a test of the robustness of the method. In this section the CIS and RPA methods are compared for naphthalene and quinoline which can be viewed as extending the pi system of benzene and pyridine by one ring. Table 37 contains the higher occupied and lower unoccupied orbital eigenvalues for these molecules. 59 Table 38 Electronic excitation spectra for Naphthalene Type CIS f fv RPA f: f EXPa (cm1) (osc.) 7r>7r* 32138 0.004 0.005 31575 0.003 0.010 32000 (0.002) 7r>r* 37034 0.154 0.008 36059 0.139 0.119 37500 (0.102) 7>7r* 45469 1.844 0.561 43304 1.300 1.131 47500 (1.000) 7>r* 44630 0.000 0.000 44392 0.000 0.000  7>r* 46153 0.000 0.000 45594 0.000 0.000  7>r* 48551 0.621 0.118 46749 0.416 0.352 49500 (0.300) a Ref. [54] The electronic spectra of these molecules is more complicated than benzene and pyridine due to the presence of excitations from the additional pi orbitals. The calculated excitation spectra for naphthalene is reported in Table 38. The RPA oscillator strength values are not only nearly equivalent, but are also in line with experiment. This allows more confidence in the assignment of experimental bands based on calculation. As with benzene and pyridine the RPA transitions are systematically lower in energy than the corresponding CIS values and the CIS dipole length oscillator strength is much larger than dipole velocity. 60 Table 39 Electronic excitation spectra for Quinoline. CIS EXPa Type (ff f RPA *fr fV (cmt) __________ (OSC.) 7r>r* 32467 0.027 0.002 31894 0.026 0.012 31900 n>7r* 34312 0.007 0.181 34139 0.007 0.198 to 7r>7r* 37525 0.108 0.002 36788 0.106 0.071 36200 (0.116) 7r>7r* 43277 0.597 0.182 42148 0.724 0.622 44300 n>7r* 45004 0.000 0.000 44968 0.000 0.000 (0.517) 7r>r* 46844 0.502 0.116 45618 0.470 0.387 49300 7r>7r* 46997 0.858 0.278 45770 0.336 0.270 7r>7r* 48963 0.457 0.089 47582 0.157 0.135 (0.743) a Ref. [55] The results for quinoline are even more complicated due to the presence of excitations from the nonbonding orbital. The calculated results are presented in Table 39. The calculated pi excitations indicate a similar structure to naphthalene with all bands allowed. The oscillator strengths for the nonbonded excitations are like those of pyridine and a similar explanation would apply here. The RPA performs better than the CIS method for these molecules in terms of oscillator strength. The inability of the CIS method to guarantee equivalence even in a complete basis is not grounds to reject its use for a model Hamiltonian. It is 61 demonstrated however, that it is not as reliable in the prediction of experimental oscillator strengths. The RPA performs as well as the CIS method for the qualitative ordering of the excitation energies and both methods do well for the quantitative prediction of excitation energies. The Diazines The diazines consist of a planar six membered ring containing two nitrogen atoms and, like pyridine, are isoelectronic with benzene. Two nitrogen atoms will generate two low lying nonbonding orbitals. The presence of the single nitrogen did not greatly perturb the band structure for the pi excitations of pyridine as compared to benzene. The presence of a second nitrogen in the ring perturbs the pi transition energies somewhat more but the band structure remains intact. There are now four excitations from the nonbonding orbitals most of which are not allowed and are not resolved experimentally. The calculated SCF energies and spectra may be found in Tables 310 through 313. The RPA is very capable of reproducing the spectra of the diazines with the INDO/S Hamiltonian. For benzene, pyridine and the diazines, both CIS and RPA predict the correct order of the lowest 7rir* and n7r* transition energies with respect to experiment. Oscillator strengths for n7r* transitions are systematically larger for dipole length than dipole velocity as previously observed. The highest energy nr* transition is predominately from the lower nonbonding orbital. For Table 310 SCF orbital energies for the Diazines. MO Energy(AU) Energy(AU) Energy(AU) 1,2diazine 1,3diazine 1,4diazine 12(n) 0.43067 0.42097 0.42352 13(7r) 0.38078 0.38745 0.40242 14(n) 0.35278 0.36630 0.36475 15(r) HOMO 0.34776 0.34754 0.33096 16(r*) 0.00299 0.00554 0.00347 17(7r*) 0.01297 0.01524 0.01930 18(7r*) 0.09987 0.09927 0.10012 Table 311 Electronic excitation spectra for 1,2diazine. CIS EXPa Type (cm ) fr f RPA fr fv (osc.) 26649 28531 0.014 0.573 28227 0.013 0.671 26 (0.006) 35271 0.000 0.000 35169 0.000 0.000 46893 0.000 0.000 46826 0.000 0.000 50865 51760 0.014 0.109 51705 0.013 0.117 5 5 51503 39500 39374 0.058 0.001 38741 0.054 0.022 (0.020) 50000 r>r* 49600 0.109 0.039 48664 0.139 0.127 (000) (0.100) 55363 0.464 0.003 53601 0.403 0.078 57300 57635 0.910 0.260 55079 0.565 0.427 a Ref. [52, 53] Table 312 Electronic excitation spectra for 1,3diazine. CIS EXPa Type (cms) f f RPA fr fv (.) (cm ) 1 (osc.) 31073 34282 0.016 0.386 34129 0.015 0.426 31 (0.007) 36640 0.000 0.000 36506 0.000 0.000 n>r 45396 0.000 0.000 45313 0.000 0.000 51143 51478 0.010 0.082 51433 0.010 0.087 51143 (0.005) 40310 39803 0.068 0.012 39187 0.062 0.043 40310 (0.052) 52340 50181 0.172 0.048 49139 0.189 0.149 52 (0.16) Rb 56271 (0.25) 57530 0.652 0.078 55533 0.509 0.167 58500 r>r 57594 0.626 0.011 55700 0.416 0.198 (1.) a Ref. [52, 53] b Rydberg transition. this transition the discrepancy in the calculated oscillator strengths is maintained but the magnitude of the error is diminished from that of the lowest energy transition. The two nonbonding orbitals are different in that the lower energy orbital is less localized than the higher. The hypothesis that more tightly bound orbitals will generate a larger difference in oscillator strengths holds for these molecules. In all cases studied thus far the RPA 7Tr* oscillator strengths are 64 Table 313 Electronic excitation spectra for 1,4diazine. CIS EXPa Type (cm ) fr fv RPA fr fV ( ) (cm_ m L f IV (osc.) 30875 30409 0.013 0.476 30164 0.013 0.540 (.01) n>r* 39056 0.000 0.000 38857 0.000 0.000 40706 0.000 0.000 40652 0.000 0.000 54538 0.000 0.000 54534 0.000 0.000 54000 37893 36671 0.177 0.033 35609 0.146 0.108 38808 (.10) 50880 48358 0.232 0.024 47030 0.207 0.126 r>ir (0.15) 60700 57694 0.253 0.081 56567 0.203 0.000 6 (~1.) 60700 60653 0.857 0.256 58182 0.552 0.420 60 (~1.) 7t>r* 57900 59305 0.000 0.000 57786 0.000 0.000 60306 Rb 55154 a Ref. [52, 53] b Rydberg transition. superior to those of CIS with respect to equivalence and experiment, and in most cases that obtained from the dipole length operator are better when compared to experiment than that obtained from the velocity operator. 65 The RPA for Extended Systems One utility of modem model Hamiltonians is the ability to handle very large systems. The INDO/CIS method has proven its relevance in this regard. In this section the RPA and CIS method are compared for a series of extended aromatic hydrocarbons. The series of molecules beginning with naphthalene (two rings) and anthracene (three rings) is extended up to a linear molecule of twenty rings. The twenty ring molecule consists of 82 carbons and 44 hydrogens for a total of 372 basis functions. The density of spectroscopic states is large for all molecules beyond anthracene; however, there are two transitions that are of specific interest. The first of these is the lowest transition and gives information about the HOMOLUMO energy gap. The second of these is the transition of maximum oscillator strength, which is polarized along the major axis of the molecule and generates a large transition dipole. Therefore as the molecule increases in length the transition dipole also increases. Data for these two transitions for the different molecules are presented in Tables 314 and 315. As the size of the molecule increases, the ratio of RPA dipole length to dipole velocity oscillator strengths for the intense transition is almost constant whereas those of CIS are varied. This indicates that for this type of transition the RPA oscillator strengths scale properly with size whereas the CIS method does not. However the CIS method does improve with size. Both CIS and RPA dipole Table 314 Lowest excitation for extended aromatic system. # rings CIS(cm1) f f1 RPA ff fV 2 32145 0.003 0.005 31583 0.003 0.009 3 28505 0.012 0.022 27900 0.010 0.036 4 24761 0.241 0.003 23383 0.160 0.151 5 21482 0.252 0.030 20003 0.155 0.147 6 19258 0.269 0.050 17707 0.158 0.158 10 12272 0.284 0.623 10147 0.123 0.145 20 11045 0.541 0.159 8743 0.226 0.279 Table 315 Most intense excitation for extended aromatic system. # rings CIS(cm1) fI fV RPA fV fV 2 45434 1.832 0.557 43276 1.293 1.124 3 40131 2.703 0.999 38319 1.917 1.728 4 36509 3.479 1.462 34982 2.503 2.319 5 33882 4.171 1.887 32624 3.073 2.834 6 31988 4.829 2.368 30952 3.641 3.372 10 27656 6.948 4.072 27214 5.735 5.250 20 25179 11.712 8.278 25003 10.998 9.697 length oscillator strengths scale linearly with the number of rings as can be seen in Figure 31. The RPA values are perfectly linear over the whole range of molecular size whereas the CIS values differ to a small degree for the smaller 67 molecules. The linear regressions for the oscillator strengths are f,(RPA) = 0.535n + 0.344 R = 0.9998 fC(CIS) = 0.531n + 1.344 R = 0.9945 (312) n = # rings The dipole length oscillator strengths scale roughly the same for the two methods with the CIS values being systematically higher than the RPA values. For the low energy transition the CIS oscillator strengths indicate no new trends. As with the other molecules studied the RPA oscillator strengths are much more consistent than those of CIS. The excitation energies for both the RPA and CIS procedures follow pre dictable trends. Each line was fitted to a general exponential model in order to extrapolate the transition energies to very large systems. The modeled transition energies are LOWEST TRANSITION(CM1) = E(RPA) = 8612 + 28040 exp(.0646n1625) E(CIS) = 10910 + 25610 exp(.0607n1654) (313) MOST INTENSE = E(RPA) = 24540 + 48420 exp(0.588n0691) E(CIS) = 24710 + 47110 exp(0.488n0751) Both modeled energies and transition energies can be found in Figure 32. 13.00 . S9.75 6.50 ^ IO RPA O aCIS .04 3.25 1 6 11 16 21 Number of rings Figure 31 Dipole length oscillator strengths for the most intense transition. The RPA and CIS methods scale similarly with size for a given transition. The difference between naphthalene and the extrapolated values for very large systems is roughly 21000cm1 for both lines and both model calculations. The RPA excitation energies for the lowest transition are systematically smaller than the CIS values. The difference between the two levels for the larger molecules remains constant at 2000cm1. For the most intense transition the CIS and the RPA transition energies converge to the same value as the chain length is increased. 69 x103 47.0I '' + CIS X RPA 37.2 CIS I ' 27.5 Most intense transition SExponential Model 17.7  Lowest energy transition 8.0 8.0 , ,,i1 1 I , 1 6 11 16 21 Number of rings Figure 32 Modeled and calculated excitation energies CHAPTER 4 THE UVVIS SPECTRA OF FREE BASE AND MAGNESIUM PORPHIN Motivation Much experimental and theoretical work has gone into characterizing the electronic structure and spectroscopy of porphines [56]. These systems and their excited states are of great interest in a variety of areas, and in particular are essential in understanding the initial photochemical event in photosynthesis. The electronic spectra of porphyrin compounds are characterized by three basic regions. The so called Q bands are relatively weak and occur in the visible region. The Q bands consist of a degenerate electronic transition for divalent metalloporphines (e.g. MgP) and two separate electronic transitions for free base porphines (H2P) separated by ~3300cm1. The intense Soret or B region occurs in the near UV and is often accompanied by a closely related N band of lower intensity. The B band for divalent metalloporphines is assumed to be a degenerate pair [57]. The higher UV bands in the third region are broad and are often of near uniform intensity. The earliest attempts to explain the Q and B bands for H2P were free electron models and were based on assuming specific resonance structures for porphin that involve an aromatic macrocycle of 18 7r electrons [58, 59]. This simple model Figure 41 Free Base Porphine correctly accounts for a weak band (Q) followed by a strong band (B). This 18 membered cyclic polyene can be assigned D18h symmetry. Within this group excitations between the highest occupied molecular orbital e4u and lowest empty 72 e5g(7r*) molecular orbital gives rise to states of 1Blu, 'B2u and 1'Eu symmetry. 1B1, and 1B2u lie lowest, and give rise to the observed Q1 and QI peaks. The Soret band, of 'E, type remains degenerate in this model. In contrast, metalloporphines can be likened to 16 membered cyclic polyenes with 18 pi electrons. In D16h, the HOMOLUMO excitations give rise to states of 'E7u and 'El symmetry, and both visible and Soret are predicted degenerate. Early studies determined that the two Q bands for H2P were polarized perpendicular to each other and that the B band was of mixed polarization [60]. Low temperature studies of the B band resolved a splitting of 240cm1 between equally intense transitions [61]. Hiickel calculations by Gouterman using a four orbital CI model proved adequate in explaining the splitting of the Q bands and predicted that the lower Q band would be polarized parallel to the inner HH axis [62, 63]. Further polarization studies showed this to be the case and the B band was shown to have parallel polarization at the low energy side of the band and perpendicular polarization on the high energy side of the band [64]. By 1972 extensive PPPCI calculations had failed to reproduce the assumed B band splitting of 240cm' for H2P [65]. The B splitting was predicted to be greater than 1500cm1 and other 7rr* transitions beyond the four orbital model were predicted to be part of the B and N bands now assumed to be coupled and spread over a region 3300cm1 wide [66]. Sundbom suggested that nr* may also contribute in this region [67]. Ab initio calculations, with scaled transition 73 energies, seemed to confirm the complexity of the Soret region suggesting as many as five contributing bands including nr* [68]. The inability to confirm the nature of the Soret region is complicated by the fact that predicted oscillator strengths in this region do not conform to the observed spectrum without making assumptions about vibrational broadening of the BI relative to BI [67, 68]. In addition, calculated oscillator strengths in both the length and velocity formalisms can differ by an order of magnitude [66]. The most commonly used dipole length formalism usually predicts oscillator strengths too large in the Soret region by a factor of two. Encouraged by the ability of the INDO/RPA method to predict oscillator strengths in addition to excitation energies for aromatic hydrocarbons and het erocycles the spectroscopy of free base and magnesium porphin is reexamined in detail. Results CIS and RPA calculations have been performed for H2P for several active space partitions. The number of occupied valence orbitals for H2P are 57. Orbitals 48 and 49 are the nonbonding orbitals located on the nonprotonated pyrrole ring nitrogens and 5064 are r. No orbitals outside of this range had significant contribution to the lowest 30 states. Transition energies and oscillator strengths 74 Table 41 Calculated and Experimental Excitation energies for H2P. CIS RPA EXPa Evb ff fE Ev fE fv Trans.c E4 13.72 0.022 0.0001 11.81 0.020 0.021 QI 15.94 16.58 0.033 0.001 15.92 0.033 0.029 Q_ 19.55 27.18 1.616 0.114 24.08 1.146 1.003 B 26.85 28.42 2.411 0.306 24.44 1.228 1.224 Bj_ 28.98 28.70  29.52 28.98  32.97 1.478 0.162 31.69 0.424 0.343 N1i 29.41 33.58 33.41  34.75 34.42  35.64 0.366 0.063 35.30 0.169 0.115 1 a Ref. [69] b Energies are reported in 103cm1. c Designations II and refer to transitions in the plane of the molecule polarixed parallel of perpendicular to the inner HH axis. for the lowest 10 transitions for an active space of 14 occupied and 14 virtual orbitals are included in Table 41. The lowest occurring nr* for the CIS calculation is at 39637cm' with a dipole length oscillator strength of 0.019 and for the RPA calculation 39600 and 0.019. In order to ensure that the 14x14 active space is adequate for the determination of this transition a 1060 configuration calculation was performed consisting of 53 occupied and 20 virtual orbitals. For CIS the Q, B and N 75 Table 42 Calculated and Experimental Excitation energies for MgP. CIS RPA EXPa Evb f fy Ev fr fv Trans.c Ev 15.77 0.025 0.002 14.64 0.023 0.025 Q 17.24 15.77 0.025 0.002 14.64 0.023 0.025 Q 28.62 2.495 0.288 24.95 1.351 1.243 B 25.64 28.62 2.495 0.288 24.95 1.351 1.243 B 28.71 28.65 CT 29.58 28.98 7r>r* 30.46 0.032 0.049 30.41 0.030 0.055 CT 31.80 31.49 r>r* 32.38 32.14 > 33.55 0.133 0.012 33.19 0.029 0.022 N 30.77 33.55 0.133 0.012 33.19 0.029 0.022 N a Ref. [70] b Energies are reported in 103cm1. c CT designates a charge transfer from occupied 7r to the Mg atom. bands were lowered by less than 600cm1 and the n7r* transition was lowered to 38375cm1 for CIS and 38254 for RPA. A small 7x7 configuration calculation was performed to check the sensitivity of the lower 7rTr* to the size of the active space. The results for the 6 lowest allowed states are qualitatively the same as those in Table 41 being generally 400cm1 higher in energy with differences in oscillator strength less that ten percent. 76 The results for the low energy portion of MgP are found in Table 42. The major features of the spectrum consist of a weak degenerate Q band followed by an intense degenerate B band. The Mg is set 0.4 A above the ring as suggested by Xray structures [71]. With the addition of the metal atom new features appear in the spectrum, namely the forbidden or nearly forbidden charge transfer bands. The positions of these bands are very sensitive to the Mg outofplane coordinate which is easily perturbed. Similarly to H2P, several forbidden 7r>7r* transitions are calculated to fall between the B and N bands. Discussion Calculations using the RPA procedure yield an explanation of the Soret region for H2P that conforms to experiment without invoking vibrational coupling to explain the predicted pattern of oscillator strengths or scaling excitation energies in a least squares manner. While the RPA formalism only ensures equivalence between length and velocity oscillator strengths in a complete HF basis the utility of the procedure to deal with transition moments in a balanced fashion is again demonstrated. This balance of transition moments afforded by the RPA is important if properties such as optical rotary strength in addition to oscillator strength are to be calculated. With the CIS procedure the two moments are very different as already noted by others [66]. In addition the RPA oscillator strengths are in better accord with experiment [70]. The experimental B band oscillator 77 strengths are estimated to be 1.1 as predicted here. The experimental oscillator strengths for the Q bands are less that 0.1 with QjI)Q1 as predicted by both the CIS and RPA calculation. The PPP Hamiltonian has to be modified in order to reproduce this result [65, 67]. The experimental N band occurs on the shoulder of the high energy side of the intense B band with experimental oscillator strength much less than that of the B band. This fact is inferred by the RPA results but not CIS. Both the CIS and RPA procedures predict the correct ordering of the Q and B bands for H2P as deduced from polarization experiments [64]. In addition both procedures predict the splitting of the Q bands in accord with experiment. The splitting from the CIS procedure is 2860cm1 and from the RPA 4010cm~1. For gas phase H2P the experimental splitting is 3610cm1 [69]. For the B band the RPA calculation yields a splitting of 360cm1 which is in excellent accord with the original low temperature splitting of 240cm1. The failure to reproduce this splitting theoretically led to the suggestion that this splitting was due to a vibrational progression and that the second "B" band actually appeared as "N", leaving unexplained the reasons for the sharp BII and broad B (N) [66]. Even for the 1060 configuration active space the CIS procedure predicts a splitting as large as 1000cm1. These calculations therefore support the interpretation that the Soret band for H2P consists of two nearly degenerate transitions of roughly equal oscillator 78 strength. The N band consists of 7rTr* beyond the four model. No evidence is found to suspect the presence of nir* below 38000cm1 or that other allowed 7rT* are present in the Soret region. No new observations can be attached the MgP spectrum aside from the ability of the RPA to balance transition moments for oscillator strength. As noted previously, the frequencies calculated by the RPA method are generally lower that those calculated from the CIS treatment. Both calculations are in very good accord with experiment. The ability of the INDO/RPA method to represent a balanced treatment of transition moments has been established through these calculations and those in Chapter 3. The correspondence with experimentally observed transition moments yields confidence in the method to predict and interpret spectra. The INDO/RPA method has also been shown to not degenerate with molecular size so that very large systems can be studied with confidence. CHAPTER 5 THE INDO/RPA CALCULATION OF NMR CHEMICAL SHIELDING The Concept of Chemical Shift The ability to quantitate and establish the nature of "chemical shift" as it relates to magnetic resonance spectroscopy has been, and is, of great interest in quantum chemistry. The phenomenological Hamiltonian most often used for this purpose is usually attributed to Ramsey [72, 73] and most commonly takes the form of a perturbative expansion of the energy of a system with respect to an external magnetic field and the field of the nuclei of interest [74]. The perturbation is small with respect to the total energy of most chemical systems; therefore, the perturbative expansion can be truncated at low order and is usually assumed to be linear. Many empirical and semiempirical models have been developed to predict specific chemical shift in terms of the chemical properties of the atoms of interest and the specific bonding environment [75, 76]. These models and rules are very useful in interpreting the nature of the local chemical environment in a molecule. Direct nonempirical calculations of chemical shift have met with only modest success and normally requires very large basis sets to obtain quantitative accuracy [77]. The failure of smaller basis sets is usually attributed to their inability to represent the gauge invariance of the origin of the externally applied 80 field. As a result, in the last few years methods to minimize the effect of gauge variance have been developed with much success [7881]. In this Chapter the local orbital, local origin method of Hansen and Bouman [81] for the calculation of isotropic shielding of the atoms in a closed shell molecule is developed for the INDO/S Hamiltonian. This method is chosen because it utilizes the unique properties of the RPA already discussed, and because the results obtained are easily analyzed in terms of the chemically useful concept of localized effects in molecules. The magnetic energy of interaction of a nucleus in an external magnetic field can be expressed as [82] U = .B (51) where i is the magnetic moment of the nucleus and B is the external magnetic field. The magnetic moment of the nucleus is quantized. The operator form of j is i= 7, I = h[(I 1)]1/2 (52) where P is the angular momentum operator and 7 is the gyromagnetic ratio of the specific nucleus. For an oriented molecule in an external field of magnitude B aligned along the z axis and with mi the magnetic quantum number of the nucleus we can express the interaction energy Uz = yhmiB (53) 81 Differences in magnetic quantum numbers represent differences in interaction energy. The absolute difference in energy for a change of one quantum number is then AUz = hv = I7yhB (54) the frequency of which is v= B = KB (55) 27 Therefore, in the absence of any other factors, energy differences for unit changes in magnetic quantum number are linearly dependent on the external magnetic field and are a function of the specific nuclei only. It is this difference in energy with applied field that is measured in a magnetic resonance experiment. If no other factors were involved, a single measurement of for a type of nucleus would be sufficient to describe all situations dealing with the same nucleus and NMR would be of very little interest in chemistry. Nuclei are not normally isolated, however, but in the presence of electrons. The external field induces a current in the electrons. This current in turn creates a magnetic field that is in opposition to the applied field. The net field (B) at the nucleus of interest is less than the applied field (Bo). Different electronic environments will modify the total field differently. Each chemically unique environment for a given type of atom in a molecule at 82 fixed external field strength will thus result in a different frequency. The induced field of the electrons is also linearly dependent on the external field so that vj = KBo(1 aj) = vo(l aj) = Vo vj (56) 'j =  V0 oj is the unitless shielding constant for environment j with respect a bare nucleus and represents the effect of the electrons in said environment. With respect to an arbitrary reference environment we define relative chemical shift parameters jf Vj Vref S= 0ref 17i = (57) Vo Since vj Vref ((vo and vo e vrf relative chemical shift is usually tabulated in terms of parts per million Sj(ppm) = 106 vi Vref (58) Vref In this formalism positive values of relative chemical shift represent an environ ment that is less shielded than the reference environment. Theory We seek a quantum mechanical description of a which, in general, is depen dent on the directional components of the applied field and the components of the nuclear magnetic moment. Equation (51) is modified to read U=l.[i ]&B= .B+j7.B (59) = Uo + Ui 83 U represents the total energy of interaction and U1 the electronic perturbation. A magnetic field can be expressed in terms of a vector potential that must be incorporated into the Hamiltonian of the system [17]. The kinetic energy portion of Equation (21) is modified to read ila 1. 1 H=1 + [iVi + Ai] [iVi + Ai] + V (510) i=1 where c is the velocity of light and A is the magnetic vector potential. The vector potential for a uniform magnetic field is x [i R] (511) with arbitrary gauge origin R. All coordinates are relative to the nucleus of interest. The vector potential associated with the magnetic dipole moment of the nucleus is expressed [20] iA = (512) ri 84 We choose the coulomb gauge so that V A= 0 and expand Equation (510) according to Equations (511) and (512) H = Ho + :' ft + ex (Fi R) + 2 (5 13) i 12 x(iR)+ r ] (513) = B {( R) xVi + 2 i=l i + ^1 1xiBx() ]2 + c gx(iI)+ r ]2 i=1 1 where []2 is a shorthand notation for [ ] [ ]. The last equality in Equation (513) represents the perturbation Hamiltonian of interest here. The terms quadratic in field are dropped due to the observation that shielding is practically linear with field according to Equation (59). The final expression for the perturbation is then n . f'= _ I(i ) x i} + 2 (i3 Vi) i=l ri 1 [X (i tx)]( [/ i] (514) L.2c2 [r = H'(1) + I1I'(2) = HB (1) + H'(1) + IH'(2) Perturbations to molecular energies due to the presence of a magnetic field are small so a low order expansion of the perturbation Hamiltonian is adequate. 85 To second order with a HartreeFock reference wavefunction [19] E' =E' + E2 = HFI + (V)HF I flH k ) k If klHF) (515) EH EHF Ek k=1 where the bk represent excited states of energy Ek. Placing Equation (514) into (515) yields n E' 2 {Bi ( HF(?i R) X ViJbHF) i=1l +2 ( FWH}(5 16) n 1 [I x (i )]. [j7xFi] + E 22(HFI 3 IFHF) = E1(1) + E'(2) and (OHF lII' k)(k HF) = (O&HFIH'(1) + fH'(2) Ibk)(k IH'(1) + H'(2) IHF) (517) = ('HF S'(1)'k)('kl'(1)lHF) + (HHF H'(2)4'k)(kJH'(2)'HF) +( HFfl'(1)lk)( klIH'(2) IHF) + (OHFIH'(2) Ik)(lkkIH'(1) CHF) Recalling Equation (59) we seek terms that are linear in both external and nuclear field. It is conventional to interpret experiments in terms of the phenomenological Hamiltonian of Equation (59), therefore is convenient to gear this development to yield numbers comparable to the interpretation of these experiments. It would be more systematic, however, to calculate the observed experimental spectrum directly and would make a logical project for the future. The first term in Equation 86 (516) is only linear in a single field and can be dropped. In Equation (517) all terms involving H'(2) are quadratic or higher in field. Dropping these terms leaves ( HF k '(1) k '(l) IHF) = (OHFIH'B(1) + HII(1)[ k )(k IB(1) + H (1)I) HF) = (518) (OHFIfB(1)0k(kIfIB(1)bHF) + (kHF IH(1)k)(OkkIH,(1)bHF)+ (OHFlH'B(1)k)(IkIHI(1)OHF) + (FHFHI (1)lk)('klIH'(1)VHF) Collecting those terms in Equation (518) that are of the proper form leaves 1 [I x (B i iR)] [ x Fl) U1 = 22 (HF x r3 Ir HF) 20 P n x( Xi n _H (519) (oo HF I E .[i] R'k)k E B Rfi ) x Vi]IHF) EV i=l i=1 c2 k= EHF Ek The second term in Equation (219) has the proper form as suggested by Equation (59). Through vector identities the first term can be rearranged and Equation (519) reads U1 = L Cc2 (HF i=l r3IHF oo (lkHF k k)(k E i[(ij R) XVi]IlHF) (520) i=1 i i=1 c2 EHF Ek k=1 87 identifying the form of shielding desired. Let e, be a unit vector along the direction of the nuclear magnetic moment and gB along the applied field. 21 (F [gB x(Fi J)]. [ xfi] (pB = 2 (HFl r3 HF) 1 (rrl Z k k B (i ) X Vi 1HF (521) ii= i=1 C2 Ek EHF Equation (521) is the basic equation for the calculation of chemical shielding [73]. The first term can be interpreted as that shielding created by the induced current around the nuclei of interest and is known as the diamagnetic component of shielding (ad). The second term is known as the paramagnetic component of shielding (aP) and can be interpreted as the degree to which the specific chemical environment quenches the induced current around the nucleus. The two components are opposite in sign and a delicate balance must be achieved in order to obtain even qualitative results as compared to experiment. The transition matrix elements in the paramagnetic portion are over anti Hermitian oneelectron operators. Being oneelectron operators the sum can be truncated at single excitations only. Transition moments for antiHermitian operators according to the analysis in Chapter 2 for a closed shell reference are expressed (bHylO = 1(Oic) [X k Y]yc,k rc A (522) ( k O HF) = V E[X_ Y]d,k dlOA() 8d 88 The energy in the denominator is the transition energy for the excitation (EK). Pre and post multiplication of Equation (245) yields, (A B)1 // (A B)(X Y) = (X + Y)EK // EK1 (X Y)EK = (A B)1(X + Y) // (Xt Yt) (523) (X Y)EK'(Xt yt) = (A B) Incorporation of Equations (522) and (523) into the paramagnetic shielding term yields PB = (  c)(A B) 7 ( 1dO 6) (524) yc6d Let Ry be the location vector of occupied orbital relative to the nucleus of interest. The field dependent transition integral in Equation (524) becomes ( dIO )=6 d (OdI( R) xV0).B = [(Idl(F 1r) x V6) + (dl(I( iR) x V 6)] EB (525) = [(dl(d Ry) x V16) + (R1 R) x (OdV6B)] eB so that 2 R aB = (0 ( lOIc)(A B)d(d( R) x V16) B 2 (526) + (I~IA c)[(R, R) x [(A B)6d(d I I)]] B yc6d According to Equation (227) and the properties of the RPA equations in a complete Hartree Fock basis (A B)lVd(dVl ) = (cir7) (527) The paramagnetic term becomes S= ( E (^0,1`c,)(A B)'cd(dl(rd RX V) 6 B ycd (528) 2 4 Z(0,16jI10c)(Th k x (qciioM)] C2 ( ,KA R R) X cW B where all references to the global gauge origin are contained in the second sum. For a closed shell reference and local origins the diamagnetic portion of shielding becomes Bd 1B x r[ Bx R)] ) [E xr orB= B 2^\^1  3 100r) S(529) +1 [B X (R7 R)] ( x [ r r0) C2 r 7 where again all references to the global origin are contained in the second sum. Combining the origin dependent sums of Equations (528) and (529) and using the vector relation A (B x C) = C (A x B) yields for the global origin dependent term 1 B x ( )] ) 7(530) 2 E [Fx V] 21 r3 )(cfl lx) [E'B x (R, R)] 7C Utilizing the orbital completeness relation [23] I,3()(YI + c)(CI 531) 7 c (531) c 7 gives for the origin dependent term 1 I [4. x ri B (R) = E (R R)] [I I O 7 2 [ [i(x V] 22^{3 T * {[WB x (R, R)]. [r ,)]} 2 [Ix ] +C IE(l5Iii ) [)(1 [B x (R, R)] C E 71 r  I ) ( r ) I X 76 The second sum can be rearranged 2(1 7{6e [FX V] } {[1B x (R, R)1. [F1)]} Ell [F ] [Ex 00]1 = [FB x (R, R)] 2(0{ x V]} r) _g)].[ IP I [ ] (( { * =[B x (R1, R)]. (0 [x ], )] (533) and combined with the first sum aB(R) = EB x (R ] 7R) {[F x r [ [f x V],r?}O1) 76 (534) (532) = [eB x (R 10 ] Consider the general relationship j (F x V)[(ji. r1")] = j." F x [V(. r)thI)] = r3 x [e10) + (E," 1 )] = ( x rI. (Vi,10)) + (j X rI. [( Re. r3 10l')] (535) (j x f)r (4 0)) = e. (F x V)[(ei l r )]( (E r)(e x [V )] = E. (r x V)[(Ei. F)1)] (ES. r)(.j rF) x [V[t,] I(ej x rx) l [Ej (Fx V), (i rf)] ) = 0 the last line of which indicates that the first sum in Equation (534) vanishes. The indices of summation for the remaining term are over all occupied orbitals and can be rewritten as e ] 1x,] [ ^oIAB (R) = 1 E'l ) B X ([F R)] +( '[x V]r)(31 [B x (R R)]l (536) 1c Z 7 [l'rx I)( I)" [(B x (RI, R,6)] 76 where all references to an external gauge origin have been eliminated from the expression for the shielding. 92 The final working expression for shielding becomes 1B = ' .(r(, g>B)(T( rRi) i g (? l )(g'B r )}ral31) ycsd 1 3x, 7 +4z; (^I1] KA q$ )(R) X Vq$) property integral and the construction and inversion of the RPA (AB) matrix [81]. The expression for shielding is a 3x3 matrix that is a function of the orientation of the molecule and the external field. With the principle exception of solid state NMR, the orientation of molecules with respect to an external field is random and shielding becomes an average of the principle components of the shielding matrix. It is this isotropic value in which we are immediately interested. The following discussion is for 13C isotropic shielding. 13C is chosen for study because the range of normal shielding values is 200 ppm and is a well characterized experimental quantity. 93 Localization and Integral Evaluation The occupied orbitals are localized according to the criteria of Foster and Boys [83]. A symmetric transformation of the orbitals is chosen such that the sum of the squares of the pairwise differences in their locations is a maximum. In other words, max (R = R Zi 612) 7)8 (538) An iterative process of successive two by two rotations is performed until all successive pairs are stable within a given threshold [84]. Local orbitals obtained in this fashion are easily identified as bonds between atoms and nonbonding orbitals. Double and triple bonds are represented by tau or banana bonds. These are identified by the program and are selectively delocalized to yield the more familiar local sigma and pi bonds. The property integrals and RPA matrices in Equation (537) are first evaluated in the canonical molecular orbital basis via transformation from the appropriate atomic integrals and subsequently transformed to the local basis. Three choices of local origins in the calculation of molecular shielding are evaluated. The first is the coupled HF (CHF) approximation in which all orbitals are assumed to be located at the nucleus of interest. The second choice assigns all local orbitals to be at their centroid location. As defined by Hansen and Bouman 94 [81] this is known as the full local orbital, local origin method (FLORG). The final choice, intermediate between the first two, assigns all local orbitals that are bonded to an atom to be at said nucleus and all other orbitals are assumed to be at their centroid location (LORG). Since in a complete basis the results are invariant to the choice of origins, the difference between results in a finite basis tests the sensitivity of the results to the basis type and choice of origins. There are many atomic integrals necessary for the evaluation of Equation (537) that are not normally associated with those required to perform an SCF procedure. Fortunately all integrals of the type used here can be put in the form of a normal overlap integral or a nuclear attraction integral that are easily evaluated. For atomic basis functions of the form [85] XA) = NA n,, ny, nz, RA) = NA zA exp(ArA)) (539) located at center A, dipole integrals, velocity integrals, and angular momentum integrals can be evaluated with overlap routines. The X component of each is outlined below. For dipole, (XAIXIXB) = (XAIXB + XBIXB) (540) = XB(XAIXB) + NB(XAInx+ l,ny,nz, RB) and for velocity. (XAIVxIXB) = NBnx(XAnx l,ny,nz, RB) (541) 2CBNB(XAInx + 1,ny ,nz, RB) 