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Pit"; "o it {11.1.1.2} 1.1a] 5mg; .rI III- ‘ \ .zriL‘flS IIIImmlllllllilllllllllmm 31293 01789 3102 LIBRARY Michigan State University This is to certify that the dissertation entitled DERIVATIVES OF MOLECULAR ELECTROMAGNETIC PROPERTIES USING NONLOCAL SUSCEPTIBILITY DENSITIES presented by EDMUND LEO TI SKO has been accepted towards fulfillment of the requirements for Ph.D. Chemical Physics degree in 71.6% I ‘ Major professor Katharine C. l-lint Date 30 July 1998 MSUis an Affirmative Action/Equal Opportunity Institution 0-12771 PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 1m chIRC/DaleDuepBS-p.“ DERIVATIVES OF MOLECULAR ELECTROMAGNETIC PROPERTIES USING N ONLOCAL SUSCEPTIBILITY DENSITIES By Edmund Leo Tisko A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1998 ABSTRACT DERIVATIVES OF MOLECULAR ELECTROMAGNETIC PROPERTIES USING NONLOCAL SUSCEPTIBILITY DENSITIES by Edmund L. Tisko This thesis presents three analytic derivative relationships. The underlying concept of these three relationships is connecting the change of a molecular property to other well- defined molecular properties. The foundation of this work lies in nonlocal polarizability density theory. The nonlocal polarizability density characterizes the polarization at one point in a molecule due to its reaction with an electric field at another point in the molecule. In the first part, the derivative of the electronic hyperpolarizability with respect to a Cartesian nuclear coordinate is related to the nonlocal second hyperpolarizability density, the nuclear charge and the dipole propagator. The derivation uses the derivatives of the wavefunctions and operators that comprise the six-term hyperpolarizability. These derivatives, which are also derived, are substituted into the hyperpolarizability to yield a sixty-term expression. The initial result is manipulated algebraically in a nontrivial way to yield the equality. A brief review of hyper-Raman scattering theory is given. Recent applications of hyper-Raman scattering are considered. In the second part, a new expression for the derivative of the polarization propagator with respect to an arbitrary coordinate is derived in terms of the quadratic polarization propagator and a sum of polarization propagators. The derivation involves calculating the derivative of a creation and annihilation operator pair with respect to an arbitrary parameter. Then the propagator derivative is calculated via the derivatives of many- electron wavefunctions. Past work calculating the derivatives of molecular properties using polarization propagator techniques is briefly reviewed. The third part concerns finding an expression for the derivative of the electronic magnetic moment with respect to nuclear momentum and its relationship to a nonlocal magnetizability density. The magnetizability density has a structure similar to the chemical shift. The relationship is found by considering the magnetic field produced by a moving nucleus as a perturbation on the electronic structure. The ground state wavefunctions are corrected to first order in perturbation theory using the nuclear magnetic field as the perturbation. The corrected wavefunctions are used to calculate the expectation value of the magnetic moment. When the derivative of this expectation value is taken with respect to nuclear momentum, its relationship with the nonlocal chemical shift density is uncovered. The result is correct when adiabatic wavefunctions that go beyond the Bom-Oppenheimer approximation are used. The physical content of the equation is interpreted as a description of intramolecular magnetic response. Other possible paths to a complete susceptibility theory are considered. One path involves substitution of 4-currents into the definition of the nonlocal polarizability density. Other paths arise from an initial consideration of the relativistic Dirac equation and subsequent analysis using the Gordon decomposition and the Foldy-Wouthuysen transformation. The heavens declare the glory of God; The sky proclaims its builder’s crafi. One day to the next conveys that message; One night to the next imparts that knowledge. There is no word or sound, No voice is heard; Yet their report goes forth through all the earth, Their message, to the ends of the earth. The people who walked in darkness have seen a great light; Upon those who dwelt in the land of gloom a light has shone. You have brought them abundant joy and great rejoicing, As they rejoice before you as at the harvest For the yoke that burdened them, the pole on their shoulder, And the rod of their taskrnaster you have smashed. For a child is born to us, a son is given us; Upon his shoulder dominion rests. iv ACKNOWLEDGMENTS My profound thanks and gratitude to God who has given me the ability and opportunity to ponder His marvelous and awesome Creation. I will always be appreciative of the great patience and generosity shown to me by my research mentor, Dr. Katharine L. C. Hunt. Her treatment of me gives me a goal for the treatment of my own students. Because of her confidence in me, another physical chemistry professor is on the loose. My thanks to Dr. Richard H. Schwendeman who has served as my second reader with much attention to detail. Thanks to the other members of my research committee, Dr. Gary J. Blanchard, Dr. James F. Harrison and Dr. James E. “Ned” Jackson who have given me encouragement, often unknowingly, during periods of great trial. My thanks to the members of the Hunt group who have helped me understand the incomprehensible: Dr. Xiaoping Li, Dr. Pao-Hua Liu, Dr. Olga Jenkins, Sandjaja Tjahajadiputra, Chetan Ahuja and Mark Champagne. My gratitude to my father, Thomas F. Tisko, whose love of learning has been infectious. I am forever indebted to my mother, Marie Ellen Tisko, for her kind encouragement throughout my entire 26 year educational career. Thanks to my brother, Thomas F. Tisko, whose own quest for excellence has been an inspiration. Thanks to my sister, Therese M. Warner, for her good humor and her faith. Finally I am grateful for my wife, Christina, who has had much to endure as I completed this research and thesis. I truly hope that its completion is a comfort to you. Note to Joey, Jamie, John and Mary(?): Daddy’s finally coming home. TABLE OF CONTENTS KEY TO SYMBOLS AND NOTATION x CHAPTER 1: INTRODUCTION The Derivatives of Molecular Properties and Nonlocal Susceptibility Densities l Hyperpolarizability Derivative 2 Polarization Propagator Derivative 3 Magnetic Moment Derivative 5 CHAPTER 2: RELATION OF HYPERPOLARIZABILITY DERIVATIVES TO SECOND HYPERPOLARIZABILITY DENSITIES 10 Introduction 10 Nonlocal Polarizability Densities, Polarization and Internal/External Electric Fields 11 Applications of the Hyperpolarizability Derivative i 14 Derivation of the Dependence of the Hyperpolarizability on Nuclear Coordinates 18 Further Applications of the Hyperpolarizability Derivative 27 The Hyperpolarizability Derivative and Hyper-Raman Intensities 27 Conclusion 30 CHAPTER 3: RELATION OF POLARIZATION PROPAGATOR DERIVATIVES TO NONLINEAR POLARIZATION PROPAGATORS 38 Introduction 38 Indistinguishability 39 vi Slatcrllctcuninants Occupation Number Formalism Foundations of the Polarization Propagator The Nonlinear Polarization Propagator The Derivative of the Polarization Propagator Magnetichnmics Elccuichcucnics Uses of the Nonlinear Polarization Propagator Derivation of the Polarization Propagator Derivative Introduction C"El£l"£ 12" Discussion CHAPTER 4: FIRST ORDER APPROXIMATION TO THE ELECTRONIC MAGNETIC MOMENT DERIVATIVE Introduction ll] 15 .11.: .. I l 1 El . E vii 40 41 46 49 50 51 53 54 55 57 61 62 65 67 68 76 76 77 78 Weapons: Molecular Electromagnetism Background Construction of the Nonlocal Chemical Shift Tensor Il 'E'llEllll C E' 1 Ci lS MC v E . El . Il . ll 1: . . Discussion CHAPTER 5: TOWARD A COMPLETE ELECTROMAGNETIC SUSCEPTIBILITY DENSITY Introduction The Special Relativity of Electromagnetic Theory B l . . . ll 1 . B l . . . El . I] Relativistic Adaptation of Nonlocal Polarizability Density Relativistic Quantum Mechanics Kl . _ 3 1 E . E' E . 3 l E . . Nonrelativistic Approximations to the Dirac Equation WW viii 78 80 80 81 82 85 88 91 98 98 100 100 102 106 108 108 109 112 113 113 lI-l"°H .1. 114 Higher Order Magnetic Susceptibilities 117 APPENDIX A: DERIVATIVES OF WAVEFUNCTIONS AND OPERATORS 123 Derivative of the Ground State Wavefunction 123 Derivative of the Reduced Resolvent 126 Nuclear Coordinate Derivative of the Hamiltonian 129 APPENDIX B: COMPLETE DERIVATION OF THE HYPERPOLARIZABILITY DERIVATIVE 132 APPENDIX C: EFFECT OF CREATION AND ANNIHILATION OPERATORS ON MULTI-ELECTRON WAVEFUNCTION S 148 APPENDIX D: RELATION OF POLARIZATION PROPAGATOR DERIVATIVE TO POLARIZABILITY DENSITY DERIVATIVE 152 ix Symbols an“ a,, a2, a3, [3 I3 Y Y To: In 72: 73 5m 'CI PB ”N KEY TO SYMBOLS AND NOTATION polarizability, polarizability density 043 component of polarizability, polarizability density noninvariant Dirac matrices hyperpolarizability, hyperpolarizability density second hyperpolarizability, second hyperpolarizability density Lorentz factor invariant Dirac matrices Kroneckerdelta 5m=1ifm=n;6m=0ifm¢n permittivity of free space arbitrary parameter Heaviside step fimction ath component of dipole, dipole density permeability of free space electronic magnetic moment Bohr magneton nuclear magneton arbitrary parameter gauge function Fll PM?) 00) canonical linear momentum, i = p - eA charge density, charge density at point f N-electron density operator single-electron density operator ground state expectation value of the replacement operator a: Pauli spin matrix nonlocal high-frequency chemical shift density basis function electrostatic potential third order susceptibility density time correlation function energy dependent density correlation function (1th component of the charge-current susceptibility density basis function frequency distinct frequencies sum of incident frequencies vibrational frequency of normal mode coordinate transition frequency 03m = (om — (00 inverse of the excited state radiative lifetime polarization propagator xi 1133:2103) nonlinear (quadratic) polarization propagator summation where ground state eigenfimction is not included summation over the n.K set of occupation numbers wavefunction transition frequency with damping Qmo = coma —i1‘,,,/2 annihilation operator creation operator replacement operator, a. = 313,. speed of light charge of an electron magnetogyric ratio of the electron magnetogyric ratio of the nth nucleus single-electron Hamiltonian operator reduced Planck’s constant it = h/21t current, electronic current electronic orbital angular momentum electron mass electronic magnetic moment, electronic magnetization density rest mass of a particle unspecified multi-electron state xii >I 3’ WI E, E0, En E; ((Di) WW) '1') F” , va (3(0)) occupation number of ith state as in nlscl mechanical linear momentum mechanical linear momentum of nucleus K distinct positions within molecular charge distribution distance between point f,- and 1"]- spin of the ith electron velocity amplitude of sinusoidal wave vector potential magnetic field, magnetic induction magnetic field of nucleus at point t complex conjugation operator, see pg. 17 electric displacement field energy, ground state energy, energy of state n z-component of electric field with fi'equency to, energy applied electric field with frequency a) at point i- electric field from the Kth nucleus electromagnetic field tensor reduced resolvent xiii Hamiltonian magnetic field spin of the ath nucleus scattering intensity in z direction from incident light in x direction imaginary part of z first-order spin-spin coupling constant second-order spin-spin coupling constant third-order spin-spin coupling constant through-space spin-spin coupling constant Lagrangian mass of nucleus K magnetization number of electrons polarization with frequency to at point i fluctuating polarization at point i permutation operator, see pg. 39 normal-mode coordinate position of the Kth nucleus reduced resolvent summation of operator exchanges to state A, see pg. 51 component OLD of the dipole propagator xiv V potential energy V vortex function, see pg. 71 Z charge in multiples of e ZK charge of nucleus K Miscellaneous 6“ , 6,. four-dimensional differentiation operator, see pg. 80 V, VI: del operator, ath component of del operator 500 = IOXOI ground-state projection operator 59% so M perrnutator of components B and y; permutator of [3, y and 5 0 ground state, DC electric field, time component in 4-vector |O> vacuum state Notations Subscripts on vectors or tensors indicate Cartesian components of the vector or tensor. Indices used for specific components are x, y, and 2. Indices used for arbitrary components are a, B, y, 5 and a. In Chapter 5, u, v and 6 refer to components in four-dimensional space, while i, j and k refer to components in three-dimensional space. XV (w IO] o) The bra-ket notation is used where (w |O| 1’) = jw‘O 4) (it and (it is the differential volume element that may include summation over spin indices and/or integration over time depending on the context. Densities are expressed as a function of positions and fi'equencies, as in the example 59,5(T,T',f";—too;w1,0) where 'r',f' and f" are positions within the molecule, «1),, is the frequency of the molecular response to incident frequencies a), and O (0 indicating a DC electric field). A The caret above A indicates A is quantum mechanical operator. The bar above r indicates r is three-dimensional vector. "II T Bold face indicates a tensor or in Chapter 5 may indicate a four-dimensional vector. The following symbols are used to label basis fimctions: i,j,k,landm l,).',k",x,rc'andx". [A,B] Commutator between operators A and B, [A, B] = AB — BA . {A,B} Anticommutator between operators A and B, {A, B} = AB + BA . G'(m) The asterisk indicates the complex conjugate of G(a)). (A) The brackets indicate an average of the quantity A. A - B The dot product between vectors A and B A x B The cross product between vectors A and B V - A The divergence of vector A V x A The curl of A xvi ‘0 HQ .1. . CHAPTER 1: INTRODUCTION The Derivatives of Molecular Properties and Nonlocal Susceptibility Densities This thesis is comprised of three parts. The underlying concept of these three parts is relating the change of a molecular property to other well defined molecular properties. In the first part, the derivative of the electronic hyperpolarizability with respect to a Cartesian nuclear coordinate is related to the nonlocal second hyperpolarizability density, the nuclear charge and the dipole propagator. In the second part, a new expression for the derivative of the polarization propagator with respect to an arbitrary coordinate is derived in terms of the quadratic polarization propagator and a sum of polarization propagators. The third part, contained in chapters four and five, concerns finding an expression for the derivative of the electronic magnetic moment with respect to nuclear momentum and its relationship to a nonlocal magnetizability density. The foundation of this work lies in nonlocal polarizability density theory. The nonlocal polarizability density characterizes the polarization at one point in a molecule due to its reaction with an electric field at another point in the molecule. The densities may be described as the distribution of a molecule’s polarizable matter. Nonlocal polarizability densities have been used to describe optical rotation,‘ dielectric properties of condensed matter,2 and light scattering in dense fluids.3 Hunt has further exploited the properties of nonlocal polarizability theory in the theory of intermolecular forces,” intermolecular electronic forces on nuclei,”8 nonadditive three-body intermolecular forces,9 zero temperature homogeneous electron gases,lo molecular softness,“ and vibrational force constants and anharmonicities.12 Nonlocal susceptibilities densities have also been used in the construction of a theory of vibrational circular dichroism.13 Closer to the purposes of this work, nonlocal susceptibility densities. have been used in expressions for the derivatives of molecular properties.“"““5 Hyperpolarizability Derivative The derivation of the nuclear coordinate derivative of the hyperpolarizability is found in Chapter 2.“5 6803, (r,r',r";—rl).,;ool ,(oz) / 6R? = Idr"'ym(r,r’,r",r'";—co.,;(o1,m2,O)ZKT55 (r"’,RK) .( 1) The derivation starts with the Orr and Ward’s expression for the hyper-polarizability.l7 Then derivatives of the wavefunctions and operators that comprise the six-term ‘ hyperpolarizability are substituted into one term of the hyperpolarizability to yield a ten- term expression. The derivatives used are derived in detail in Appendix A. The initial result is manipulated algebraically in a nontrivial way to yield the equality in Equation (1). The form of the second hyperpolarizability density in Equation (1) was adapted from the expression for the second hyperpolarizability in Orr and Ward.17 The complete derivation involves a sixty-terrn expression which has been included in Appendix B. The nuclear-coordinate derivative of the hyperpolarizability yields intensity information for nonresonant vibrational hyper-Raman scattering.l8 Hyper-Raman scattering is the nonlinear analog of Raman scattering. In a hyper-Raman scattering event, two quanta are absorbed in taking the scatterer from its initial state to a virtual electronic state. Then, as in Raman scattering, one quantum is emitted as the scatterer relaxes into an excited or deexcited vibrational state. A brief review of hyper-Raman scattering theory is given. Recent applications of hyper-Raman scattering are also considered. Polarization Propagator Derivative Chapter 3 contains the derivation of the derivative of the polarization propagator‘m'21 that was found to be related to the quadratic polarization propagator”22 and a sum of polarization propagators. 1,1' "" ———a““'(“’) = [Eh—0] Interim») 6n an 1." + EACmI-Iafi'(€0)+ ZACx'mI-IQE‘IUD) ' (2) + k2 , ctr. Héztiw) + .2 Cir PM (0))- The relationship between the polarization propagator derivative and the quadratic polarization propagator was prompted by a relationship found by Hunt et a1." between the nuclear coordinate derivative of the electronic molecular polarizability and the electronic molecular hyperpolarizability density. A general relationship between derivatives of linear response properties and quadratic response properties was suggested by this specific relationship between polarizabilities. The polarization propagator is employed in the calculation of linear response properties.21 Polarization propagators can be constructed in a conventional fashion as a sum-over-states or in a second quantized fashion using a sum over creation and annihilation operators and the techniques of “superoperator” algebra.23 In the derivation of the derivative of the polarization propagator, the second quantized version is used. The advantage of second quantization comes from how the many-electron wavefunction is managed. The antisymmetrical nature of the many-electron wavefunction follows from the change of the wavefunction’s sign when two electrons are interchanged. In second quantization, the antisymmetry of the wavefimction is ensured by the use of anticommutation relations between creation and annihilation operators. The relationships between the anticommutation relations and Slater determinants are discussed in this chapter. The derivation involves techniques found in Appendices A and C. However, the novel portion of the derivation involves calculating the derivative of a creation and annihilation operator pair with respect to an arbitrary parameter. The derivative is calculated via the derivatives of many-electron wavefunctions. Throughout the derivation, the second quantization formalism is used; however, portions of the derivations are done in parallel with the Slater determinant formalism for pedagogical reasons. The advantage of Equation (2) is proposed to be in the calculation of derivatives of molecular properties such as molecular gradients and Hessians.24 The derivatives of many other molecular properties have been computed using analytical and numerical techniques. Past work calculating the derivatives of molecular properties using 4 polarization propagator techniques is briefly reviewed in Chapter 3. Different possible applications of Equation (2) are discussed as well. Magnetic Moment Derivative In chapter 4, the derivative of the electronic magnetic moment with respect to nuclear momentum is found to be related to the molecular high-frequency (“paramagnetic”) chemical shift density from the theory of nuclear magnetic resonance.”26 5E0. (f') '4 Z6 - -r _ _ap:_=—#ijg$(r,r )VB dar. (3) The relationship is found by considering the magnetic field produced by a moving nucleus as a perturbation on the electronic structure. The ground state wavefunctions are corrected to first order in perturbation theory using the nuclear magnetic field as the perturbation. The corrected wavefunctions are used to calculate the expectation value of the magnetic moment. When the derivative of this expectation value is taken with respect to nuclear momentum, its relationship with the nonlocal chemical shift density is uncovered. The result is correct when wavefunctions that go beyond the Bom- Oppenheimer approximation are used.27 A brief review of research using nuclear momentum derivatives, especially with application to vibrational circular dichroism spectroscopy, is given. Equation (3) demonstrates that within a molecule the electronic response to an internal magnetic field occurs via a molecular property that describes the response to an external magnetic field. The analogous equivalence has been shown by Hunt at al.""5 for intramolecular electronic response to internal electric fields. Chapter 5 considers possible routes to a more complete theory of intramolecular electromagnetic response. The first consideration involves classical electromagnetic theory in the context of special relativity. In the relativistic formulation of classical electrodynamics, the electric and magnetic fields are not two different vector quantities, but, rather they are both components of a second-rank four-dimensional tensor. The relativistic current density is a four dimensional vector that includes as the “time” component, the charge density. Similarly, the electrostatic potential and the magnetic vector potential are integrated into a four-dimensional vector potential. Molecular polarization and magnetization fields are not distinct, but rather the components of a four dimensional second-rank tensor. All of the quantities and laws of electrodynamics can be formulated very compactly in a relativistic formulation. The four Maxwell equations that fundamentally describe all electromagnetic phenomena become two equations in the relativistic formulation. Maaskant and Oosterhoffl originally formulated the nonlocal polarizability density in terms of current densities. Hunt" reformulated polarizability density in terms of polarization operators. However Hunt’s reformulation is derived with the assumption that the applied field is obtained from a scalar potential. It is suggested that a complete nonlocal electromagnetic susceptibility density theory could be found from the removal Of this restriction and use of four-dimensional vector calculus. A relativistically consistent four-dimensional quantum theory for electrons was constructed in the early days of quantum mechanics by Dirac.”29 In addition to 6 increased accuracy in the calculation of atomic energies, the theory incorporated, in a fundamental way, the spin of the electron. However, a disadvantage of the theory was the appearance of energy states associated with the antimatter analog of the electron, the positron. In the description of electronic interactions with electric or magnetic fields, the positron states can not be neglected. Positrons in the description of low-energy interactions such as molecular interactions are difficult to conceptualize. Fortunately, for low-energy interactions, techniques have been found that can remove the “positron” portion of energies in exchange for a series expansion of “electron” energies. The first and most commonly used expansion is the Foldy- Wouthuysen transformation.”“ In this nonrelativistic formulation of the Hamiltonian (nonrelativistic because it no longer involves positrons), several hyperfine terms are found that depend on the nuclear momentum and the nuclear magnetic moment. It is suggested that such terms be included in an application of perturbation theory to find new magnetic phenomena and novel explanations for discovered phenomena. REFERENCES ‘ W. J. A. Maaskant and L. J. Oosterhoff, Molec. Phys. 8, 319 (1964). 2L. M. Haflcensheid and J. Vlieger, Physica 75, 57 (1974); 79, 517 (1975). 3 T. Keyes and B. M. Ladanyi, Molec. Phys. 33, 1271 (1977). 4 K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983); 80, 393 (1984). 5 Y. Q. Liang and K. L. C. Hunt, J. Chem. Phys. 98, 4626 (1993). 6 K L. C. Hunt, J. Chem. Phys. 92, 1180 (1990). 7K. L. C. Hunt and Y. Q. Liang, J. Chem. Phys. 95, 2549 (1991). 8 P.-H. Liu and K. L. C. Hunt, J. Chem. Phys. 100, 2800 (1994). 9 X. Li and K. L. C. Hunt, J. Chem. Phys. 105, 4076 (1996). ‘° R. Nirnalakirthi and K. L. C. Hunt, J. Chem. Phys. J. Chem. Phys. 98, 3066 (1993). " P.-H. Liu and K. L. C. Hunt, J. Chem. Phys. 103, 10597 (1995). '2 K. L. C. Hunt, J. Chem. Phys. 103, 3552 (1995). ‘3 K. L. C. Hunt and R. A. Harris, J. Chem. Phys. 94, 6995 ( 1991). " K. L. C. Hunt, J. Chem. Phys. 90, 4909 (1989). '5 K. L. C. Hunt, Y. Q. Liang, R. Nimalakirthi and R. A. Harris, J. Chem. Phys. 91, 5251 (1989) ‘6 E. L. Tisko, X. Li and K. L. C. Hunt, J. Chem. Phys. 103, 6873 (1995). '7 B. J. Orr and J. F. Ward, Molec. Phys. 20, 513 (1971). ‘8 D. A. Long, in Nonlinear Raman Spectroscopy and its Chemical Applications, edited by W. Kiefer and D. A. Long, NATO ASI Ser. C 93 (Reidel, Dordrecht, 1982), pp. 99-130, 165-179. '9 J. Linderberg and Y. Ohm, Propagators in Quantum Chemistry (Academic Press, New York, 1973). 2° J. Olsen and P. Jorgensen, J. Chem. Phys. 82, 3235 (1985). 2‘ J. Oddershede, Adv. Quant. Chem. 11, 275 (1978). 22 H. Hettema, H. J. A. Jensen, P. Jorgensen and J. Olsen, J. Chem. Phys. 97, 1174 (1992) 23 P. Jorgensen and J. Simons, Second Quantization-Based Methods in Quantum Chemistry (Academic Press, New York, 1981). V 2‘ Geometrical Derivatives of Energy Surfaces and Molecular Properties, P. J orgensen and J. Sirnons, eds. NATO ASI Series C: Vol. 166 (D. Reidel Pub. Co., Dordrecht, 1985). 2’ N. F. Ramsey, Molecular Beams (Oxford University Press, Oxford, 1956). 26 C. P. Slichter, Principles of Magnetic Resonance 2nd ed. (Springer-Verlag, Berlin, 1978). 27 L. A. Nafie and T. B. Freedman, J. Chem. Phys. 78, 7108 (1983). 28 P. A. M. Dirac, Proc. Roy. Soc. Lon. A117, 610 (1928); A126, 360 (1930). 29 R. H. Landau, Quantum Mechanics II, 2nd ed. (John Wiley and Sons, New York, 1996) 3° L. L. Foldy and S. A. Wouthuysen, Phys. Rev. 78, 29 (1950). 3’ R. E. Moss, Advanced Molecular Quantum Mechanics (Chapman and Hall, London, 1973) CHAPTER 2: RELATION OF HYPERPOLARIZABILITY DERIVATIVES TO SECOND HYPERPOLARIZABILITY DEN SITIES Introduction In this chapter, a new analytical result is derived for the nuclear coordinate dependence of the electronic hyperpolarizability B(-we;m1 .002) , which gives rise to nonlinear optical processes such as frequency-sum and difference generation."”'3""5'6 In earlier numerical work, derivatives of B with respect to nuclear coordinates have been estimated serniempirically7 and calculated ab initio.8'9"°’” This chapter focuses on the interpretation of the [i derivatives via their connection to a different molecular property. The derivative of [EX—0),, my, .002) with respect to coordinate fix of nucleus K is (shown to depend on the second hyperpolarizability density y(f, f', f", f’"; .496 ;ml .012 ,0) for the electronic state, on the nuclear charge, and on the dipole propagator T(f'"',fi") from i“ to f’" . This result holds because the electrons within a molecule respond to changes 5?“ in the Coulomb field of the nucleus, due to an infinitesimal shift in the position of nucleus K, via the same nonlocal electrical susceptibilities that characterize their response to external electric fields. 10 Nonlocal Polarizability Densities, Polarization and Internal/External Electric Fields Nonlocal polarizability densities or(‘r',f’;—0);00 and nonlocal ) 12.13.14.15 hyperpolarizability densities B(f,f’,f”;—co;m — 00 ',a) ') and y(f,f',f",i'"’;—0);m — (0' — 01",0) ’,00 ") '6 describe the distribution of polarizability within a molecule. When an external field F‘(f,0)) is applied, the electronic polarization P(f,0)) at point f is“5 P(i,00) = Po(i,a)) + I <1? or(i';f',00) - F(f',(0) 1 co +5 ldco'ldf'df" B(i';f’,0) —r0',f",c0'):F(f’,0)—00')F(i'",0)') "° (1) 1 co co . +_ I dw'ldm"1(fi'(fi"(fim Y(I‘;IJ,CO"(D' -OJ",I'",O)',I'"',(D" IF(I",(D—CD' _0)n) 6.0 -Q x F(f",00') F(f”',a)")+- -- . In equation (1), P0 (E00) denotes the polarization in the absence of the external field, and symbols such as -, :, and 3 indicate tensor contractions. The convention used by Orr and Ward17 and by Bishop18 is followed in showing the frequency dependence of the susceptibilities. The polarization in equation (1) is related to the charge density by V . P(f,(0) = —p(f,00) (2) and the same equation relates the polarization and charge density operators, P(f) and 13(f) . Thus the polarization contains the full information about the electronic charge redistribution of a molecule in the applied field, and not simply information about its dipole density. 11 Ill The nonlocal polarizability density, or(f,f’;—(0 ;00) , is a tensor quantity which 15.16 describes the linear response to the applied field, and it can be defined as 9803') = (0| 13.. (0909131: (F)! 0) + (0| Ps(?')G'(-G'(-m.)e.(0no>+- (10) Full permutation symmetry of Bum exists only if damping is neglected. Equation (9) is analogous to Equation (43b) for the spatially integrated hyperpolarizability Bap, (—0)a;01,,(02) given by Orr and Ward.17 To find the derivative of Bum (f,f’,f”;-ma ;03 1 ,co 2) with respect to the nuclear Cartesian coordinate RE , the derivative of the ground state wavefunction with respect to an arbitrary parameter 1. is used.” BIO) 63C Evanglm. (11) For consistency with the approximate treatment of damping, as in the work of Orr and War ‘7 the imaginary components of the eigenvalues in off-resonant denominators are neglected; then G*(0) can be replaced by G(O), and the derivative of G(m) with respect to 1. satisfies23'9‘ 996%me )L—Sca; 596(0) )+goo-:C‘G(O)G(w)+G(0)G(O)—500 (12) Also94 a——P;(’)=< (OI—G(O)P. ()|0>+ 03) If A = R}? , the 6 Cartesian coordinate of nucleus K, then22'23’9" 19 63C m K m m an." —Idf z P.(r )T (r ,R"). (14) The change in the inverse lifetimes of the excited states due to an infinitesimal shift in RI,‘ has been neglected. In Equation (14), Te,(r"',1_{x) is the dipole propagator, which determines the field at 1"" due to the polarization at fix ; in general, Tap (if) = Va WEE-{$1}. (15) The Einstein convention of summation over repeated Greek subscripts is followed in Equation (14) and below. From Equations (9) through (13), each of the six terms in Equation (9) for Bw1(f,f',f";-00 a ;00 , ,0) 2) generates ten terms when differentiated with respect to RI,‘ . The contribution to the hyperpolarizability derivative from the first term (taken as representative) is 6;.<0|P.(P)G(wa)P °1) wz T (m) ’ (OIP.(r"')G(0)P.(r)G (wa)P°(f")G(mn)Pt(f')IO> +<0|P.(r)G(coa)Pi’(r "’)G(coo)P$(r ")G(wa)Pt(r' )I 0) -0< IP.(r "')G( O)<0|P..(r)| o)G(s.) P‘.’(r' ')G(oox)Pt(r' )l0> -(0| pa(f)G((oa)G(0)138(f"')l 0)(0|P:(r-")G(m.)13,(r')| 0) -<0|P.(r"’)G(0)P.(r")I 0><0lP.(P)G(wo)G(con)Pt(i')|0>, -< > < -< -< < (16) x< OIP.(r")G(0)P.(r'")I 0><0lP.(f)G(wa)G(wu)Pt(i')l0 0|Pa(r)G(a>o)P3(r")G(con)P °(r'")G( + OIP.(r)G(wa °(r")| 0><0lP.(r"')G(0)G(con)Pt(f’)l0) OIP.(r)G(wo 36 ")G(wn)<0|Pt(P')| 0>G(0)P.(f"’)|0> )P )P .+ 0an (r)G(0).,)P, 0(1‘")G(CDI) P06 )G(O)Pe (rm)l O) J 20 The fourth term of this expression contains the fluctuating polarization operator 13: (f") which can be changed to a total polarization operator 13, (f") through the identity"5 (0| 13a (f)G(a).,)G(O) 13, (f "')| 0)(0|P: (r- ")G(o)1) 13, (r')| 0) (17) = (0| P. (r)o(m,)c(0) P. (f "’)| WI P. (f")G(co 1) Pt (f’)| 0)- Similarly, the eighth term has a fluctuating polarization operator, which can be converted to a total fluctuation operator. Also with the definition of the fluctuating polarization operator, the first and third terms within the brackets in Equation (16) combine to yield (0|P.(f"')G(0)P.(f)G(mo)P$(f")G(an)Pt(f')l0) -(0|P.(f"')G(0)<0lPAP)!0)G(mo)P$(f")G(mn)Pt(f')|0) (18) = (0| P. (f'")G(0) P2(P)G(m.)13$(f")G(m,)fa,(r')| 0). Similarly, the ninth and tenth terms within the brackets of Equation (16) combine to give the matrix element <01macho»:(t~)e<4.)tsnew) 0). 09> From Equations (16), (17), (l 8) and the expressions generated by differentiating the remaining terms in Equation (9) with respect to K}? , we obtain BBQ,” (f,f',f";—(oo;a),,a1 6R}? so” (0|1340600.)132(f"')G(w.)P3(r")G(co.)13.6)!0) '8 +(0lP,(f')G'(—cm)P‘,’(f")G‘(—mo)P3(f)G(0)P.(f"')|0> l<0| P. (f)G(mo)G(0) P. (f"')| 0><0l P. (i ")G((°1)1315 (P)! 0) +<0| P. (P)G(wo)G(cm) Pe(f')| 0><0| P.(f'")G(0)P. (f")| 0) +<0| P.(P)G(wo)G(wn)Pt(?')l 0><0| P. (T'")G(0)P.(f'")l 0) +<0| P.(f"’)G(wz)G(0)P. (P') 0><0| P. (f)G(wa)Pt(f')| 0) < 2) = ldfm ZKTss(f""-R_K)X [1+ C({COi} —) {-(01l)] (20) Xt -5037 + 0|P.(f")G'(—coz)G(0)P.(f"')l0><0|P.(P)G(wn)Pt(f')|0> ,+ 0|P.(f")G‘(—coz)G(m1)P.(f')|0><0IP.(f"')G(0)P.(f)I0>_, 21 where 501m denotes the sum of all permutations of 1'5fl (f') , p1 (f”) , and 13, (f "'), simultaneously with 0),, 0), and 0 (respectively) in the expression that follows. The operator C({mi} —> {—m,» takes the complex conjugate and replaces each (0i by -c0i; the operation applies to 0),, a), and 006. Again damping has been neglected in off-resonant terms; it is reiterated that 6(0) and G*(O) are interchangeable at this level of approximation. In order to connect the hyperpolarizability derivative to the second hyperpolarizability, the following identity is used,” (0| P. (f)G(wa)G(m 1) Pp (T')l OXOI P. (P'")G(0) P. (f")| 0) + <0! Pa (P)G(wa) Pt('r")| 0><0| P. (f'")G(0)G(wz) P. (f")l 0) = (0| Pa(f)G(coa)G(cm)Pt(i’)l 0><0| P.(f'"')G(wz)P. (f")l 0) + (0| P.(P)G(91)Pt(f’)l 0><0| P.(i"')G(0)G(wz)P. (f")| 0) , (21) which is proven by converting both sides S1 and S2 in Equation (21) to the explicit sum over states form S) = $2 _—_- 71.2 2! 2! (0| 13“(f)l mel 13136.,» 0>
    (Qmo + Qjo "' (1)1 " (02) 22 at j (Qua - (DaXQmo ‘ 001) Q10(01'0 ' (92) ( ) where am, = mm, -i1‘,,,/2 and the primes on the summations over m and j imply that the ground state is excluded from the sum. We also use the complex conjugate of the identity in Equation (21). Then Equation (20) can be simplified using the permutation operator @1513? 22 prr [1+C({m,} Fl + will 0lP..(r)G(m )G(O)13(r")’|0(0|p1(f")G(m,)139(i"’)|0) +<0|P (r)G(w )G(wn)Pt(r' )IOXOIP.(f"')G(0)P.(f")|0> +<0lP.. (f)G(m.,)G(m,)f>B(f')|O)(O|137 (f”)G(0) P.(f'")| 0) +< < < 0|P.(P"’)G(w2)G(0)P.(i")l0>(0|P.(P)G(wa)Pp(i')lO) 0le(f")G'(—92)G(0)P.(i’")|0><0lP.(i)G(wn)Pt(f’)l0) _+ on. 6091-4460013. 0) 0><0|P.(P"')G(0) r». (0| or =[1+c(ec P +<01e. ' Therefore 6Ba57(f,f',f";—0)U;0),,m2) x5097: 6R}? .1 =14r~'z*<'r..(r~znx)x[1+c({s.}—>{-80} ’(0lP.(f)G(u.)P2(f"')G(coo)P$(f")G(cm)P.(f')l0) +<0|Pt(f')G'(-con)P3(f")G’(-w )P° (r)G(0)P (r" ')IO> -<0|P.(P)G(wo)G(0)P.(f" '0)| ><0|P.(r' )'G(mu)Pt(r' )|0> _-<0|P.(f)G(0)P.(f"')l0><0|P.(r' ’)G (-a>2)G(wn)Pt(f' 10), The quantity in the brackets of Equation (24) can be compared to the second (23) (24) hyperpolarizability density. The equation used for the second hyperpolarizability is taken from Orr and Ward’s article on nonlinear optical polarization.l7 The equation is directly quoted. 23 Pa)" = K(-Coo;(.0n(1)2 ,coa)h'311.2.3 1 ’ <9. cerium)... (at). + .

    ,... (HP), 2' (918 - maXng - 011- m2XQm - (Dr) (Dig + 003)(ng - 031- wZXQng - (01) .. + at)..

    ... + .crane-Prue. _ (Qig + (1)1)(ng + (DJ '1' (1)2)(0113 "' (.03) (Gig '1' 030(ng + (01+ (02)(th + (Do)_ r. (25) < 1

    .,....um)..., P>..... ‘ _,., (n..-s.)(n..-s.)(n.,—s.) (am.—e.)(n;.+s.)(n.,—s,) m" 91"”)...mg..., ,..(H'”2>.,,.

    n _ (0m, + (09(th + 010(th + (01) (0..., + (1)3)(Qng - (1)2)(th + 0);) A J L Orr and Ward’s equation for the second hyperpolarizability is identical to the expression in the brackets of Equation (24), though explanation is needed to relate the two equations to each other. First the frequencies are compared; the (03 in the second hyperpolarizability corresponds to O in the hyperpolarizability derivative. The matrix element notation translates as (em), = <11 - 1W?) P291044 m> —> 01132601 m> lm = (II — lH'Ps(r) E93(f)df| m) —> (1|fi,(r"')| m). Note that the expression from Orr and Ward includes the incident electric field to obtain (26) the net polarization whereas in this formulation, flexibility is maintained to multiply by the electric field subsequently. The correlation is complete when it is remembered that summation on the index of the polarization operator over all the Cartesian coordinates in Equation (24) is necessary. The 11”,,3 operator is the same permutation operator as 50 p7, and the h in the denominator of the equation of the second hyperpolarizability can be distributed through the frequency expressions to yield the energy expressions of the equation for the hyperpolarizability. 24 One can use the definition of the reduced resolvent to simplify the sum-over-states in the equation for the second hyperpolarizability. Then the frequency conjugation operator C({m,} —> {-0).}) can be factored from the second hyperpolarizability equation to simplify further. The first and fourth, second and third, fifth and seventh, and sixth and eighth terms of the second hyperpolarizability equation are frequency conjugates of each other. Finally, the coefficient K(—a).,;0)1,—m2 ,-003) depends on the number of zero frequencies and the number of repeated frequencies in the set (11,, (02. (03 and 0),. The value of K becomes clear for specific frequencies when integrations over the frequency values are taken. Integration over a single (0 will yield a different number than three integrations over three distinct frequencies. Thus being aware of all of these equivalencies, the equation for the second hyperpolarizability can be written as ym(r,r',r",r'";—o);c0,,m,,013) 1011340914)...» 3(r'")G(wa)P$(f '419( )n(f')10> -[1+9(11co.1-n1)]xn riOIPa(f')G ‘1—4.)p$p.< "11> <27) ”" -<0|P.(P)G(coa)G(0)P.(r "')l 00>< IP (P’ W )P .6 )'l 0) _-<0|P.(r)G(0)P.(r"')|0 ><0|P.( ")G -{02)G((01)135(l")|0>d Relating Equation (27) to Equation (24) yields the goal of the derivation. 513.910 f'r";-<01Ps(f')G(-wn)G(w2)P,(f")10> + (OIP.(P)G(wo)G(0)P.(T"')l 0)<0|Ps(f’)G(-w1)P.(f")l 0) =n (so) +1>.<0l Pp (ml 00 l Py (PM Ollflmo + 910 “ (02) ((21:10 - (Do) Qmo (Qjo + (DIXQjo - (1)2) = h-erzr m J and the analog of Equation (30) with the roles of {139 (f’), 0),} and {137 (f"), (02} interchanged. Equation (30) corresponds to Equation (21) with BB (f’) and 13.1fm) interchanged, G(O) replaced by G(—m,), G(O),) replaced by G(O), and damping neglected. 26 Further Applications of the Hyperpolarizability Derivative The analytic result given in Equation (28) is needed to relate quartic anharmonicities of potential energy surfaces to electronic hyperpolarization energies,96 and to relate nonadditive three-body forces to the three-body polarization.97 Equation (28) is potentially useful for analyzing the origins of hyper-Raman scattering on the intramolecular scale, and for making qualitative predictions about hyper-Raman intensities. In ab initio calculations with a given basis set and method, a direct evaluation of the left-hand side of Equation (28) is expected to be more efficient computationally. However, use of the form on the right may assist in basis-set optimization by indicating which regions of the molecule contribute most to the hyperpolarizability derivative. The Hyperpolarizability Derivative and Hyper-Raman Intensities The derivative of the net molecular hyperpolarizability Bum (-c0 0301,00,) with respect to the vibrational normal-mode coordinate Q is 55m,(-we;031,w2)/5Q = IE2"},jdfdf’df"df"'ymay,(i‘,f',f”,f"';—00.,;001«102,O)ZKT,,5(f"',RK)25%. (31) Intensities for vibrational hyper-Raman scattering depend upon the derivative given by Equation (31), with 001 and c0; both equal to the frequency of the incident radiation (1),, and 0),, = 2cm .28'29'30 This connection'0'29'30’ms for hyper-Rarnan scattering of incident light plane-polarized along the space-fixed Z axis, initially propagating in the -X direction toward a scatterer at the origin in the space-fixed axis system (X, Y, Z) is reviewed. 27 Scattered radiation of frequency a), is detected along the Y axis. The intensity (per unit solid angle) 122(0),) of the scattered radiation plane-polarized along Z, due to an isolated molecule that undergoes a transition from vibrational state m to n is given by40 2 2 1.4.0.): 37231411181 - (32) The intensity 1xz(o),) of scattered radiation plane-polarized along X satisfies Equation (32) with uz replaced by 11x. At the level of the Placzek theory as applied to hyper- Raman scattering, uzand ux are identified as the electronic dipoles induced by nonlinear response to the electric field of the incident light, so 11.. =1/2 szz(—2031;G)1,(Di)EZ(COJ)Ez(wi)a (33) where 132(0) 1) is the electric field of the incident light (and similarly for uz). Below, szz(-’2C0i;0)i,0)i) and Bm(_2m,;m,,m,) are abbreviated as BXZZ and B712, respectively. The hyperpolarizability components BUK (IJK = XZZ or ZZZ) are expanded as series in the normal mode coordinates Q,, about the equilibrium nuclear configuration (denoted by the subscript eq), flux = Bun (34) Hyper-Raman scattering occurs at frequencies 00, shifted from 2014 by (Em — En)/h , with intensity Ixz(m,) given by40 1x201) 5") ——"5ml< lel ”ll [aBXZZLxJ Ez(0)i)4 (35) 128:21toe c3 an 28 and similarly for 122(0),) with BXZZ -> Bm. For transitions between the ground vibrational state and the state with one quantum of excitation in the mode QV , in the harmonic approximation, the strength of the transition is 2 ll = h/va ’ (36) where 03. is the frequency of the normal mode Qv. For a sample of N freely rotating molecules, with probability Pm to occupy the initial state |m) , the intensity In ((1),) of hyper-Raman scattering is related to the isotopic average of the hyperpolarizability derivative, <(aB XZZ/an [eq)2> by40 =__l <[ an Lg >9 (37) and similarly for 171(0),) . In Equation (37), 10 is the irradiance,4o defined by 1, = 1/2 csogE,(m,)’, (38) and g is a coherence factor.98 The space-fixed tensor components c‘BBUK/BQv Lq are related to the molecule-fixed components 6Bijk/ (3Qv L] by a .. 66%”: Lq .-_- 13%|: an an 310%: L, (39) where 3,, is the direction cosine between the molecular axis 1 and the space-fixed axis 1. Since Equation (3 7) relates observed hyper-Raman intensities to the derivatives of the hyperpolarizability, by Equation (31) the hyper-Raman intensities are also related to the second hyperpolarizability density y(f,f’,f”,f”'; —2c0t;00i ,0» ,0) . 29 Conclusion The principal result of this work is contained in Equation (28), which establishes a link between the derivative of the lowest-order nonlinear response. tensor BaBY(—coa;031,w 2 ) , taken with respect to the position of nucleus K, and the nonlinear susceptibility density yaw (f,f’,f”,f"';—2m, m), ,m, ,0) of the next order. The second hyperpolarizability density determines the change in the effective value of the hyperpolarizability when a static external field is applied to a molecule (cf. References 24, 25 and 31 for cases with uniform applied fields and spatially integrated values of y and B). When nucleus K shifts infinitesimally within a molecule, the electrons respond to the change in the nuclear Coulomb field via the same nonlocal susceptibilities that characterize their response to applied electric fields. This result is illustrated in Equation (28), since the change SF? in the Coulomb field resulting from the shift in the position of nucleus K is given by ZKTfi, (f'", R“ ) 6R}?- Thus the second hyperpolarizability density determines the change in the hyperpolarizability due to small distortions of the molecular geometry. Based on Equations (31) and (37), the second hyperpolarizability density also determines band intensities for vibrational hyper-Raman scattering. 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Schneider, G. Brehm and P. Freunscht, Phys. Status Solidi B - Basic Res. 189, 37 (1995). 35 8° W. H. Yang, J. Hulteen, G. C. Schatz and R. P. VanDuyne, J. Chem. Phys. 104, 4313 (1996). 8' C. K. Johnson and s. A. Soper, J. Phys. Chem. 93, 7281 (1989). 82 G. Mayer, C. R. Acad. Sci. B 267, 54 (1968); F. Hauchecorne, R. Kerherve and G. Mayer, J. Phys. (Paris) 32, 47 (1971). 83 R. S. Finn and J. F. Ward, J. Chem. Phys. 60, 454 (1974); J. F. Ward and I. J. Bigio, Phys. Rev. A 11, 60 (1975). 8“ M. Stahelin, C. R. Moylan, D. M. Burland, A. Willetts, J. E. Rice, D. P. Shelton and E. A. Donley, J. Chem. Phys.98, 5595 (1993); E. A. Donley and D. P. Shelton, Chem. Phys. Lett. 215, 156 (1993). 85 R. W. Terhune, P. D. Maker and C. M. Savage, Phys. Rev. Lett. 8, 404 (1962). 86 J. D. Jackson, Classical Electrodynamics 2nd ed. (John Wiley & Sons, New York, 1975). V 87 C. H. Lee, R. K. Chang and N. Bloembergen, Phys. Rev. Lett. 18, 167 (1967). 88 R A. Bradley, R. Georgiadis, S. D. Kevan and G. L. Richmond, J. Chem. Phys. 99, 5535 (1993). 89 P. R Fischer, J. L. Daschbach and G. L. Richmond, Chem. Phys. Lett. 218, 200 (1994). 9° J. S. Horowitz, B. E. Kohler and T. A. Spiglanin, J. Phys. Chem. 89, 1574 (1985). 9‘ L. D. Ziegler and J. L. Roebber, Chem. Phys. Lett. 136, 377 (1987); Y. C. Chung and L. D. Ziegler, J. Chem. Phys. 88, 7287 (1988). 92 C. C. Bonang and S. M. Cameron, Chem. Phys. Lett. 187, 619 (1991); 192, 303 (1992). 36 93 C. C. Bonang, S. M. Cameron, J. D. Getty and P. B. Kelly, Chem. Phys. Lett. 209, 35 (1993). 9‘ Appendix A. 95 Appendix B. 9‘ K. L. C. Hunt, J. Chem. Phys. 103, 3552 (1995). 97 X. Li and K. L. C. Hunt, J. Chem. Phys. 105, 4076 (1996). 98 D. A. Long, Raman Spectroscopy (McGraw-Hill, New York, 1977). 37 CHAPTER 3: RELATION OF POLARIZATION PROPAGATOR DERIVATIVES TO NONLINEAR POLARIZATION PROPAGATORS Introduction In quantum chemistry, calculation of molecular properties involves two quantities, operators and wavefunctions. Using perturbation theory, the value of a property is often found by operating a single operator upon a smn-over-eigenfunctions (sum-over-states). For example, the dipole polarizability can be calculated from the second-order perturbation expression‘ (1) (ols.ln>+<01s.ln> EO_En+hm EO-En—hfl) , (143(0)) = ‘E'{ where the In) are the wavefimctions, pa is the dipole operator of the ath Cartesian coordinate and 2' indicates the sum over the excited states only. In general, accurate excited-state wavefunctions are more difficult to calculate than ground-state wavefimctions. Also for accurate property calculations, a large number of excited-state wavefunctions may be needed. Thus the sum-over-states method is often impractical, since its accuracy is dependent on the accuracy and number of the excited-state wavefunctions. Use of the polarization propagator avoids the problem of needing excited-state wavefunctions. In a polarization propagator calculation, only an accurate ground-state 38 wavefunction is necessary. To calculate a property, a sum is still needed; however, the sum is not over a complete space of excited-state wavefunctions but over a complete space of operators. The operator sums are found by applying the equation of motion of the polarization propagator. The equation of motion2 for the polarization propagator <(B;A)) that describes the effect of operator B on the molecular property of operator A is hm<w.(e)]. <4) The w.(r,) are the individual one-electron wavefunctions. If the two particles are exchanged by switching electronic coordinates r. and r2 , the total wavefunction changes its sign. Therefore, this wavefirnction is antisymmetric. Using the definition of a determinant from linear algebra, the total wavefunction is written identically as w.(rx) w.(r2) 1 “"‘mo w..(r2) ,5 . (5) 40 This determinant is an example of a Slater determinant. Similarly, an antisymmetric three-electron wavefunction can be written as “RED; SHED; “AER; . \P=_ Wb 1'1 Wb r2 Wb 1'3 - (6) J3 v.6.) w.(r2) v.6.) In general, a n-electron wavefunction is written as the following Slater determinant v.01) v.02) w.(r~) 1 81,01) whim) whim) “m f i '- . i ' ‘7’ WNGI) WN(r2) WN(TN) Writing a multi-electron wavefunction in this fashion allows the Pauli exclusion principle to be satisfied. When any two columns of the determinant are exchanged, signifying the exchange of two particles, the magnitude of the determinant remains the same; however, the sign is changed. 3. 13.1.].[1 IS 10 .. The properties of creation and annihilation operators found in harmonic oscillator analysis‘, many-body solid state theorys'6 and quantum field theory7'8 are exploited to construct antisymmetric wavefunctions. Relating such operators to electrons in molecular systems,”9 a creation operator operates on the electronic wavefunction of a molecular system and creates an electron in a specific quantum state. As a simple example in an atomic system, the creation operator al(1s) is applied to the vacuum state lg) as follows: 41 3*(15) IO) = lls) . (8) The creation operator, a*(ls), creates an electron in the 18 state. The annihilation operator destroys an electron as in the following examples. a(1s ) [152) = lls') or a(2s ) lls‘zs‘) = —|1s') . (9) The negative sign in Equation (9) is a consequence of antisymmetry. Both examples in Equation (9), the initial state had two electrons. In both cases, the annihilation operator destroyed an electron so that the final state had only one electron. The creation and annihilation operators used above only had spatial labeling. Since the antisymmetry prOperty is a function of spatial and spin coordinates, spatial and spin labels should be used to describe the electronic state. The examples in Equation (9) become 31(18u)|@) = Ilsa), a(1s..) Ilsa 15B) = “80 and _ a(2s..)l1s..2s.,) = -| Isa) - (10) What happens when we try to create an electron that is already present or annihilate an electron that is not present? The implication of the Pauli exclusion principle is that the maximum number of electrons in a given quantum state is one. In addition, the minimum number of electrons in a given state must be zero. Therefore, trying to create an electron in a state that is occupied or annihilating an electron that does not exist is not physical. Such operations are defined as zero, e. g., a*(1sa)llsslsp>=0 and 3(25a)l1sslss>=0- (11) 42 rrl These definitions ensure that the Pauli exclusion principle is satisfied. The operators can also be defined in an alternative, but equivalent fashion, using anticommutation rules to ensure antisymmetry. For states m and n, the anticommutation relations are {ahsal} = aLal + aIaL = 0 {airman} zaman+anam=0 (12) {almau} = aLan + anal. = 8...... These anticommutation relations demonstrate that when m = n, the same electronic state can not be created twice nor the same electronic state be annihilated twice. A two-electron ground state that satisfies the antisymmetry requirement in terms creation operators is written as, e. g. at (186)8'(1sa)| g) = llsfi 15a) - (13) An n-electron antisymmetric wavefunction is written as a*(n1m)~-a*(1s,)a*(1s.)|o>=|n1m,...1s,13.). (14) An excited configuration of a multi-electron system can be constructed using a combination of creation and annihilation operators. First the annihilation operator destroys one of the electrons in the Hartree-Fock ground state, then the creation operator places an electron into an excited state. For example, creation and annihilation operators can be used to form an excited configuration of helium, a*(28a)a(1sh)| 139 151) = '23: 15(1) . (15) Because of the anticommutation relations of Equation (12), the order of the operators is very important. Appendix C explains in more detail the consequence of applying creation and annihilation operators to a multi-electron state. 43 Operators in the second quantized formulation of quantum chemistry are products of creation and annihilation operators with specific coefficients. The operators can involve any number of electrons, though only one and two electron operators are regularly used. A one-electron operator has the form, 9 = Eleuaian where e... = <¢.‘(i)l6(h .h)|¢.(i)>- (16) The coefficient 0” is the matrix element of a more traditional quantum mechanical operator with the basis functions 4’1 and 41,. The index i indicates the ith electron in the system. Examples of quantities described with one-electron operators are the kinetic energy, momentum, angular momentum, magnetic moment and polarization. Two electron operators have the form, V = “£3; 3i am an <¢k (i) ¢| (j)lv(fi ’ f'1' ’ Pi ’ fij] ¢n (0 $111 0)) ' (17) The indices i and j refer to different electrons, i. e. i and j can never refer to the same electron in the calculation of a single matrix element. The two electron operators are used for the calculation of Coulomb and exchange energies. Occupation Number Formalism When a basis set is used to describe the multi-electron wavefunction of a system, the wavefirnction for a given state is written in terms of whether a particular basis function is used or not. For clarity of presentation, the one-electron spin-dependent hydrogenic wavefimctions are chosen as an example of a basis set. This basis set is {15“, 185, 28,1, 25,3, 2pm, 2pm, 2pm, , }. The multi—electron wavefunction is derived from antisymmetrized products of these functions. In the occupation number formalism,“9 the wavefunction is constructed by counting the number of electrons in each state, e. g. W) = Inlsa 11155112311 nzsp' ' °nn1me nulmfi) 9 (18) where the r1“1m denotes the electrons with the basis function nlma, where nlmu are the four quantum numbers needed to fully describe the hydrogenic single-electron state. Thus the Hartree-Fock ground state of the helium atom is rewritten in the occupation number formalism as 1,1,0,0,0,---,0). (19) Ilsa 159) = The ones in the right side of Equation (19) indicate that the lscl and 1s,3 basis functions are occupied while the zeroes indicate that none of other basis functions are occupied. An excited configuration of helium can be rewritten as '15:: 25a) = I 1909190903. ' ' 90> - (20) A sum over a complete set of occupation numbers spans a complete multi-eliectron space; therefore, the completeness relation for multi-electron wavefirnctions can be written as Zii> = a*(lsu) a'(1ss)"'a'(nlms)l ®)- (22) Further properties involving the importance of ordering of creation and annihilation operators in multi-electron states are found in Appendix C. 45 Foundations of the Polarization Propagator The polarization propagator has its origins in the time correlation function of the density operator. The density operator is a statistical description of the state of an ensemble of quantum systems. One text describes the density operator as an averaging operator of single quantum systems over an ensemble.4 A density operator can represent a mixed state, which does not have a specified wavefunction that can be constructed from a basis set. Therefore, to find numerical results involving a large number of quantum systems, the density operator is used. In conventional formalism, the N-electron density operator is expressed as o(t)=|‘P(t)><‘P(t)|- (23) For calculations with single-electron operators, the N—electron density Operator contains an overabundance of information. Therefore, the simpler single-electron density operator is used. In the language of second quantization, the single-particle density operator can be expressed as 963) = Z¢:(f)¢,-(f)a!(t)a,- (t) where the 6,, 4),- are basis functions. (24) The relationship between the densities at two different times is found in the density correlation firnction constructed by Zubarev10 and considered by others"'”. Correlation functionsl3 describe how one quantity changes in response to a perturbation that couples to a second quantity. As a simple example, consider that an arbitrary charge distribution will deform in the presence of a time-dependent external electric field. The density time correlation firnction relates the charge distribution at one instant of time to the charge 46 distribution at another instant of time. This charge density correlation function may be written using the Heaviside step fimction“ as so)=—ie(t-t'){-}- (25> Since the correlation function is written using the Heaviside step function, a charge distribution in the future never has an effect on a charge distribution in the past. Thus causality is ensured since the effects of a charge distribution propagate into the firture, not into the past. The Fourier transform of the time-dependent correlation firnction is the energy- dependent density correlation function 1,905)” E =lim x°°( ) Z E—hm.o+ie E+hw.o-i8 Johanna) shaman} where hm,,=E,-E,, (26) In terms of the points f and f' , the time-dependent density correlation function is rewritten as f,f";E = lim , , pr( ) Z E-hw.o + re E+hm.o -18 {WWI n)(nlp(f')|0> (OWN n> The energy-dependent density correlation function can be rewritten in terms of the polarization propagator by substituting Equation (24), the definition of the density operator, into Equation (27). r(0l¢§(f)¢,-(i)ai(t)aj(t)|n)(n|¢l(f')¢.(?')al(t)an(0W) ‘ pr(f,f';E)= Z lim21 E—hm.o+18 ((29) 1.18.1 “01-4) (0| 4); (f') d), (f') a}. (t) a. (0i n)(n| 4): (f) 9,“) £0) 31' (0i 0) , E+hco..o - i8 J Rearranging Equation (29) yields the energy-dependent density correlation function in terms of basis functions and the polarization propagator x..(f,f':E) = £ng 9:6) ¢,-(r) 92 (PM. (r') x}: (E), (30) where x: (E) is defined as the polarization propagator. . o t . t o o t J .0 . m): {< la.a.|n> (larallanlaraJl >}=,;(.,). (31) no E—hcono +18 E+hcono - 18 Note E=hco is substituted to give the fi'equency-dependent polarization propagator. The imaginary infinitesimal portion of the propagator shall be dropped since only off-resonant response will be considered. The reduced resolvent Operator R(co) is defined to further simplify the expression. R(co) = (4,0, who, )"(1—|0)(o|) where 1: z|n)(n|. (32) Therefore, the polarization propagator becomes 48 X216”) = ’(OI alarm-(0)31 aJIO) ‘ (0| 31': a. R((0)aia,-I 0) - (33) Defining a}‘ as 3? = a; a; (the single-electron replacement operator) modifies the polarization propagator to become x226»)=-—. (34) The polarization propagator is rewritten as x226»)=—— (35) or n:;::(m) = —(o|a: R(m)a::|o) - (clay R(-m)a:| o) . (36) It is also possible to include the damping of excited states in the definition of the polarization propagator; however, such off-resonance damping is assumed negligible. The Nonlinear Polarization Propagator The polarization propagator is a general linear-response function meaning that it can be applied to any situation where the state density responds linearly to an applied perturbation. Specific examples include the polarizability as the linear-response function measuring the response to an applied electric field, and the paramagnetic susceptibility measuring the response to an applied magnetic field. When linear-response firnctions are inadequate to describe the response, nonlinear-response functions are used in addition. Nonlinear-response fimctions have been constructed by Olsen and J PJrgensen16 and have been reinterpreted using second-quantized operators by Hettema et. a1.17 and Moszynski et. al.18 in terms of nonlinear polarization propagators. The nonlinear polarization propagator has the form 49 l (OlatlanKafii — p.21) m> (mm... - he. - hsz—hmo. - hon) (0|a23ln) (3 7) (-l"l0)oh + hmzx-hmm + 71031 + hmz) (0| ah I n)(nl(afi - 9:) mel a3: I 0) \ (-h(Don + hmrxr-hwon - from) 11:37:22: ((01 9032) = (1+P12) 20‘ where the permutator P12 permutes the following variables (019032 P12 = ’9' ('9 )9" (38) K’HK" and the operator of is defined as p: = (0| afiIO) . The nonlinear polarization propagator expressed in terms of the reduced resolvent operator becomes (one: R(-03. - e.)(a:: - p::)R(-e.)a::|o> +‘ H:::::::(0)1,C02) = (1+P12)4 (0| at: R(w2)(a§3 - 9:3)R(m. + m2) a:|0) + “(0| at: 8(a).)(s: - p:)R(-e.)a:: | o) (39) V I J The propagator that is Equation (3 9) is used to determine nonlinear molecular properties such as the hyperpolarizability. The Derivative of the Polarization Propagator The structure of the polarization propagator in the form of Equation (29) shows that for contour integration with respect to energy, its poles yield excitation energies of a molecular system while its residues yield transition matrix elements. Thus, the values of molecular properties can be found not only by applying the polarization propagator but also by finding its poles and residues. 50 The polarization propagator has been used to calculate a variety of molecular properties. Oddershede12 mentions in his review article the following properties: oscillator strengths, Rayleigh scattering cross sections, photoionization cross sections, excitation energies, radiative lifetimes, static and dynamic polarizabilities, hyperpolarizabilities, dipole moment derivatives, potential energy curves, nuclear spin- spin coupling constants, nuclear magnetic shielding constants, magnetic susceptibilities, Verdet constants, spin-rotation constants, magnetic rotatory strengths, force constants and CS van der Waals coefficients. Most of these properties are calculated as single-point calculations at the equilibrium geometry of the molecule. However, in many cases, it is important to know the value of a property at several different molecular geometries about the equilibrium geometry. Also nuclear-coordinate derivatives of the properties are often calculated using numerical differentiation techniques.” I l . E . The derivatives of magnetic properties with respect to a parameter have been examined, all using numerical methods. Several studies have examined the relationship of various orders of spin-spin coupling constants to changes in molecular geometry. 1J (H, C) and 2J (H, H) surfaces have been calculated for methane and perdeuteromethane using 49 distinct geometries.20 Lazzeretti, Zanasi and Raynes also created surfaces from the contributions to the spin-spin coupling constants: Fermi contact terms, spin-dipole terms and orbital paramagnetic terms. The surface construction of 'J (H, C) and 2J (H, H) for methane has been repeated more recently using 51 distinct geometries.21 Calculations of 2J(H, H) have been done for CH4, SiH4, GeH4 and SnH4 at multiple geometries.22 The 51 calculations necessitated finding the normal-coordinate derivatives of the spin-spin coupling surface so that vibrational averages could be calculated. 1J (H, C) and 1J (H, N) have been computed for HCN and HNC at the equilibrium bond (distance r,, and at re 5: 0.1A.23 Third-order, i. e. vicinal, spin-spin coupling constants 3J (H, F) in substituted fluoroethanes have been determined as a function of the torsion angle between the hydrogen and fluorine atoms.24 The through-space spin-spin coupling constants TSJ(P, P) and TSJ (Se, Se) in diphospho-methanes and diseleno-methanes have been calculated as a function of torsion angle using a simplified polarization propagator technique.” (The through-space contribution accounts only for that which is due to overlap of the lone pairs of the phosphorus or selenium atoms.) Spin-spin coupling constants have also been computed as a function of hydrogen-bond distance.26 Other magnetic properties have been calculated as a function as internuclear distance. The magnetizability and 13C nuclear magnetic shielding surfaces for methanehave been determined using 59 distinct geometries.27 The spin-rotation constant has been computed in GaH28 and AlI-I29 as a function of internuclear distance. The nuclear magnetic shielding constant and spin-rotation constant of various isotopomers of second row hydrides have been calculated as function of internuclear distance.30 Surfaces of the nuclear magnetic shielding of 17O and H, the spin-rotation constant of 17O and the rotational g—tensor in the oxonium ion H3O+ versus normal vibrational coordinates have been constructed.31 Vibrational averages for Verdet constants have been found for N2, H2, CO and HF .32 52 E] 'E . Electric properties that have been calculated using polarization propagators include Raman intensities, which depend on the normal-mode derivative of the polarizability, for CO, N2, HCl and C12.33 The intensities were calculated using numerical differentiation techniques on polarizability calculations at three bond lengths. Potential energy curves of the ground state and various excited states of BH have been constructed.34 Vibrational averages of the ground state energy, dipole moment and different Cartesian components of the polarizability, cram) and organ), and hyperpolarizabilities, 0mm), 0mm) and 0mm), have all been calculated using polarization propagator techniques.” The hyperpolarizabilities were computed by applying the finite-field technique36 to calculations of the polarizability. Vibrational averages of the second hyperpolarizability y(co,o),0) of N2 were found using finite-field techniques, applied to the polarizability calculated with the polarization propagator. The vibrational contributions to the hyperpolarizability and second hyperpolarizability of linear polymethine dyes (push-pull polyenes) have been determined and have been used to examine the change in nonlinear optical properties versus bond-length alternation ('BLA).37 Additionally, polarizability surfaces for 12CH4 and 12CD4 have been constructed using 49 distinct geometries.” The effect of vibronic coupling in the K-shell x-ray spectra of ethylene has been recently examined via calculation of normal-mode potential energy derivatives as vibronic coupling constants.39 Oscillator strength sum rules of H2 have been computed at 21 geometries to find their internuclear coordinate dependences.4o 53 Uses of the Nonlinear Polarization Propagator The nonlinear polarization propagator has been used in quadratic-response function t11eory'6"7to calculate quadratic-response properties and linear-response properties of excited states. Quadratic-response functions have been used to calculate quantities that are dependent upon vibronic coupling such as vibronic coupling constants, phosphorescence lifetimes and forbidden dipole-transition strengths. The nonlinear propagator is used to calculate the mixing of singlet and triplet spin states due to spin- orbit coupling. This spin mixing allows spin-forbidden dipole transitions and phosphorescence. Spin-forbidden transitions and phosphorescence lifetimes have been calculated for formaldehyde.“42 The vibrational structure of ground-state excitation bands has been determined by use of vibronic coupling constants for H20, NH3, CH4,” ethylene,“ and pyrrole." The effect of vibronic coupling on the two-photon spectra of benzene“5 and pyramidine“7 for dipole-forbidden two-photon transitions has been computed. Spin-orbit effects on the Auger spectrum of water have been examined.48 Potential energy curves of the n* state of the cyclopropenyl cation C3H3+ have been calculated including the effects of vibronic coupling on the transition from the ground state to the 1t* state.49 Derivation of the Polarization Propagator Derivative Introduction The derivation relating the derivative of the polarization propagator to the first order nonlinear polarization propagator is suggested by the relationship found by Hunt et al.50'5' 54 pa ma Tn. can1L1: between the nuclear-coordinate derivative of the polarizability and the hyperpolarizability density, 6aw(f;i',m)/6R:‘ = 16mm, (r;r',m,r",0)z"T,, (rat—1"). (40) The polarizability density or“, (f; f',(o) can be calculated using the polarization propagator a...(r.r';w>=-[p(r).]:[p(r'>.]:metre) where [p(r>.]:=<>~lp(r).lx>, (41> whereas the hyperpolarizability density is calculated with the nonlinear polarization propagator B...(nf'.f";wuw")=[pat]:[p(r').]:[p(f"),]:11:22:20»aw"). (42> The matrix elements [p(f)m]:L = (1|p(f)a | K) may have a dependence on an arbitrary parameter and, as shown below, the replacement operators that comprise the propagators may also depend on the same parameter. Therefore, when the derivative of the polarization propagator is considered, the derivative must contain terms that differ from the nonlinear polarization propagator. These “extra” terms will be shown to cancel the terms from the derivatives of the standard matrix elements [p(f)a]: in Appendix D. The idea is emphasized that the extra terms arise in both the standard matrix elements and the propagators because the basis functions that comprise the basis set for the calculation are allowed to vary with respect to the arbitrary parameter 11. I l . ES‘ l-l 1!! f1 . The derivation for the derivative of the polarization propagator is accomplished by calculating and manipulating the derivative of the ground-state wavefunction, the 55 derivative of the reduced resolvent and the derivative of the Hamiltonian.‘2 Finding the polarization propagator derivative involves also involves the derivatives of the creation and annihilation operators. The dependence of the creation and annihilation operators on an arbitrary parameter, such as nuclear coordinate, can be demonstrated by examining the derivative of the ground state wavefunction in the language of second quantization. The simple case of a one-electron ground state is now considered. |0>=a3|®> (43> The derivative of the ground state is found to be52 9'9)— =-G(o)93£Io> (44) an 5n The reduced resolvent, G(O), can be expanded and simplified. 9'22“ _ _ -4 _ 51C 5'1 — (1 goXSC E0) (1 go)an|0> =-(sc-E.>“((-e)%lo> (45> =—R(O)%C-IO> A new definition of the reduced resolvent is applied to Equation (45) R(m) = (SC—E0 + hm)-'(1- so) (46) The derivative in the language of second quantization becomes a 0) 1 63C 1 l I a — — 21 an 0 R(O) ao aoI Q) (47) 63c ‘ 63C . . . . . where 3:]- = (K I El 7») are the matrix elements of the Hamrltonran denvatrve and a: = a1 a, is defined as the product of a creation and an annihilation operator. 56 This simple analogy between the ground state derivative in the second quantization formalism and the ground-state derivative in the standard formalism holds only for one electron systems. When two or more electrons comprise the ground state, the commutation relationship between the creation and annihilation operators becomes important and the simple analogy fails.’3 The antisymmetrical nature of the multi- electron wavefrmction is taken into account by using occupation number wavefunctions. Thus the differences between the derivatives of Hunt et. al.’°'Sl and this work lies in the nature of the polarization propagator which is defined in terms of single-electron basis states for a many-body system. Since the property is defined with a single-electron basis, the anti-commutation relationship between the single-electron states must be taken into account. I: . . E l E I . . E The polarizability density, in the language of second quantization, is expressed as’4 aw (irate) = —[p(f)a]:[p(f')fllr 11:32:61)) where [flan]: = (Mpfimx), (43) To take the derivative of the polarizability density with respect to an arbitrary parameter, we need to consider what elements of the polarizability density may have parameter dependence. In this treatment, the electronic coordinate is the only parameter that will not be considered. First the matrix elements [15(17):]: = (Mp(f)u|x> are examined. Both the (XI and the In) wavefunctions may have parameter dependence that is easily calculated in terms of sums-over-states with single-electron states. 57 All of the elements of the polarization propagator may have parameter dependence. The derivative of the polarization propagator can be written as _;_a“¢¢(‘°)=- °' *R(w)10-|> (0| aa,,;R(m>:Io>- Pie—4611“” |°> 6n 5'1 a) a or Vim-fl" -O) 1 - anaa" —m a1 (49) (new) 'l0)-<0| :R()a: an mam >a..I(>> ). ea;_ .1, _,,.§I__o>an <0Ia: an (RIO)- (Ola: R( )anl0> (Oh. R(— )a The derivative with respect to any parameter 1] of the ground-state wavefunction and the derivative with respect to any parameter 11 of the reduced resolvent can be shown to be52 .5102 _ _ Bic (9n — R(O) 5n IO) (50) 9%(‘192 = -—R(co) figmm + R(m)R(O) (3% go + 50 %R(O)R(w) (51) where p = IOXOI. Upon substitution, the polarization propagator derivative becomes alIfifiKm) A 63., —5n= +(0l— an SCR(O) as R(@) a.‘ l-O) (O 5n + (OlatR(o>) d—anT-JRRO) 21 IO) (Ola: R(m)R(0)%n-Joae |0> - (Olate %R(O)R(:) a 1—|0> (ma: ‘2‘" + (0| aéR(o>) a5 R(O) —|0> + (OlgR(0) a5 R(-— (o) at|0> (52) an an —+<0lat3R(—(o)(Xi—mm R(— e) - - A3|0) allo) > +<0Ia:1R(— e>a:R(o>%Io> — (OI a5 R(- 58 h)...“— Terms (1, 5), (4, 7), (8, 12) and (l l, 14) in Equation (52) are combined using the definition p: = (OIafiIO). As an example, terms 1 and 5 are combined. 53C . 63C . (0| 73? R(O) a: R(m) art-Io) — (01 at (o ER(0)R(6I)) art-Io) = - (Olatl0) = (olagnSERw) a: R(m)a:1l0) — (OI%R(0)(OIa§IO)R(w)a§:I0) = (OI%R(0)(at - p:)R(e) a::Io> Completing the combination of terms (1, 5), (4, 7), (8, 12) and (11, 14) yields (53) 6112.500) an if: R((o)at1|0> + (OlaiR(m)2(%-1i°) R(w) 215 I 0) — (0| a: R((n) = +<0I%R(0)(ai - 6311(0)) afi1|0> - (oI aati an +(OlatR(m)(at3—piI)R(0)%SCIO>+(0|%R(0)(at1—923)R(- m)a.’;l0> (54> (:3:R(-o>)a:|0>+ if Io> + (Ola:1R(- e)(a: - Pi)R(0)%SCIO> IO) -(0| - (Olafi R(- (0) We limit consideration to the set of parameters 11 such that aSC/an can be written as a sum of one-electron operators ah/an , so that the Hamiltonian derivatives become 2%: ET" A: 55 an Ian 2was ( ) and a(EEO-BO) Iahon” r» A" _ = _. .~- .~ 56 an an ”(a P I ( ) 59 The expression [@I is a multiplicative term that relates the nonlinear polarization propagator to the quadratic response function of a specific molecular property. After substitution of Equations (55) and (56) into Equation (54), the polarization propagator derivative becomes antzfilw) = [fl " an x ((0Ia::R(0>(a:-0:)R(0>a::I+ 0+) (2:0Ia R(0)(a: -0. “0)R(0)a I0 +(OIa: R(‘Dxaii ‘ 9911(0) afiZI I0)+ (OI a..- R(0)(ae4 - Pr IR(-(a.: - p10)R(-0>a I0>+ <0Ia0=R( (—0>(a:- 0::)R(0>a =>>I0 -<0I3fR(0>a:rI0>—<0Ia:R(0>—“—'I0> —(0I7;I‘R(4>>a:I0>-(0Ia:: R( an)- The first six terms of the polarization propagator derivative have a structure analogous to the hyperpolarizability’5 (3.3. 65304930) = (0| 0.. (f)G(00)P3(f")G(00)Pa(f')l0) +(0|Ps(f')G(-0>)P$(f")G(-0>)Pa(f)l0) +(0lPa(f)G(w)P8(f')G(0)Pv(f")|0) (58) +(OIPI(f")G(0)P8(f')G(-00)Pa(f)I0> + (OIP.(f")G(0)P3(f)G(w)Ps(f')|0> +(0|Ps(f')G(-0>)P3(T)G(0)P4(f")|0>. In fact, the hyperpolarizability can be expressed in terms of the nonlinear polarization propagator. Bap,('r3?',f";w',w")=IP(f),]:IP(').,:IIP(r"),:HI11:23:1100. (59) 60 However, Equation (5 7) contains more than just the nonlinear polarization propagator. The terms which depend on the derivatives of the replacement operator must also be considered. 1: . . E] E l 3 To consider the derivative of the replacement operator at = a}, a: , the operator is transformed into the occupation number basis set. First, two different complete sum- over-single-electron-states are inserted, one preceding the operator and one following it?3 a:=a;a‘= z 2 In'ln'2”.n'1".n'xn.> {nk} in'kl x (11,111,2"11'1"'n'x"'laxaxInlnf'qh.”'nx”°> k X (n'ln'2."n'l...n'x'..|nln2...nl+1...nK _1...> Ink} {n'k} (61) x 5D,”)! 5n'2-n2" '5n'pnx-l ”511,101+!" .(m n; ...m.. .m .. .I_ Summation over the primed occupation numbers yields 3:: {z}(_l)sl~sx50,n151’nxIn]mum +1---n.. _1...) a=7_§? : . (64) 0.0.) mm) --3 I...) 62 The multi-electron wavefunction is a sum of products of single-electron wavefunctions. Therefore, the derivative of the multi-electron wavefunction is a sum of sums of derivatives of single-electron wavefunctions. (The first sum is represented by the determinant notation.) WI“) W002) d111213"'1NON+lON-+2"'>_ 1 W203) W202) ''''' an Jig 5 E 0.0.) 0.0.) ----- 3 WI (1") W102) """ W101“) W261) W202) 6‘45“”) +_1_ 5.” a." 5." J1?! : : : WN (1") WM (r2) """" WN (TN) WI (1'!) WI (r2) """" W10") Mn) w2(r2) '''''' wz(r~) 1 5 5 5 “LT/ii s s s WNGI) aW~(r2) 601,6”) 6n 5'1 0n 6W1 (W) W2 (In) WN .(TN) (65) The derivatives of the single electron wavefunctions in Equation (65) can be expressed as a sum-over-states. 6M an and: 63 = Cka‘“) (66) The prime on the summation of Equation (66) indicates exclusion of the Ik) function. Substitution of Equation (66) into Equation (65) yields III “’2‘”? “I“? dIIIZIBH'lNONHONQ'”) 1 W2.“ “12.1.2 “12:1“ 00 0R5“ : : ' . : WN (r1) WN (r2) """" WN (TN) “’1 (1'1) “’1 (1'2) """" WI (”1) 1 (”IA“) Wm(f2) ...... Wm(TN) +7.01%.sz E f f WNII'I) WN'(1'2) ' ° ’ ' ' WN .(YN) Mn) w.(r2) ------ w.(r~) 1 Wzlrn) W2.(r2) °°°° Wzng) +fim§NCNm : : '. w...(n) WI”) ..: Walk”) (67) Note that if there is a determinant where the Im) functions are included in the set of single electron functions used to construct the multi-electron configuration, i. e. Im) e {Ip}: p = 1,2- - -N} , then that determinant is zero. In addition, no assumptions have been placed upon the determinants; they may be ground or excited configurations. Before converting Equation (67) back into an occupation number representation, the Im) fimctions in the determinant must be put into standard order. The ordering is done by exchanging the Im) wavefunction row with as other rows as necessary. Each exchange of rows introduces a sign change, giving an overall factor that can be symbolized as (—1)S""Sk . The quantity Sm-Sk indicates the number of row exchanges. Equation (67) is 64 written as a double sum, the k sum indicating the sum of determinants and the m sum indicating the sum of single electron states. W1 (In) W: (r2) """" W1 (In) 611.1213---1;:N.10N.2~->= % §m§k(_1)sm—skcm w..(r.) Wm(r2) w...(r~) . (68) Wn (r1) Wu (f2) ”' '” WN(1’N) In the occupation number formalism, Equation (68) becomes a1I1213”'1NON+lON+2°°') an = {I 2‘.k(-1)S“'"S"50,um 61% len1 mmnr -1'"nm +1"-nn) (69) 2"E1El E II" Equation (69) is the derivative of the multi-electron wavefunction. Thus Equation (69) is substituted into Equation (63) to continue calculating the derivative of the replacement operator. aa: — - an = {2 } E: m§k (__1)Sm Sk (_1)S2L S‘ 50’”; 61'”: 80mm 51,1“ m; (70) x{ Chnlnln2"'nk‘1"'n1+1”'nx '1'”nn+1‘”nN>— <78) (0 0'6; R('C°)a: - mg. Cr... (0 I a: R(CD) a?" I 0) - kg CL: (0 I 3?; R((D) a? I 0) - ”231 CA’m (Olaf? R(-0)) a: I 0) - :33" Ch (0 I at’ R(-CO) at I 0) ‘ EA'Cm(OIafiIR(—w)a?|O)—k§ CHIOIaizm—(DMHOI- (79) The terms are rearranged and the definition of the polarization propagator, Equation (36) is applied to yield the final result. 67 1.x " an.,..(m) = [2112] n:;::;::(m,o) an an . - 21cm<0larR(w)aé1|0)- 21Cm<0|ai3R(-‘°)33I°> — ZA’C,.m(OIa§R(a))a2‘oIO)- EA'me(0Ia.'§‘IR(-O))aélo) —k2'C;m(0IafiR(m)at'I0)-kZ,C2r(0Iai'R(-°3)33I0> _ k2 c;.(0lat R(m) a: | 0) - kr: Cir (Ola? RH”) filo) 5110 It" 1. 1' A" = — nx:x':x" ((0,0) I an .. + EXCmH:;%’((D)+ zlcxmnmm) (80) + k2‘. ' C21. Ht? ((0) + k2 Cid: 11:33:10)) - Discussion The result in Equation (80) demonstrates a new relationship between response functions. The equation of motion for the linear-response function, Equation (2), shows a relationship between linear and quadratic-response functions; however, the relationship is not a derivative relationship. (Parkinson56 has used this relationship to calculate the dipole polarizability of H20 with the quadratic-response frmction. Since the polarizability is more easily calculated using the linear-response function, using the quadratic-response function is not advantageous.) This work demonstrates that the calculation of molecular properties from energy derivatives with respect to an electric or magnetic field such as hyperpolarizabilities or hypermagnetizabilities can be calculated without finite-field techniques. Only response functions are used. Since the relationship in Equation (80) is general, the equation may 68 suggest an efficient method for the calculation of the parameter dependence of electromagnetic properties allowing for frequency dependence. The chief advantage of Equation (80) is in the calculation of derivatives of linear- response properties via the nonlinear polarization propagator. The calculation of energy derivatives with respect to nuclear-coordinate molecular gradients and energy second derivatives with respect to nuclear-coordinate molecular Hessians is essential in the calculation of molecular structure?7 The derivatives are also important in the calculation of vibrational energies via harmonic and anharmonic force constants. Much effort has been used to find efficient methods to calculate these quantities.""'59 Though the calculation of the derivative of the polarization propagator via calculation of the second-quantized one-electron replacement operator appears to be novel, the calculation of derivatives of individual creation and annihilation operators is not. The derivative of the creation operator appears first in the paper by Bak et 01.“? where the authors calculated first-order nonadiabatic coupling matrix elements necessary for accurate accountings of phenomena such as A-doubling61 and spin-orbit coupling. The theory was also applied in the calculation of atomic polar and axial tensors of Stephens62 in work on the rotational strengths necessary for describing vibrational circular dichroism."3 The result of this chapter differs from the work of Bak et. al. in that the derivative of the number-preserving replacement operator has been found rather than the derivative of the number-changing creation and annihilation operators and that the result of this chapter is expressed in terms of polarization propagators rather than molecular gradients. 69 The result in Equation (80) was suggested by the relationship found by Hunt et (11.50'51 relating the derivative of polarizability to the hyperpolarizability density. In published work“ and Chapter 2 of this thesis, a similar relationship has been found between the derivative of the hyperpolarizability and the second hyperpolarizability density. This relationship suggests a relationship between the derivative of the quadratic polarization propagator and the cubic polarization propagator. Such a relationship would be useful as cubic polarization propagators constructed for the random phase approximation are already being used to find various cubic electric response tensors such as those responsible for third harmonic generation, DC-electric field induced second harmonic generation, degenerate four-wave mixing, etc.“ The cubic propagators" and mixed analytical-numerical techniques“ have also been used in the calculation of hypermagnetizabilities which are responsible for magnetic field induced birefiingence or the Cotton-Mouton effect, a magnetic analogue of the Kerr Effect. In their review, Rizzo, Rizzo and Bishop67 mention that the calculation of the vibrational corrections to nonlinear properties, such as the hypennagnetizability, remains a nontrivial problem. The extension of the results of this chapter to the next order could aid in the calculation of such corrections. 7O REFERENCES ‘ Peter W. Atkins, Molecular Quantum Mechanics, 2nd. ed. (Oxford University Press, Oxford, 1983). 2 Roy McWeeny, Methods of Molecular Quantum Mechanics, 2nd ed. (Academic Press, New York, 1981). 3 Poul Jorgensen and Jack Sirnons, Second Quantization-Based Methods in Quantum Chemistry (Academic Press, New York, 1981). ‘ Claude Cohen-Tannoudji, Bernard Diu and Franck Laloé, Quantum Mechanics, Vol. I (John Wiley and Sons, New York, 1977). 5 Alexander L. Fetter and John Dirk Walecka, Quantum Theory of Many-Particle Systems (McGraw-Hill Publishing Co, New York, 1971). 6 Richard D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, 2nd. ed. (Dover Publications, Inc., New York, 1992). I 7 Michio Kaku, Quantum Field Theory (Oxford University Press, Oxford, 1993). 8 Michael E. Peskin and Daniel V. 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Phys. 90, 4909 (1989). 5' K. L. C. Hunt, Y. Q. Liang, R. Nimalakirthi, and R. A. Harris, J. Chem. Phys. 91, 5251 (1989). 52 Appendix A. 53 Appendix C. 5‘ x. Li and K. L. C. Hunt, J. Chem. Phys. 105, 4076 (1996). 55 B. J. Orr and J. F. Ward, Molec. Phys. 33, 513 (1971). 56 William A. Parkinson, Int. J. Quant. Chem. Symp. 26, 487 (1992). 74 57 Attila Szabo and Neil S. Ostlund, Modern Quantum Chemistry (McGraw-Hill, New York, 1992). 58 Geometrical Derivatives of Energy Surfaces and Molecular. Properties P. Jergensen and J. Sirnons, eds. NATO ASI Series C: Vol. 166 (D. Reidel Pub. Co., Dordrecht, 1985). 59 H. B. Schlegel, Adv. Chem. Phys. 67, 249 (1987). 6° K. L. Bak, P. Jorgensen, H. J. A. Jensen, J. Olsen, and T. Helgaker, J. Chem. Phys. 97, 7573 (1992). 6' Jack D. Graybeal, Molecular Spectroscopy, lst Rev. ed. (McGraw-Hill, New York, 1988). 62 P. J. Stephens and M. A. Lowe, Annu. Rev. Phys. Chem. 36, 213 (1985). ‘3 K. L. Bak, P. Jorgensen, H. J. A. Jensen, J. Olsen, and T. Helgaker, J. Chem. Phys. 98, 8873(1993) ' 64 E. L. Tisko, X. Li and K. L. C. Hunt, J. Chem. Phys. 103, 6873 (1995). ‘5 P. Norman, D. Jonsson, O. Vahtras and H. Agren, Chem. Phys. 203, 23 (1996). 6" S. Coriani, A. Rizzo, K. Ruud and T. Helgaker, Chem. Phys. 216, 53 (1997). 67 C. Rizzo, A. Rizzo and D. M. Bishop, Int. Rev. Phys. Chem. 16, 81 (1997). 75 CHAPTER 4: FIRST ORDER APPROXIMATION TO THE ELECTRONIC MAGNETIC MOMENT DERIVATIVE Introduction Intramolecular response to internal electromagnetic fields has been described by the same electromagnetic response tensors that describe intramolecular response to external electromagnetic fields. The first example of this equivalence was derived by Hunt1 who showed that the derivative of the electronic dipole moment with respect to nuclear coordinate is connected to the nonlocal polarizability density. an,3 /6R§ =Ididf’orm,(f;f',0)ZK Tm(f’,RK). (1) The physical interpretation of this connection can be discerned by examining the balance of electric fields in the molecule at its equilibrium geometry. The electric field at the nucleus in a molecule at equilibrium is zero since the electric field from the other positive nuclei in the molecule must balance the electric field from the negative electronic charge distribution. When a nucleus is perturbed away from equilibrium, the Coulomb field changes throughout the molecule. The electronic charge distribution responds to this change in the electric field via the nonlocal polarizability density. This response is weighted by a distance relationship between the position of the nucleus and the point in the electronic charge distribution where the change in the nuclear Coulomb field is computed. This distance relationship, known as the dipole propagator, is defined as 76 1 J = 3(fa‘r'axrb“r'BI’5aBIf'f'I2 _115 _ _ 6 5f—f'. (2) |r_rr| lf-f's 3 “P ( ) Tab (f,f’) = Va Vfl[ The nonlocal polarizability density that characterizes the intramolecular response is the same tensor used to describe the electronic molecular response to an external electric field. Other such relationships have been found and will be discussed, subsequently. ll l lS 1.1.]: .. The distinction between the polarizability and nonlocal polarizability density should be clarified. The nonlocal polarizability density yields the response of the molecule at a single point due to an applied field at another point. When an electric field interacts with a point of charge distribution, the charge distribution at the field point becomes polarized. This polarization field in turn polarizes the charge distribution at the response point. To find the total response at a single point in the molecule, the effect of the field at all field points in the molecules must be summed. To calculate the total collective response, i. e. the total polarizability, from all points, all the response points must be summed. Thus, the relationship between the polarizability and nonlocal polarizability density is that the polarizability is equal to the nonlocal polarizability density integrated over all field and response points. (1043(0)) = Idfdf'aagfij'xo). (3) 77 IntramoleculaLElectriLResnonse The intramolecular responses of the polarizability’ and hyperpolarizability3 due to an applied electric field are related to external response tensors. . c301,,y (0))/5R: = J'dfdf'df" B,.,(r;r',m,r",0)z" T50(f",R"). (4) 60am (—co.,;(0',(0")/ 6R: _ (5) = Idfdfrdfrrdfrrry any: (T;T',(D r, 110,0) n, frrr,0)zK T55 (fur, RK ). Equation (4) shows that the electronic polarizability responds to an internal electric field via the nonlocal hyperpolarizability density, whereas Equation (5) demonstrates that the hyperpolarizability responds via the second hyperpolarizability density. IntramoleculatMagnetiLReannse Derivatives of the electronic magnetic moment with respect to nuclear linear momentum have been related to nonlocal charge-current susceptibility densities.4 I drdr' Im<0| p(r')G((D)G(- co)[fxi(f)L|0>V§|1—,—xl—— -‘f’l %_ hZKe apg‘ MKc (6) In Equation (6), G((o) represents the reduced resolvent from standard perturbation theory. G(m) =(1-geX8c-Eo-hm)’1(1-p),where go =|0)(0|, (7) ZK is the charge and MK is the mass on nucleus K while VI,‘ is the gradient Operator with respect to the nuclear coordinate R“. p? represents the [3th Cartesian coordinate of the linear momentum of nucleus K while m3 is the 01th Cartesian coordinate of the electronic magnetic moment. Equation (6) demonstrates that the change of the electronic magnetic 78 moment due to an internal electric field is related to a charge-current susceptibility density, x2“ = (0|p(r')G(0J)G(—m)[ij(f)L | O). This susceptibility tensor is the nonlocal density analog of the rotational strength, the quantity calculated to find the intensities of the transitions associated with vibrational circular dichroism, VCD. VCD5’6'7 is the phenomenon when light of one circular polarization has a different degree of vibrational absorption than light of the other circular polarization. The effect occurs only in chiral molecules or molecules with chiral crystal symmetry. Heuristically, VCD can be understood by considering the electric field of the nuclei during a vibration. As the nuclei move, a time-dependent electric field is produced. The electric field produces a time-dependent deformation of the charge density that induces a magnetic moment within the molecule. The intensity of the absorption is determined by a quantity called the rotational strength that couples the electronic electric-dipole transition matrix element to electronic magnetic-dipole transition matrix element. The charge-current susceptibility x3” used in the theory of VCD relates a change in the electronic magnetic moment at one point in a molecule to the change in the polarization due to an applied electric field at another point in the molecule. Thus, the derivative of the electronic magnetic moment with respect to nuclear linear momentum can be calculated from a change in the current density due to electric field perturbations. The work in this chapter also calculates the derivative of the electronic magnetic moment with respect to nuclear linear momentum. However, the result differs because the susceptibility density used to calculate the response is different. The response of the 79 magnetic moment due to an applied magnetic field is considered, rather than an applied electric field. In this chapter, the magnetic moment derivative with respect to nuclear linear momentum is related to the paramagnetic nonlocal chemical shift density. The nonlocal chemical shift density describes how a magnetic field at one point in the molecule affects a magnetic moment at another point in the molecule. Magnetic moments are created in the molecule when magnetic fields induce the charge density to circulate. This induction of circulation is termed the magnetization. In macroscopic terms, the magnetization is defined as an average of magnetic dipole moments just as the polarization of a molecule is defined as an average of electric dipole moments.8 One can understand a magnetic dipole as a loop of current in the same way that an electric dipole is pictured as two oppositely charged particles separated at a distance. Molecular Electromagnetism Eackmund ‘°’” provide an introduction to molecular Many texts9 and monographs electromagnetism, i. e., the interaction of molecules with electric and magnetic fields. Molecules interact with magnetic fields differently than they do with electric fields. Viewed figuratively, the electronic charge distribution of a molecule ‘stretches’ when perturbed by an electric field. When a magnetic field is applied to a molecule, the electronic charge distribution becomes ‘twisted’. The twisting of the electronic distribution produces two different effects. Diarnagnetism is produced when the twisting 80 causes circulation of the electrons in currents. These induced currents circulate as to decrease the total magnetic field. Pararnagnetism may be produced when the applied magnetic field torques the electrons that cause their magnetic moments to align with the magnetic field. The alignment of the magnetic moments increases the total magnetic field. In molecules without unpaired electrons or net orbital angular momentum, the response to an applied magnetic field is diarnagnetic. However, the description of the response is inherently quantum mechanical and has an unequivocal dependence upon an arbitrary function named the gauge whose value changes the quantum mechanical description of the magnetic response but not its actual value. The next section discusses the basic theory of the gauge function and its relationship to the vector potential and magnetic fields. I! E .13 E . lll 'E'll The simplest magnetic object found in nature is a magnetic dipole. This contrasts with the electric case where the simplest particle is a monopole. Because magnetic monopoles do not exist, the magnetic field can not emanate from a point source, thus the divergence of a magnetic field is always zero. V - fi = 0. (8) When the divergence of a vector is zero, the vector can be described as the curl of a second vector.“13 In the case of a magnetic field, this vector is named the vector potential. B=VXA. (9) 81 The curl of a gradient of a scalar function is always zero. Therefore, the vector potential can be parameterized with the gradient of a scalar function. §=VX(X'+V}\.), where X=A'+V}.. (10) This scalar function A is known as a gauge function. The gauge function does not affect measurable quantities but may ease their calculation. 5 1.1” 'E'll “11.1 . The application of a magnetic field to a molecule changes the Lagrangian of the molecule. If the nuclei are fixed and there are no spin interactions between the nuclei and the electrons, the Lagrangian9 of a single electron within a molecule with applied electric and magnetic fields in SI units“ is 1 ,2 , _ £=§m¢r +V+e¢—er-A (11) where 4) is the electric scalar potential and A is the magnetic vector potential. The canonical linear momentum associated with the Lagrangian is defined and subsequently calculated as 5.53 , pk=—ar—=m¢rk-eAk° (12) k The canonical linear momentum is seen to depend on the vector potential. Using this canonical momentum, the Hamiltonian of the molecule becomes SC=—1—(p+eX)2 +v—e¢ (13) 2m. 82 When p is replaced by p = -inV , the Hamiltonian after expansion becomes -h2 2 eh_ eh e2 2 =—-—V +——A V+—V A+—A +V— e0. (14) 2111. rm. 2imc 2m, For convenience in calculation, a gauge for the vector potential is often chosen such that V - X = 0. This gauge choice is referred to as the Coulomb gauge. After applying the Coulomb gauge and assuming that only magnetic fields are acting on the molecule, the Hamiltonian becomes eh EJC=—V2+——A V+—A2+V. (15) 2me 1m. 2m. The Hamiltonian can be examined in orders of the vector potential where 2 30°) =—V2+V, 3C“) =—A v and SC”) =2—e—A2. (16) im. me The two different magnetic effects mentioned earlier can be seen with the division of the Hamiltonian into first and second-order terms. The first-order term is the paramagnetic term. Its energy depends on the alignment of magnetic dipoles with the magnetic field thus increasing the total magnetic field away from the molecule. In molecules with zero spin angular momentum, the magnetic dipoles originate from the net orbital angular momentum of the molecule. The second-order term can be considered the diarnagnetic term where the applied magnetic field induces electronic currents in the molecule. These currents produce magnetic fields that oppose the applied magnetic field and thus decrease the total magnetic field away from the molecule. The distinction made between paramagnetic and diarnagnetic effects is not rigorous. While the first-order paramagnetic effects are independent of the gauge, the second-order 83 paramagnetic effects are dependent on the choice of the gauge. How the total second- order magnetic effects are described can be changed by adjusting the gauge of the vector potential. Because of choosing the proper gauge, the total second-order magnetic effects can be described using different combinations of first-order perturbation theory applied to SC“) in Equation (16) and second-order perturbation theory applied to SC“) in Equation (16). Therefore, division of magnetic effects in a molecule into diamagnetic and paramagnetic effects is illusory, but conventional. Adjusting the gauge will not affect the second-order energy of the system. In this work, only the first-order effects are considered, therefore the specific choice of gauge, within the general Coulomb gauge, is of minor importance. Construction of the Nonlocal Chemical Shift Tensor ll 'E'llEllll : A nonlocal magnetizability tensor will be constructed in this chapter via finding the derivative of the electronic magnetic moment with respect to nuclear linear momentum. The theory presented will consider only the zeroth and first-order parts of the Hamiltonian from Equation (16). The second-order contribution will not be included. To find the magnetic moment derivative, the expectation value of the magnetization operator is calculated. However, the expectation value is calculated with wavefunctions from first-order perturbation theory, where the perturbation is the first-order portion of the Hamiltonian from Equation (16). 84 When a charged particle moves rectilinearly relative to a fixed reference fiame, the particle produces a magnetic field.“ §(§)__)queRxV__p0ZeRxfi - 47: [if ’ 47rM lfif ' (17) In equation (17), no is the vacuum permeability associated with SI units, R is the vector distance from the particle, Ze is the particle’s charge, M is the particle’s mass and p’ is the particle’s mechanical linear momentum. When a nucleus inside a molecule moves, it creates a magnetic field. The electronic charge distribution changes in response to the magnetic field of the moving nucleus. The effect on the ground-state wavefunction is calculated by applying standard nondegenerate time-independent perturbation theory. E' _ 1 3 l S III E . The first-order correction to the ground state wavefunction from standard perturbation theory is found to be’ “1“) = 2, | i) 1 5.3-8. (18) where the III/i) are the unperturbed wavefunctions from the zeroth-order Hamiltonian, the Si are unperturbed energies and the prime on the summation indicates summation over all states except the ground state. When the first-order perturbation of Equation (16) is substituted, the ground-state perturbed wavefunction becomes (WiliXPI‘VoI 5045‘ IV.) (19) IMP>=Iwo>+2i' 85 At this point, the nature of the perturbation A - '13 needs to be considered. For a uniform applied magnetic field, the vector potential can be written as _ 1 _ A f = — B x f . . 20 0 ,1 > < > Substituting Equation (20) into the perturbation A - 1‘) yields X-p=-1—(§xr)-p. (21) LE, (22) where the electronic angular momentum operator has been defined as l = f x p. The electronic magnetic moment operator is defined as H = - :21— —e—l. Thus for a uniform m magnetic field, the first-order perturbation of Equation (16) can be written in terms of the electronic magnetic moment operator. 3C“) = -j.I-B (23) One can arrive at a similar result if, for the magnetic interaction between a nucleus and the electronic charge distribution, the perturbation Hamiltonian used is'5 3C“) = -I 3,. Xndr (24) In Equation (24), the e and n subscripts represent electronic and nuclear quantities, respectively. This form of the interaction Hamiltonian is written in terms of the electronic current density rather the current. The density formulation is vital to the construction of the nonlocal magnetizability density. The divergence of the electronic 86 current density is zero for steady currents, i. e. where the charge density is not permitted to fluctuate over time. Thus, the current density can be written in terms of the magnetization density: j,=Vx m.. I (25) Note that the magnetization density in, is written as a vector whereas the electron mass m, is written as a scalar. The magnetization in Equation (25) is more than only the magnetic dipole density, because it includes all the information about the electronic current distribution under the influence of an applied magnetic field. Substitution of Equation (25) into Equation (24) yields SIC“) = -I(me,)-X,dr. (26) The vector potential due to the linear momentum of the nucleus and the electronic magnetization density are finite for a finite molecule. Therefore, for a surface surrounding the molecule at an infinite boundary, the following integral is zero. -I (A. x m.) - d8. = 0. (27) By application of the divergence theorem, Equation (27) becomes ~jV-(anm,)dr=0. (28) Adding Equation (28) to Equation (26) and using the vector identity v-(a x E): B-(Vx §)-5-(Vx 5) yields SC“) = —IV .(A’, x ‘rfi,)dr -I(me,)-X,dr = —Im.-(v x X,)dr = —Im,-B.,dr (29) where BI, is the magnetic field due to the motion of the nucleus. 87 The interaction Hamiltonian of Equation (29) can be substituted into the expression for the first-order ground-state wavefunction of Equation (18) to yield ,161mm.e.(r)|y.>d3rwi>. 50 _ 51 (30) IT>=IW0>-Ei Substituting Equation (17), the expression for the magnetic field produced by a nucleus undergoing rectilinear motion, into Equation (30) yields era-(“r“) , pole MI W lwo)dr IWI=IWO>+WEX 50_5i IV.) (31) Equation (31) has the implicit assumption that the origin in the three-dimensional space considered is at the nucleus. This choice is arbitrary and differs from the choice of origin for the electronic coordinates which affects the gauge of the magnetic field and is often intentionally varied for ease of calculation.""7"8"9 This wavefimction is the new ground state wavefunction corrected to first order for magnetic fields produced by a moving zero- spin nucleus. El '1[ '11 I" The wavefimction found in Equation (31) is now used to calculate the expectation value of the electronic magnetic moment density at point f'. The expectation value is computed by calculating the ground state matrix element of the magnetization operator 88 (‘Plfiu(f')I‘P)=<\VoI‘n1(T')I\l/o> mm? x P") + Hole 2’ Id3f<‘Vi'Ifi_le(f')IWi> In) .2; (e.,—ere-..) ' The derivative of Equation (32) with respect to the nuclear linear momentum is is performed while also neglecting the fourth term. The electronic wavefunctions used are the complete adiabatic wavefunctions formulated by Nafie and Freedman,20 which only have parametric dependence on the nuclear momentum 1’)". Therefore, the only quantity in the expression for the electronic magnetic moment with functional nuclear linear momentum dependence is the quantity i x p“. As a lemma, the derivative of f x f5" with respect to the z-component nuclear linear momentum is calculated. f x PK 5 e e e 6( 5p: ) = apf [(YPZ‘ - 2p?) + (2P;< - XP§)J + (xp‘f - Ypf)k] (33) =fi-fi=—V 89 In Equation (33), the vortex firnction V has been defined9 as V = -yi + xj - Applying this result within the derivative of the electronic magnetic moment yields 6171.0”) ___ drlm.(r')I~P> _ 5P. 61?2 e V rm. - w>< Im—‘i—thr " “028 ' 34) 4n M i 50 _ 51 ( V W oIm°(:—_r——Z Iwi)(wilfi.(f')lwo)d3r _HoZez. II 4n M i 80 _ 5i Because the vortex function is a function of three dimensional space but not of electronic coordinates, the matrix elements in Equation (34) of the form (wiI—m° (r) . V ITIJ Iv/o) can be rewritten as ( “If? )lwo)V . Thus, Equation (34) becomes 6116') __lloZe 513, - 41rM ' _ _, ah) ah) _ _, ‘ (Woler )IW.)+(n|j,,(f')|0> (Ollb(f')|n> “”8““ F6730 5 -6 4m + 6 -6 +726 0 n 0 n (23) 1 --m—,§00(r)5(f-r')8,p. In Equation (23), the current density is defined quantrun mechanically as . - 1 Cl- _ _ h (l- _ _ h . <0l1.(r)|n>=§[<0|§:j (r-r.);v..Im>+l <24) and QM?) is defined as q? 6.6) = <0Iz—i.6(t—n)l0>- (25) J mj The index j indicates both electrons and nuclei with charge qj, mass m,- and position rj. Hunt demonstrated that the nonlocal polarizability density formulated in terms of 106 current densities could be reformulated in terms of polarization operators as long as the field incident upon the molecule could be derived from a scalar potential.’ _ <0|Pa(f)|n> (0|P(f')|n)(ana(f)|0> a“”(r’r"°)’2{ {go—6,422) + B6.—6,+rtm i (26) nato Since the electric field is derivable from a scalar potential, Hunt’s formulation of the nonlocal polarizability density is able to describe the molecular response to extemal9"° F and internal“12 electric fields. A complete theory of nonlocal susceptibility densities would describe electric, magnetic and mixed electric-magnetic response. 0'... One approach to extending nonlocal susceptibility density theory is to substitute into Equation (23), as an ansatz, classical 4-current densities. _ _ 1 (0|j,(f.t)|n> (olj.(r'.t')ln> “"43”“??? 2. 6 -6 + 6 -6 (D n: 0 n 0 n (27) — —(:—,§,,O(r,t)5(r — r',t - t')5,,. Substituting the 4-current densities does not make Equation (27) relativistically rigorous. Such rigor would need to be developed from the first principles of quantum electrodynarnics. This approach is discussed further in the next section. Since the zeroth component of the 4-current density is the electric charge density, the (zoo component of the “relativistic” nonlocal polarizability density of Equation (27) would include Hunt’s formulation of the nonlocal polarizability density plus additional terms associated with the time dependence of the charge density. The applications of the 4-polarizability density may follow from analogies with Hunt’s earlier work. However, such analogies would be indirect since the tensor calculus of relativistic electrodynarnics is different from the calculus of Coulombic electrostatics. Application of the four-dimensional 107 tensor calculus upon Equation (27) may uncover interesting and unknown relationships between the responses of electric and magnetic fields with molecules. Four-dimensional susceptibility tensors have been used by McLachlan to describe retardation effects in the theory of long-range intermolecular forces.“ }. (28) auv(f,f';(o)= n¢0{ 80-8n—hm + 80—6n+hm This form of the susceptibility density describes the response of the relativistic interaction perturbation V = - I 1,,(f) A" (f,t)df (29) where the current densities are assumed to be time independent. McLachlan’s susceptibility density was constructed from the generalized susceptibility discussed by Landau and Lifshitz." The susceptibility of Equation (28) is simpler than the susceptibility of Equation (27) since the 4-currents in Equation (28) are time-independent. However, McLachlan’s equation suggests the ansatz used in the creation of Equation (27) is not too unphysical. Relativistic Quantum Mechanics K] . _3 l E . As coordinates in the space-time of relativistic mechanics, space and time belong to the same class. This equivalence contrasts with 3-space where the x, y, and z coordinates belong to the same class but the time coordinate is different. The energies of most non- relativistic phenomena can be calculated with the Schrodinger equation. 108 2 i—az—‘P+V‘P=ih§-‘P (30) 2m 6x3. at The Schrodinger equation treats the spatial and time coordinates differently as it has a second order spatial derivative and a first order time derivative. A relativistic equation would have identical orders of derivatives for the time and spatial coordinates. The Klein-Gordon equation""17 is a relativistic equation that has second order time and space if ' I derivatives. E i h2[i2325-—@2—2--§23—323)W—m3c2\P=0. (31) c at 6x ay 62 The negative signs in the spatial derivative arise from the special relativistic tensor calculus needed to describe invariance of the space-time interval, Equation (5). Also, note that in Equation (31), m0 is the rest mass. (In special relativity, the total energy includes the kinetic energy and the potential energy of the rest mass, i. e., E2 = (p2 + m3 (9)8 or for a particle with no kinetic energy, E = mocz.) Analysis of the Klein-Gordon equation demonstrates that it is an inadequate description for particles with half-integral spin such as the electron but it provides an excellent description of exotic massive zero-spin particles of high-energy particle physics. E . E . Dirac formulated a relativistic wave equation based on first order derivatives 6 a a a th‘a \P = [—ihC(a13x’ + (1,2 ‘6? + a3 32') + Bmocz :lLP (32) 109 where the 0ti and B are coefficients. Dirac put the condition on the solutions of Equation (32) that they must satisfy the Klein-Gordon equation as well. For Equation (32), the Dirac equation, to have well defined solutions, the 0ti and B coefficients must be 4 x 4 ,. matrices. 0 0 0 1 0 0 0 —1 0 0 1 0 l 0 0 0 _0010 _00 0 _000-1B_0100(33) " a"0100’°‘”0-io “”1000 ’00-10 1000 1000 0-100 000-1 These matrices can be written as matrices of the Pauli matrices and the unit matrix i: an = . To make the Dirac equation truly Lorentz invariant, the Dirac matrices y” are used rather than the 0ti and [3 matrices. i" “l . - -(‘ ”l 35 71'" -O'i 0 an Yo-B- O -l - ( ) Using these matrices in the Dirac equation yields solutions that are vectors. These vectors are four-component spinors. To examine some of the consequences of having these four-component spinors as solutions, let us investigate the solutions for a fiee electron with a specified momentum. The four solutions to the Dirac equation are 110 ( 1 f O \ l LPr = CE ' P Calif—E!) , ‘Pz = 0 eilfi'i's‘) 9 mo c2 +E CE ' P \ O \mhc2 +E ) f 1 \ l 0 ) 0 1 ‘1’; = cE-P 6‘3““ and ‘114 = 0 6‘5“). (36) mo c2 -E CE ' P K 0 J \ moo2 —E ) ‘11, and KP; are called the positive energy solutions and ‘11, and \P, are the negative energy solutions. When Dirac first published his equation, the interpretation of a negative energy state for the electron was mystifying since the negative energy state of an electron is unobservable. With the discovery of antimatter, the negative energy states were reinterpreted as positive energy states of antielectrons or positrons. Within each solution there are two sets of components. The upper two components are called the large components and the lower two components are called the small components. At low energies, i. e., most chemical energies, the rest mass energy moo2 is much greater than the kinetic energy E. Therefore the quantity fl— within the mac2 +E lower two components is much less than one. Thus the lower two components are much smaller than the upper two components and their labeling as large and small components is natural. At first glance, ignoring the negative energy states and the small components might prove to be useful. However, the Dirac equation is actually four coupled wave equations. One could start with a solution from the free Hamiltonian that neglects the small 111 components and positron energies. However, when an interaction Hamiltonian is added, the small components and positron energies reappear. In a fully relativistic calculation, the electron energies are inextricably connected with the positron energies. E l E . . In relativistic quantum mechanics, the current density is written as j=c¢-yu.\r (37) where §=Lyl - yo and II” is the Hermitian conjugate of ‘1’. Using substitutions of the Dirac equation and the calculus of 4-component spinors, Equation (3 7) can be rewritten 18,19 as '= 53—6—6173! 7,310+ flfi-‘E — EAEJKP + ¥(—a— - £AJ‘P} 2m 6x,l “ 2m 6x" hc 6x” hc (38) = jun + jeonv - In Equation (38), jim is interpreted as an internal current analogous to the internal currents and charges that arise from magnetizations and polarizations. The convection current density jam, describes crurents analogous to the currents created by moving charges. The currents in the Gordon decomposition are still fully relativistic; that is, the currents calculated involve positron states in addition to electron states. However, the technique is included in this chapter for consideration when trying to separate applied fields from their molecular responses. Nonrelativistic transformations would still be needed for chemical applications. 112 Ff r-.n_ .. Nonrelativistic Approximations to the Dirac Equation Chemically, the concept of electrons having positron ‘shadows’ is mysterious. Even for relativistic effects that become observable in molecules with high-Z nuclei, chemists are not accustomed to searching for positrons within molecules. Fortunately, methods exist where the positron portions of relativistic equations can be approximated by expansions of the electron portions of the equations. One technique, the F oldy- Wouthuysen transformation, will be briefly examined. WW For molecular systems, fully relativistic calculations are impossible since the interaction Hamiltonian between electrons is not well defined. Even under relativistic approximations molecular calculations quickly become intractable. In a relativistic calculation, each electron in the system does not have just four components but instead has 41’ components where n is the number of electrons within the system. For example, calculation of energies of the helium atom involves 16 x 16 Dirac matrices as well as 16 component spinors. Calculation of energies of the lithium atom uses 64 x 64 Dirac matrices and 64 component spinors. This problem is avoided by finding a transformation that removes the effects of the positron states so that relativistic energies can be calculated from a series expansion using only electron states. The expansion that is considered in this thesis is the Foldy-Wouthuysen transformation?” Operators within the Dirac equation can be classified according to how they couple the components of the wavefunction and their parity. Operators that couple the electron 113 ‘EFIII ~ - «I and positron states have odd parity. Operators that couple the electron states with other electron states (or positron states with positron states) have even parity. Thus uncoupling the effects of positron states on the electron states involves finding a transformation for the odd operators in terms of even operators and hi gher-order odd operators. The Foldy- Wouthuysen transformation achieves this uncoupling. The method involves a unitary transformation where the generator of the transformation is a specific odd operator. The essence of the transformation involves commutation relations between even and odd operators. li-l”’H .1. Before stating the molecular Hamiltonian fiom the result of a Foldy-Wouthuysen transformation on the Dirac Hamiltonian, the two-body interaction between charged particles must be considered. There exists no fully relativistic closed expression for the two-body interaction. However, relativistic approximations such as the Breitequation""21 can be made for low energy systems such as molecular systems. Moss presents a nonrelativistic expression based on the Foldy-Wouthuysen transformation of a single electron system, a relativistic expansion of the vector potential and the requirement that the interaction be symmetric with respect to the interchange of electrons. Moss states that his ad hoc Hamiltonian is accurate to mc20t5 where a is the fine structure constant and that it agrees with the Hamiltonian rigorously derived by Itoh22 from quantum electrodynamics. In the Hamiltonian below, nuclei are treated as Dirac particles; however, their anomalous magnetic moment is considered in an ad hoc fashion. 114 Moss20 gives the nonrelativistic molecular Hamiltonian as H =E{mc2 -C¢I + n?/ 2m + 8P3(§i ' fit) the electron’s rest mass energy, the electron’s interaction with external electric potential, the electron’s kinetic energy, Zeeman interaction of e’ spin with magnetic field, - (ng/4mc2) s, - [‘i, x E - E x iii] interaction of electron spin with magnetic field + (eh2/8m2 c2)V . i=3 -(1/8m’c’)n? - (gun/Zm’ c’)(st - E) u? + $678113. n, -(e2/167teom2c2)7't1'(rfilit'j) caused by movement through external electric field, the Darwin term, an effect of the positrons, relativistic correction to electronic kinetic energy, relativistic correction to Zeeman interaction, Coulomb interaction between electrons, orbit-orbit interaction, - (e2/1 61t80 m2 c2)(fit ° Tij) r53 (fij - fij) retarded orbit-orbit interaction, - (eng/ 81t80 mc2 r131) §1 '(fu‘ x 17i) + (eguB/41reo mc2 r13) §1 '(Tij x it) spin-orbit interaction of an electron’s spin interacting with magnetic field arising from motion relative to the electric field of a another electron, spin-other-orbit interaction of an electron’s spin with the magnetic field created by another electron’s orbital motion, 115 .-.—-.....a +(gzl-13/87tjocz) ['1 5’ I 3(81') r11) (r11' S1)/r1’1 —(g 111/3811“) (r11)(81'§1) -(e 112/821m c)5 (r11)+“°} +§imacz +Za e¢a + 713/ 2% - 611116.. - 1‘31.) +B§Q[ZuZBez/8Tt80rap +"'] —Za ez/ 411280 not + (Z... e2/87reomma c2) 7‘11 '(r'E in) +(Zae2/81160n1rhc2)(7c1 'rta)r"1§(7'ta ' in) - (Zn eng/8‘tt80 me2 tin) §i ' (fiat X 171) dipolar spin-spin interaction, Fermi contact term of spin-spin interaction, relativistic correction to Coulomb interaction, E ;. rest mass energy of the nucleus, § the nucleus’ interaction with external electric I potential, I the nucleus’ kinetic energy, Zeeman interaction of nuclear spin with external magnetic field, nucleus-nucleus Coulomb interaction, electron-nucleus Coulomb interaction, electron-orbit-nuclear-orbit interaction, retarded electric-orbit-nuclear-orbit interaction, spin-orbit interaction of an electron’s spin interacting with magnetic field arising from motion relative to the electric field of nucleus, 116 -(Zaeg11LB/4momuc2 r3.) 6., (is, x Ea) spin-other-orbit interaction of an electron’s spin with the magnetic field created by a nucleus’ orbital motion, + (ega H111 / 47:60ch fig) L - (n, x E) spin-other-orbit interaction of a nucleus’ spin with the magnetic field created by an electron’s orbital Pl motion, - (66.. 111. uN/4nso c2) (§, - in) / r131. dipolar electron-spin-nucleus-spin interaction, X - 3(§i ° fia)(l'ia ' 1.3/13. + (8811. pg uN/3go c2)5(f,a)(§i - in) Fermi contact term of electron-spin-nucleus spin interaction, -(Za 62 hz/81Som2 c2)5(i-,a) +~ - -} +~ - -} relativistic correction to Coulomb interaction, etc., +3CQ + - -- nuclear quadrupole interaction, etc., (39) where the i and j indices indicate electrons, the or index indicates nuclei, g, is the electron spin, in is the nuclear spin, g is the magnetogyric ratio, H13 and 11111 are the Bohr and nuclear magnetons, respectively, and the momentafi are canonical momenta, i=5—eA. Higher Order Magnetic Susceptibilities Vibrational effects can be very important in chemical shifts of light nuclei.23 Rovibrational effects on chemical shifts have been calculated for lH"’F,24 “N25 and 13C.26 These calculations are done readily using numerical differentiation techniques. However, 117 during the course of a vibration. A higher-order magnetic susceptibility density may offer such interpretations. Intramolecular response due to mixed electric-magnetic effects such as the effect of hydrogen bonding on the chemical shiftmg'29 may also find a they lack a physical interpretation in terms of the electromagnetic phenomena occurring ‘ physical explanation. n A physical interpretation of vibrational averaging of magnetic properties may include the effects of nuclear motion via the nuclear canonical momenta. The terms in equation (39) that involve the nuclear canonical momenta are a I 3C ' = 2i2§+ ni/Zma + (2., e2/8tteomm,ll c2)[fii ° (r12 fia)+ (it - itch-EGO, i110] 40 - (Z11 egllg/ 47:811mac2 r131.) §1 -(fta x in). ( ) These terms describe the nuclear kinetic energy, the retarded electron-orbit-nuclear-orbit interaction and the spin-other—orbit interaction of the spin of an electron with the magnetic field created by a nucleus’ orbital motion. The creation of a higher-order susceptibility density that would have the ability to describe intramolecular magnetic response to moving nuclei could involve Equation (40). This Hamiltonian could be used in a first-order perturbation theory similar to that used to create the magnetic susceptibility density found in Chapter 4 or in a second-order perturbation theory. Relativistic effects have been incorporated into ab initio calculations in the chemical shift of high-Z nuclei, ””Sn,30 WW,” and non-zero spin isotopomers of the halides.32 Relativistic theories of the chemical shift by Pyykko”, Pyper’" and Zhang and Webb35 have been formulated. These theories consider the relativistic effects on the core electrons of heavy atoms within molecules. The effects arise fi'om the high velocities that 118 are created through interaction with the large electric fields that originate from the high-Z nuclei. Effects of the nuclear momentum on the molecular magnetizability are not specifically considered. 119 REFERENCES ' P. W. Atkins, Molecular Quantum Mechanics, 2nd. ed. (Oxford University Press, Oxford, 1983). 2 W. Rindler, Introduction to Special Relativity, 2nd. ed. (Oxford University Press, Oxford, 1991). 3 E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (W. H. Freeman and Co., New York, 1991). ‘ H. C. Ohanian, Classical Electrodynamics (Allyn and Bacon, Boston, 1988). 5 J. D. Jackson, Classical Electrodynamics, 2nd ed. (John Wiley and Sons, New York, 1974) 6 Refer to the Appendix in Jackson for a discussion of units. 7 R. Becker, Electromagnetic Fields and Interactions (Dover Publications, New York, 1982) 8 M. H. Nayfeh and M. K. Brussel, Electricity and Magnetism (John Wiley and Sons, New York, 1985). 9 K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983). ‘° K. L. C. Hunt, J. Chem. Phys. 80,393 (1984). “ K. L. C. Hunt, J. Chem. Phys. 90,4909 (1989). ‘2 K. L. C. Hunt, Y. Q. Liang, R. Nimalakirthi and R. A. Harris, J. Chem. Phys. 91, 5251 (1989) ‘3 W. J. A. Maaskant and L. J. Oosterhoff, Molec. Phys. 8, 319 (1964). '4 A. D. McLachlan, Proc. R. Soc. London, Ser. A 271, 387 (1963); 274, 80 (1963). 120 ‘5 L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, 3rd ed. (Pergarnon Press, Oxford, 1980). '6 W. Greiner, Relativistic Quantum Mechanics: Wave EquatiOns (Springer-Verlag, Berlin, 1990). '7 M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, MA, 1995). '8 R. H. Landau, Quantum Mechanics 11 (John Wiley and Sons, New York, 1996). ‘9 H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One- and T wo-electron Atoms (Academic Press, New York, 195 7). 2° R. E. Moss, Advanced Molecular Quantum Mechanics (Chapman and Hall, London, 1973). 2‘ M. E. Rose, Relativistic Electron Theory (John Wiley and Sons, New York, 1961). 22 T. Itoh,Rev. Mod. Phys. 37, 159 (1965). i 2’ D. B. Chesnut, Chem. Phys. 214, 73 ( 1997). 2“ D. M. Bishop and S. M. Cybulski, J. Chem. Phys. 101, 2180 ( 1994). 25 C. J. Jameson, A. C. de Dios and A. K. Jameson, J. Chem. Phys. 95, 1069 (1991). 26 A. K. Jameson and C. J. Jameson, Chem. Phys. Lett. 134, 461 (1987). 27 M. J. Stephen, Molec. Phys. 1, 223 (1958). 28 T. W. Marshall and J. A. Pople, Molec. Phys. 1, 199 (1958). 29 J. A. Pople, Molec. Phys. 1, 216 (1958). 3° C. C. Ballard, M. Hada, H. Kaneko and H. Nakatsuji, Chem. Phys. Lett. 261, 1 (1996). 3‘ M. Hada, H. Kaneko and H. Nakatsuji, Chem. Phys. Lett. 261, 7 (1996). 121 ’2 C. C. Ballard, M. Hada, H. Kaneko and H. Nakatsuji, Chem. Phys. Lett. 254, 170 (1996) 33 P. Pyykkd, Chem. Phys. 74, 1 (1983). 3‘ N. C. Pyper, Chem. Phys. Lett. 96, 204 (1983); 96, 211 (1983). 3’ Z. C. Zhang and G. A. Webb, J. Mol. Struct.-Theochem. 104, 439 (1983). 122 APPENDIX A: DERIVATIVES OF WAVEFUNCTIONS AND OPERATORS A. Derivative of the ground state wavefunction In deriving the derivative of the ground state wavefunction with respect to an arbitrary parameter, the orthonormality of wavefunctions that are eigenfunctions of a Hamiltonian is exploited. The proof begins by considering the following off-diagonal Hamiltonian matrix element in a basis of eigenfunctions of the Hamiltonian. n¢0 (nlflCIO)=Eo1++E" 5(nl0) 6n( Shim +n( |%_O>_O . (A5) 61:6 $601 é=l0> _(n I510__)_ Using Lermna 1, the derivative of the matrix element becomes _<__>6n|€1C|0 __)alo 6'ch a). l+671. a). 610) 63C (A6) -1 :6 I— a, =(1~:.-1=.o) (nl 0, l0). Multiply Equation (A6) on the left by In) and sum over all the excited states 2 nan 61 0) .. 63c >:In> (A7) Recall that the complete sum-over-states of projection operators |m)(m| is one. Z|m>(m|=1- (A8) Now insert one, as a complete sum-over-states into the equation between the energy denominator and the Hamiltonian derivative. 2I11><11 521—9:gayn>. (A9) In the summation over m, the m = 0 term vanishes. Therefore, the sums can be rewritten 2I11><0I)(E.-Eo—1w)"(1—Io><0I). (A11) Thus by using the zero frequency expression for the reduced resolvent, Equation (A10) becomes 811161913)=—(1—Io>(oI)(E.-Eo)‘1(1-Io>(0I)—Io> a) - —G(0)—-agc l0) (A12) “ a). ’ To find the derivative of the ground state, the left side of Equation (A12) must be manipulated. First, the sum-over-excited—states is rewritten as sum-over-states minus the ground state term. é'lanl-OEL §|n) 59"” = 5(0'|0)+(0|54—0—>- = 0 (A14) 0“). 5’1 (mgr-@590). 125 If the ground state is real, (not complex), then 0 ((01%— ))= é—af'w) (A15) Thus substituting Equation (A15) into the last line of Equation (A14) yields (OI%=-() . (A16) The only time a complex number Z = -Z* is when Z is purely imaginary; however, we are assuming that the ground state is real. Therefore Z = -Z 2'» Z = 0. (o old—02:0 (A17) 11’ Using Lemma 2, Equation (A13) becomes 0 0 0 0 |><|d>al>l)<|51>d>- (ms) Substituting Equation (A18) into Equation (A12) yields the result 6! _0_>_ m— -G(0) —| 0) (A19) B. Derivative of the reduced resolvent The proof starts with the definition of the reduced resolvent from Equation (A1 1), noting that the Hamiltonian operator 3C can be substituted for the energy En when the wavefunctions are eigenfunctions of the Hamiltonian operator. The symbol go is used to represent the ground state projection operator l0)(0|. 126 G((o) = 23" 2!" I n)(oI 336(0) = dogma-gem). Lemma} @(1- 6)) = EIOXOIGmeI - IOXOI) = EIOXOImel ~|0>(0|0>(0| = §|0>61n(ml-|0>(0| = IOXOI-IOXOI (B4) = 0. 127 Lemma (1- (0X1 - (a) = §(lm)<0I) = gglmelanl-Im)<0l -I0><0In)a...6.o‘<0l—IO)13h,(V%r—211119(——r)__Z +Idrdr'dr——)——;—plr) HES—Ll (C1) 2me i K=12m x-I If- Rxl Ir -r'| K-bKlRK —Rli The terms of the Hamiltonian describe respectively the electronic kinetic energy, the nuclear kinetic energy, the electrostatic potential energy between the electrons and the nuclei, the electrostatic potential energy between the electrons and the electrostatic energy between the nuclei. Taking the derivative of Equation (C1) with respect to nuclear coordinate i“ is straightforward. Only the third and fifth terms of the Hamiltonian have nuclear coordinate dependence. aSCKo__ x K 1 _ J L J an, Idrp(r,)zVI_ RI EgszI_I_§LI. (C2) 129 In the Bom-Oppenheimer approximation, the second term is an additive constant. Since the derivation in the thesis concerns only electronic response, the second term will be ignored. Therefore the nuclear-coordinate derivative of the Hamiltonian to be considered is 653C 1 °=- df ' z“v§——_—. C3 The first manipulation is a substitution using the identity foI l - - Va I—1—— where - —K- _ —fl r-R l r-R Vf,‘ refers to a gradient with respect to the nuclear Cartesian coordinate and V,l refers to a gradient with respect to the electronic Cartesian coordinate. Thus Equation (C3) becomes 559’- Idfp(r)Z‘V,I_ 1m. ,_ 6R5 - (C4) At this stage an integration by parts is performed to yield the final result. Necessary for the integration by parts is the electrostatic identity V - P(1") = -p(f). Considering the form of integration by parts to be In dv = uv —f v du , the integration proceeds as du = p(f) = —V - 136) :> u = -F(f) 1 1 _—K =V———— d =VV———-= ,R C5 v lf-§K| :> v lf-fixl fir ) ( ) =P V7 w— m1) Z‘VV _ . 130 The boundary term in Equation (C5) is zero since the polarization of a molecule at . . . . . . . — 1 mfimte distances lS zero. The defimtlon of the dipole propagator T(f,RK) = VV[| _KI] f—R is substituted to yield the result. 6H0 afi“ = -1 a; Hf) z“T(f ,R’K). (C6) 131 APPENDIX B: COMPLETE DERIVATION OF THE HYPERPOLARIZABILITY DERIVATIVE Introduction The proof of the result of Chapter 3 begins with the expression for the hyperpolarizability taken from Orr and Ward’s article on nonlinear optical polarization.‘ The polarization operators are adapted to be polarization density operators so that the expression becomes a hyperpolarizability density. Ban. (“Sf"rw. mm) = 5091K0|13a (f)G(wo)13$ (f")G(mn)f>s(f’)|0> +

      a(?')|0) +<0|13.(F)G(coo)133(?')G(coz)f>.(f")l0> B...(f,f',f";-w.;w..w2) = +<0l1:.(:”)G (-w2)AI;p(:')G.(-coa)r>.(:')|0> . (2) +.(r ')G (—wo)P.(r)|0> < < + 0|13,(F")G‘(—mz)133(F)G(wn)1‘>.(?')l0) _+ 0|fia(F')G‘(—m1)fi3(f)G(w2)13.(f")|0> ..J 1B. J. Orr and J. F. Ward, Molec. Phys. 20, 513 (1971). 132 In the above expression, three types of quantities depend on the nuclear coordinate: the ground state wavefunction, ( , the reduced resolvent, G(m ,) , and the fluctuating polarization operator, in: (P’). Note that the total polarization operator, 1'50l (f) , has only electronic coordinate dependence. The derivative of the ground state wavefunction for an arbitrary parameter is calculated to be2 510) 653C 3;: = -G(0) filo) . (3) The derivative of the reduced resolvent is2 aG((°) -G( )a(gc___"__E0) 63C m (0 a). G(m )+ +500— 6). —G(0)G(w)+G(m)G(0)a—-m (4) The derivative of the fluctuating polarization operator is2 91?.“ 0|— idem-m +<0|1‘>.(r)G(0)-—l0> (5) When the arbitrary parameter is specified as the nuclear Cartesian coordinate, i. e. A=R§ , the derivative of the Hamiltonian becomes2 _gRgcg ___ _J‘drm ZKPg(rm)T35 (- ng K). (6) Taking the derivative with respect to nuclear coordinate R? of all six terms of Equation (2) and substituting Equations (3) through (6) yields for the hyperpolarizability derivative, the sixty term expression of Equation (7). 2Appendix A 133 “Ira- F(0|P (F)G(ma )P$(F' ')G(m,)13,(r' )I 0) +<0|P (T)G(wo )Pg(r )G(c02)Py(" No) _a_ +(0lP.(F' ')G (—w2)P3(F' )G (—m .)P (F)l0> 0R? +(0lPo(F' )G (—wn)P‘.’(F ")G H) .)1‘> (F)l0> +<0lP. (F ')G (—a>2)P° (F)G(an)Pp(F’ )l 0) +(OIP:(ZF' )6 (—w:)P° (F)G(m2)P.(F' )l 0) j =IdF “R(F'" R) ><{(0lP.(F' ')'G(0)P.(F)G(P.(F)G(m.)P3(F')G(m2)P.(F")lO) +<0|P.(F)G(wo)P‘Z(F"')G(mo)P3(F")G(an)Pp(F')l0) +(OIP. (F)G(coo )P° (F" ')G(m a)P3(F' )G(co2)P.(F' ’)l 0) -(0|P(F'")G(0)(OIP. (F)! o)G(...)f>3('r-")o(m.)13,(r' )I 0) -(0lP (F" ’)G(0)(0|P. (F)! 0>G(m )P3(F')G(co2)P.(F ")l 0) ~(0lP.(F)G(wa)G(0)P.(F'")l00>(0|13°(F' ")G((0|P§ (F' )G(wz)P.(F' ')IO) - 0|P.(F"')G(0)P.(F")IOXOIP. (F)G(w )G(con)Pa(F' )l 0) - OIP.(F"')G(0) Pp(F')l0>(0|P.(F)G(coa)G(wz)P. (F ')10>- - OIP. (F")G(0) P.(F"')l 0>(0|P. (P)G(w.)c(m,)13,(r')| 0) - Ole(F')G(0)P.(F'")|0X0!P.(F)G(coo)G(w2)P.(F")|0) 0|P.(F)G(wa)P$ (F”)G((0lP.(F"')G(0)G(wx)Ps(F’)l0) - 0|P.(F)G(wa)P3(F')|0>(0|P.(F"')G(0)G(m2)P.(F")IO) - 0|P.(F)G(ma)P3(F")G(mn)(OlPa(F')|0>G(0)P.(F"')|0> - 0|P.(F)G(wo)P§(F')G(G(0)P.(F"')|0) OIP.(F)G(wa)P3(F")G(mn)Pa(F')G<0>P.(F"’)|0> + 0|P.(F)G(wa)P§(F')G(w2)P.(F")G(0)P.(F"')|0> 134 +<0IP.(F"')G(0)P.(F")G‘(-a>z)P3(F')G‘(-coa)P.(F)l0) +<0|P.(F'")G(0)P.(F')G'(—a>n)P‘.’(F")G'(-wa)P.(F)|0) (0|P.(F")G‘(-mz)P2(F"')G‘(—w2)P8(F')G‘(-wa)P.(F)I0) (0|139(F')G'(—co1)P3(F"')G'(—co1)13$(F")G'(-ma)13a(7)|0) 0|132(F'")G(0)<0|P,(F")l0)G‘(-coz)P3(r')G‘(-m.)r>.(F)!0) 0|P2(F'")G(0)<0IP.(F')l0)G‘(—<0IP3(P')G‘(-w.)P.(F)l0> OIP.(F')G‘(—w1)G(0)P.(F'")|0><0IP$(F")G‘(-m.)P.(F)I0) 0|P.(F'")G(0>P.(F')|0><0IP.(F")G‘(-o»2)G‘(—co.)P.(F)l0) 0|P.(F"')G(0>P.(F")IOXOIPB(F')G'(-mx)G‘(—coo)P.(F)|0) 0|P.(F')G(0>P.(F'")|0><0|P.(F")G’(-m2)G‘(-wo)P.(F)l0) OIP. (F")G(0) P.(F'")I0><0IP.(F')G‘(—w1)G'(—wa)P.(F)!0) ( ( ( ( ( ( ( ( +(0lP.(F")G'(-wz)P§(F')G°(—wa)P3(F"')G'(-wo)P.(F)|0) ( ( ( ( ( ( ( + + + 0|P.(F')G‘(—w.)P3(F")G'(-co.)P3(F"')G'(-wa)P.(?)|0) 0|P.(F")G'(—w2)P§(F')|0><0|P.(F"')G(0)G‘(-wa)P.(F)|0> 0|P.(F')G‘(-wn)P‘.’(F")l0><0|P.(F'")G(0)G‘(-ma)P.(F)I0) OIP.(F")G’(—co2)P3(F')G‘(—wa)<0|P.(F)I0>G(0>P.(F'")|0> 0|P.(F')G‘(—cou)P$(F")G‘(-G(0>P.(F"')|0> o!P.(F")G'(-mz)P§(P')G'(—m.)P.(F)G(0>13.(P'")|0> + 0|P.(F')G‘(-w1)P$(F")G‘(-wa)P.(F)G(0)P.(F"')|0> (7) + 135 “WI: +(Ol13..(?"')G(0)Py (7")G’(-w2)133(F)G(wn)13.3510) (0|1'5,(f"')G(O)13I,(F')G‘(-cm)132(F)G(a)2) 137 (in) 0) (0'131(f")G.(-032)132(7'")G.(-02)132(f)G((01)133(F)|0) +(0I139(?')G’(—m1)132(T"')G°(—col)13:(F)G(mz)13,(?")|0) -(0|132(7"')G(0)(0lI3,(7")|0)G°(-C02) 132(F)G(m1)13,,(F')10) -(0|132(7"')G(0)(0|135(?')l (”GT-cot) 13:.(?)G(002)1'57 (WHO) "(W 131 (7")G.(-032)G(0) 13.1me OXOIISZ (36901) 135 (PM 0) —(0|f>,,(F')G‘(—m,)G(O)13,(F"')|O)(0|132(F)G(m2)1’5,(F")|0) ‘(0l13.(7"')G(0)l3a(7)l 0><0|137(7")G.(—032)G(C°1) P.(F')l0> -( "< -( + + 0|P.(F'")G(0)P.(F)|0><0|P.(F')G‘(—wn)G(coz)P.(F")|0> 0|P.(F)G(0)P.(F'")|0>(0lP,(F")G‘(—wz)G(m1)PB(F')IO> 0|P.(F)G(0)P.(F"')|0><0|P.(F')G‘(— +<0IP.(F")G‘(-mz)P3(F)G(a>n)P‘.’(F"')G(wn)Pp(F')|0> +<0|P.(F')G‘(-w:)PZ(F)G(m2)P§(F’")G(w2)P.(F”)|0> -<0|P.(F")G’(-m2)PZ(F)I0><0|P.(F"')G(0)G(wn)PB(F')|0> -<0|P.(F')G‘(-mn)P2(F)l0><0|P.(F"')G(0)G(co2)P.(F")|0> -<0lP.(F")G'(-w2)P3(F)G(mn)<0|PB(F')|0>G(0)P.(F"')10> -<0|P.(F')G‘(—wn)PZ(F)G(w2)(0|P.(F")|0>G(0)P.(F"')lO> +<0lP.(F")G’(—w2)P2(F)G(cm)P.(F')G(0)P.(F'")l0) < + 0|P5(?')G°(-wn)P3(F)G(w2)P.(F")G(0)P.(F"')| 0)}- 136 Lemma: Proof of fluctuating polarization operator identity Terms 7, 8, 15, 16, 27, 28, 35, 36, 47, 48, 55 and 56 have matrix elements of the form (0| 13a (f)G(m°)G(O) 136 (f"')[ 0)(O|13: (f”)G(m,) 13,(r')| 0) . The fluctuating polarization operator is changed to a total polarization operator by proving the identity (0|13a(F)G(mo)G(0)i5,(?"')l0><0|§:(F")G(wn)fia(?')10> =<01macaque)134?"):o>,(r")e(mm(f')lo>. (8) The proof of the identity (8) begins by substituting the definition of the fluctuating polarization operator, Equation (9) into the left side of identity (8) to yield Equation (10). 1336") = 13,(‘r'")-<0|1’5,(F")|0> . (9) <0mGGaIo>p(i-")Io> =(Olfia(?)G(wa)G(0)138(F"')|0><0|13,(?")-<0|13,(?")|0>G(wx)fip(?')10> (10) =(0|13a(F)G(ma)G(0)I3,(F"')|0><0|13,(?")G(cm)1’5¢(?’)10> -(Olfia(?)G(wo)G(0)1356'")!0><0|<0|1‘>,(?")|0>G(m:)1‘>s(?')|0)- Substituting the sum-over—states definition of the reduced resolvent, Equation (B11), yields Equation (B12) after little manipulation and recalling that (0| n) = 50,. In><0| lanl—IOXOI G(wl)=§ SEEM] =§ 0W0] . (11) 137 <0I13a(r~)c(ma)c(0)126-"'1o><0If>2(f-")G(cm)m(f')lo> =(Olfia(?)G(mo)G(0)Pe(P"')I0><0|P,(T’")G(con)f>a(?')I0) -gmg]:(0' =GI 0)<0|13,(?")G(w11‘>p(?')IO) -(0|13.1(7')Gv(ma)<3(0)1‘>e(f-"')I0>(0|1‘>,,(r")I0){:21“(OI(”IP"(r)|0%_mI 0'1“") =(OIPa(?)G(ma)G(0)Pe(F"')I0><0|P,(?")G(col)1‘>p(?')|0) -<0If> ( 196(0) 6(0),: (r )|o olp (-)l0>§2“5°‘1(01133(?')I0) a a c 1 nos-(DI =<0Ii>,(r-")Io><0'P“(")'::'_<:'IP“(")'°> =(OlPa( r)G(wa) G(O)P.=(r" “IO<01P1("")G((01)Pp(r')|0> -<0IPa( r))G(mur )G(0) P,(r" '))IO (0|§,(r )IO) x0 (12) = (OIPa(P)G(wa)G(0)Pe(F"’)I 0)<0I1‘>,(?")G(m1)13,(r')| 0). M" )0) Thus, the identity is proven. Continuation of proof The first and fifth terms in Equation (7) are simplified by applying the definition of the fluctuating polarization operator. <0Ifi,(?"')G(0)fia('r')G(wa)P$(F")G(w1)f>s(F')I0) -<0I13e(f"')G(0)<0IM?)0>G(coo)13$(F")G(con)Mr") 0) (13) = (0|13e(f"')G<0)13f:(7)60»)P3(F")G(con)fia(f')l0) 138 Similarly, the following pairs of terms in Equation (7) can be simplified: (2, 6); (19, 17); (20, 18); (21, 25); (22, 26); (39, 37); (40, 38); (41, 45); (42, 46); (59, 57) and (60, 58) to yield Equation (14). 6130431 (r,f',f";—co°;o),,c02) 6R? x{$(f")G(m1)f>B(F')| 0) +<0IP,(F'")G(0)fii(‘r’)G(ma)P8(f')G(coz)P,(F")I0) +I0|13.,(F)G(a(f')l0) IOI13a(F)G(wa)PSIF'")G(mo)P3(?')GIw2)P, (F")IO) -<0I13.,(f)G(coa)G(0)Pe(‘r'"')I0><0I1‘>,(P")G(wn)f>a(f')|0> —<0|13,(r)G(ma)G(o)is,(r'")|o>r(f"')G(0) P, (f")IO><0IPa(?)G(coo)G(wn)fip(f')I0> -<0|13=(?"')G(0)Pp(f')l 0><0I13a(f)G(wa)G(mz)P,(f")I0) -<0|13,(F")G(0)1‘>,(F"')I0X0113a(F)G(wa)G(mn)fip(f')I0) -<0IPp(P')G(0)Pe(f"')I0><0IPa(F)G(ma)G(coz)P,(f")|0> OIPa(F)G(wa) 13$ (P")G(m1) P2(f"’)G(wx)f>a(f')IO> 0|13a(f)G(coa)1’58(F')G(coz)132(f"')G(wz)P, (F")IO) 0|ii..(f)G(ma)f>,(F")I0)<0IPe(f"')G(0)G(wn)Pp(f')I0) 0|13..(P)G( OI13a(i‘)G(wa)P$(f")G(a>1)P§(P')G(0)PAP")0) 0|13..(i")G(coa)PEIF')G(coz)P$(F")G(0>Pe(?"')|0> I I I I I I +I0|P3I?'")G(0)133(F")G'I-w2)133(?')G'(-coo)f>a(f)|0) I I I I I I I =Idf"'z"rs(r"',fi“) + + + + + + Olfis(f"')G(0> 133(f')G.(—‘Dl)133(7")G.(-030)13a(filo) 0|1‘»,(f")G‘(—$(F")G‘(-coa)13a('r')I0) Olin(f")G‘(-w2)G(0)134?")0><0IPB(F')G‘(-mo)fia(F)l0> 0|Pp(f')G‘(—mn)G(0)f>e(‘r'"')| 0><0|13, (F")G’(-wo)i5a(f)l 0) 0|13,(f"')G<0>fia(f')I0><0lf),(f")G‘(-w2)G’(—wo)m?)0) 0|13e(f"')G<0>f»,If")!0><0|PBI‘r")G‘(—w.)G‘(—wo)1301f)!0) + + 139 OIPBIF')G(0) P,(P"')IO)I0|1‘>, (f")G‘(-co2)G‘(-coa)f>a(f)l 0) Olfi,(‘f")G<0>P3(P"')IO><0|Pa(f')G‘(-cm)G‘(—wo)f>..(f)l0) 0|13,(f")G‘(-coz)P3(f')G‘(—wo)132(?"')G‘(—wa)f>a(f)l0) ol§,(r')c°(—m,)ra3(F')c‘(—m.)ls:(t"')c'(-mo)no)o> 0|I3,(F”)G’I-w2)1‘>s('r")|0)I0|Pe(f"')G(0)G'(-wa)13a(?)|0) 0|P5(f')G‘(-wn)f>, (7")! 0><0I135(?'")G<0> G‘(-wa)i5a(i‘)| 0) OIP, (?")G‘(-mz)132(r')c'(—mo)132(r)6(0) 13,(‘r'"')I 0) 0|MP)G°(-co.)i‘>$(f")G‘(-wa)132(f)G(0> 13,('r'"')IO> 0|138(f"')G(0)13$(?")G‘(—coz)PZIF)G(wn)Pa(F')I0) 0|Pc(?"')G(0)133(7')G°(-mn)P3(f)G(w2)131IF")I0) 0|13,(f")G‘(-wz)132(F"')G‘(—m2)fi2(P)G(con)f>p(f')I0) 0|1%(f')G‘(-con)f>i’(f"')G‘(—con)P2(f)G(mz)i‘>,(f")|0) I I I I I I I I I I I I IOIP,(F")G'(-coz)G(0)Ps(?"')I 0>I0|Pa(‘f)G(mn)Pa(?')IO I I I I I I I I I I I + + + + + + + + > - 0|Pp(f')G‘(-con)G(0)1347'")0><0Ifia(F)G(coz)f>,(?")IO> 0|in(f"')G(0)13,.('f)I0><0I13,(F")G‘(-co2)G(wn)fip(F')I0) 0|1‘n(f"')G<0) 13,. (f) 0><0IPB(?')G’(—wn)G(mz) 13, (F")IO> Olfia(f)G(0> 13,(f"')l0)(0|I3,(F")G'(-m2)G(m1)139(F')IO) 0|13,.(F)G(0)r>,(f"')|0><0lfip(f')G‘(—mI)G(w2)P,(f")I0) Olii, (f")G‘(—..(f)I0><0l13,(F"')G(0)G(wn)fia(7')I0) 0|fifl(f')G‘(-m:)r>a(f)|0><0I1‘5,(F"')G(0)G(m2)13,(f")l0) + Olin, (f")G‘(—m2)132.(f)G(m1)1‘>3(r')G(0) 13, (‘r‘"')| 0) + 0|135(f')G.(’ml)132(f)G(C°2)f’$(f")G(O)13¢(fm)|0>} (14) + + The expression within the {} brackets in Equation (14) is compared to the expression for the second hyperpolarizability density adapted from Orr and Ward.l 140 ym(f,f',f",f"';—(oo ;w 1 ,a), ,0) = (OI 13a(?)G(mo)1‘>2(F"')G(wa)fi$(f")G(m1)1‘>,(f')I 0) +(OI fa. (?)G(wa) P2 (F')G(coz) 132 (F "')G(co2) 13, If") 0) +(OI fa. (F)G(wo) 13$ (f")G(co1)f>§(F')G(O> P, (F")I 0) +(OI Pa (F)G(ma) 1’52 (F "')G(w a) 1’5: (f ')G(o>2) 13, (P ")I O) +(OI13..(P)G(mo)13$(f")G(m1)132(?"')G(co01356)O) +(OI 13.. (F)G(coa)1‘>8(f')G(wz) 13$ (F")G(O> 134?") 0) +(OI 1‘», (F'")Gm) 133 (?)G(wo) 13$ (F")G(ml)133 (F')! O) +(OI1‘5,(F"')G(O) 13f: (F)G(coo) P2 (F')G(co2) P, (f")I O) (OI 1‘5, (‘r‘")G(o>.) 13.36") O>e(?"')l O) (OI 13, (F")sz) 1‘», (f")| O>(OI 13.. (r)G(m,)G(w,) 13,(r')| O) OI 1% (f ')GIO) 1’5, (F "')I O>(OI 13a (r)G(m,)G(m,) 13, (f ")I O) 0| i5, (r 06(0):) i5, (‘r‘")I O>(OI 13.. (F)G(coa)G(O) in (?"')I O) OI 13e(f"')G(con) 1336") O>(OI 13., (r)G(m,)G(m,) is, (“f ")I O) OI 13, (f ")G(O) 134?") O>(OI 13.: (f "')GG(m2) ii, (?")I 0) OI Pa (f)G(co2) 1‘5, (f")| O>(O| i5, (f"')G(O)G(wn) 13, MI O) (OI 13., (f')G‘(-cm) 1’53 (F")G' (-coo) 1’52 ('r')G(O) in, (F'")I 0) OI 13, (F') G‘(- 13, (F") O) ( ( -I -I -I -I -I -I + + < + oIf»,(r')c°(—m.)is:(r~)c‘(—m,)132(rm)c'(_m.,)1w)o) 0113,(max-m)f»:(r'~)c‘(_m2)133(r')c‘(_m,)13,(r)|o) +(OI13,(?"')G(O>133(7')G‘(—mu)i5$(F")G‘(—mo)f>a(f)lO> (0|13,Mex—m)13;;(r')G'(—m,)132(rm)e'(_m,)13,(r)|o) +(OIPIIF')G‘(—w)132(f"')G‘(-wn)f>$(F")G’(—wa)f>a(f)lO> +(OI13,(F"')G(O>13$(F")G‘(—co2)fi3(F')G‘(-coa)fia('r')lO> + + 141 -(O|fia(f')G‘(—wx)fi,(f")lO>(OIfie(f"')G(O>G‘(-coa)f>a(f)lO> —2)f>e(f"')lO>(O|Pa(f')G‘(-cm)G'(-wo)f>a('r')lO) -(O|P...(‘r"")G(O>Pn(f')IO>(OIin,(f")G‘(—w2)G‘(—ma)PAP)!O> -(OI1‘>, (F")G’(—mz)fia(F')IO>(OIfu(‘r"")G(O>G’(— -(OIPB(F')G‘(—cm)f>...(f"')lO)(O|f),(7")G‘(-ooz)G‘(—mo)fia(f)|O) -(OIfa(F"')G(O)13,(F")IOXOIOB(F')G‘(-mn)G‘(-ma)13,6)O) -(O|PBIF')G‘(—wx)fia(f)IO>a(f)lO> (OIPp(f')G‘(—con)132(F)G(m2)f>2(F"')G(wz)f>, (f")I 0) (0113, (‘r’")e’(-m2)132(f)G(mx)f>3(f')G(O) 13,(f"')IO) +(OI13, (f")G‘(-w2)f>2(F)G(co:)P3I‘r‘"')G(cm)Pp(f')IO) (OI135696140013:(P)G(wz)f>3(f")G(0) fi,('f"')I O) (Olin(f")G‘(-w2)f>2(F"')G'(-co2)P2(f)G(wn)f>a(F')IO) (OIIOe(f"')G(O)P3(f')G‘(-mn)fi2(F)G(wz)fi,(P")IO> + +(OI1‘>¢(F'")G(O> fi$(f")G‘(-coz)f>2($660134?)O> -(Ol13a(?)G(O)13e(f"')IO>(OI13,(F")G‘(—wz)G(wI)f>a(F')IO> -(OIfia(F)G(OI fia(f')G‘(-m1)G(mz)P+ (f")|O> -(Ol134666913,(f")IO>(OIPp(f')G‘(-co)G(O)f>e(F"')|O) -(Olf>a (?)G(wx)f>p(F')IO>(OIP, (F")G'(—coz)G(O)f>e(f"')I O) -(OI1‘>s(f"')G 13a(?)l0)I0I135(f')G.(-(01)G(o)2)13, (7")IO -( -( -( + + + + + (15) ) Olin (F"')G(O) 13.. (f)! O)(OII‘>, (f")G‘(-w2)G(G(con) 13,, (F') O) OI PBIF')G’(—wn)13a(f)l O>(OI Ps(f"')G(0)G(mz)Py (F")I 0) Proof of a sum-over-states identity The expression in the brackets {} of Equation (14) is equal to Equation (15) if term by term they are equal. Forty of the terms are equal to each other; however, eight terms are not. The following convention is adopted for an ordered pair: (nth term of (14), mth 142 mm ... radium term of (15)). In this convention the following terms are similar but not equal: (7, 10); (8, 13); (13, 16); (14, 15); (21, 31); (22, 32); (25, 29) and (26, 26). The equality is found once the terms are grouped into two distinct sums and the sum-over-states definition is substituted for the reduced resolvent. The substitution for the reduced resolvent is needed since the equality is proven by manipulating the energy denominators. The algebraic manipulations below will show that the sum of terms 7, 14, 21 and 26 of Equation (14) (labeled 3,) equals the sum of terms 10, 15, 26 and 31 of Equation (15) (labeled 3,). Similarly, the sum of terms 8, 13, 22 and 25 of Equation (14) (labeled 9,) equals the sum of terms 13, 16, 29, 32 of Equation (15) (labeled 8': ). S1 = -(0|P,(?'")G(0) P, (F") 0>I0IP,(?)G(wa)G(wn) P, (P')I 0) - IOIP,(F)G(coo)PB(?')I 0>I0lP,(?"')G(0)G(w2) P, (7")I 0) (16) - (OIP,(?")G’(-w2)G(0)P,(F'")I 0)I0 135(P')G.(-C°o) Pm (7)| 0) - IOI P, (?")G(0)P,(?"')I 0)I0|P,(?')G'(-OO:) G’I-ma) Pa(?)| 0)- :vszO'f’eIF"')|m>(nIP,(r')|0) m n sz(nu‘moXQng-O)l) _ 2' 2, (0|13a(f)lm>ImI133(7)0)(0I13,('f'")|n)(n|py(i“")|0) m n (gins—03°)Qns(ans-C02) 17 _ , , (OIP,(?")Im>(mIP,(F"’)|0)(0|P,(P')In>(an,(F)IO) ‘ I m n ((21,, +co2)nm,(a;, + on) g En' (0| P, (F")Imel P,(F"')| 0>(0|P,(F')I n)(an,(F)|0> — — 0mg (0;; 4' (1)1)(0; + (03) Recalling that the summation labels m and n are arbitrary, we exchange m and n in the second summation and sum like terms together over a common denominator. 143 IOIPeIF'") m)IIIIIP, (7") o)IOI Pa(?)| n)InIPn(7')I 0) 0mg (9,, - moXQng - (on) '2' (0| PaI?)| n)Inl Puff") 0WI 134?")! m)Im|P, (F")l 0) m n (nag—mo)ng(ng-m2) (0| P, (F")I m)ImlPJF'") 0)IOI PBIF'N n)Inl Pa(?)| 0) (52;, + (02)ng (0;, + cos) {3; 2' IOI P, (F")I WI“! | Pe('f"')l OXOI 13,6") n)Inl Pa(?)l 0) om,(r2;,+m1)(n;,+ma) ' S] = “2'2' (18) _EIZI (OIP (r )Im) m(lp,(r )I0)( (0|Pa(r) [(n.,-n,)+(a,,-m,)] QmIQns‘ coaXQng-wIXQm-mz) , ,Ions,(n")Im><0In,(n')In>[(n;,+n,)+(o:,+n,)] -2 2‘. . . . m n (m+w2)flm(flng+mn)(m+ma) 2; 2, (0|f)£(f'")'m><0l13a(f)|n>(nmg + gag “(1)1 -032) m n malng- maXQg- (DIXng-(DZ) -X' 2' (OlPr(r Mm) (m'Pe(r" )OX 0|P5(r Mn) (nlpa(r)|0)(Q m+Q;g+c01+m2) ... ,. (unmannmwnxmuna) SI=-E'2n (19) The same manipulations are performed on the sum, S, S: = -I0IP,(?’")G(w2)P,(F")I 0>I0|Pa(?)G(wa)G(wx)Pn(f')I 0) -I0|Pa(?)G(OJ1)Pp(?')I 0)(0|1‘>,(f"')G(0)G(m2)13,(F")|0) -(0IP,(?")G‘(-co2)PEIF'") 0)(0IPn(?')G'I-mn)G’(—coa)Pa(F)I0) -IOIPpIF')G’(-wn)Pa(?)l0)I0|P,(7")G'(-CO2)GI0)PAP") 0)- (20) , (0|P,(?'")|m>(mlP,(F")I0)(0IP,(F)In>(nIPn(?')I0) (9mg - (sznng - maXQng - cm) ,IOIP,(?)|m)Im|Pn(?')| 0)I0|P,('r'"')| n)InIP,(F")I0) . m n (am-ml)nng(nng-m2) , IOIP,(?")I m)(m|f>,(r"')| 0)I0|Pn(?')l n)IIIIP,,(?)I0) m n (Qing+m2)(Q°g+mI)(Qig +(Da) (0IPnIr' )lm) ImIPGIr)|0 0)(IP,(r")|n) )InIP,(r" ')I0) ”2 n n (:2:,,,+nm)(n:.,+nnz)og S2=-2'2 (21) 144 Next the summation labels, m and n are exchanged in the second and fourth summations. s, = _ z, 2. <0|Pe(?"')|m><0In.(r)In> m n (0m, - onXQng - maXQng - on) _ 2,(OIP.(?)In)(n|P,(F')I0>(0IP.(?”')Im)(mIP,('f")l0> m u (an, - 031)Qm8(§2m, - (02) (22) , (0| P, If") m)Im| P,(?"')I 0)IOI PBIP'N n)Inl Pa(?)|0) - X . . . m n (Om, + (DZXQng + (DIXQng + (00) _ , ,(0|P,(F')In>(nIP..(?)I0>(0|P,(?")|m>(mlP,(?'")I0> n n (9;, + m,)(Q;., + (1)2)ng ' S _ , , (OIP.(F"’)Im)(mIP, (F")I 0>(0|P..(?)In)(an,(F')I o>[(am,) + (a, — no] 2 - ' 2 E ' m n (am-m2XQu-maXQu-wn)9ms _ Z ' (0' 13, (“f")lmxnfl 135G"). OXOI 1313“,). n)In| 13016:» O)[(Qms) + (9;: + (90)] m n (Qing+mz)(flig+ml)(flig+coa)flmg (23) IOIP, (F")ImeIP, (7")1 O>(mIP,(F')IO>ImIP,(F")I0>I0IP,(?'")In>InIP,(‘r")I0> 2 m n (flung-(Do)Qng(Qng—ml) ,(0IPBIF')Im>ImIP.('r'"')I0>(0|P,(F")In)(an.(F)I0) X 2 . . m n (ng+a)1)ng(Qng +030) 2,I0IP.,(II'))Im(mIP Ir” '>I)I0 >I0IP,(r' ’)In)In|P..(r)I0) m n ng(a;8+(02XQng +000) (25) S, hm?(0|Pe(?"')Im>Im|PnIP’)0)I0IP.(P)In)(nIP,(?")l0>[(oq-ImIP,(F"')I0>I0lP,(?")In>(nIP.(F)IO>[(a;,+m2)+(o:.,+m,)] m n (m,8+(01)§2(m(fl;g+ma)(fl;8+mz) (26) _ , ,(OlfieIPHNmIImIHI?)O>IOIPQI7)ID)IHIP,IF")I0X0»;+0“-cm-a)2) --E X m " QWIQOO'WXW'WXQm 0):) , 'IOIfDBG'XmeIp (r" )IO) I0IP,I- )ln) (nIPa(P )I0)(Qm+Q.g+mx+m2) -2 2 n n (Q;,,+co1)flm(££g+coa)(§£g+coz) Next the 3'; sum is manipulated. I0|P.(?"')I m)ImIG(coI)Pp(7')I O)I0lP.,(?)IH)IOIP,(7")I 0) 8,2 = _ a, 2“, (0mg - (00(an - waXQng - (02) _ 2' 2' (0|Pa(?)|m)Im|P,(F")I 0)I0|P.(?"')I n)InIP,(F')I 0) m n (am, — wz)Q..,(Q.., - an) (27) _ 2' 2, I0IPp(?')I m)IImIP.(F'")I 0>I0|P,(?")| n)IIIIP,,,(?)I 0) m O (0;, + (00(5):, + 09(5):, + (00') _ 2, 2' I0|P,(r' ')Im>ImIP.,(r) )I0)I I0|Pn(r' )In) IOI 13. PI n>(nIi>,(r)|0>(n.,+9.3,+0,+,,,,) '“ “ (d,+n,)(o;,+mz)(9;,+m.)n,, Result The preceding work has shown that the 48 terms inside the {} brackets of Equation (14) are equal to the 48 terms of the second hyperpolarizability density of Equation (15). Thus, substituting the second hyperpolarizability density into the brackets of Equation (14) yields the result: 63“”, (f,f',f";-coa;a),,m2) 6R? ' (29) ... jd‘r’mzxymfij',‘ rrr,frrr;_m a “D I ,(D 2 ,0)T65(?m:§x) ° 147 APPENDIX C: EFFECT OF CREATION AND ANNIHILATION OPERATORS ON MULTI-ELECTRON WAVEFUNCTIONS Multi-electron wavefunctions can be represented as Slater determinants or as occupation number wavefunctions. When an occupation number wavefunction is used, the single-electron states that comprise the multi-electron wavefimction are placed in a conventional order. In this work, the ordering of the single electron states will be in terms of energy. The occupation number wavefunction can be written as a product of creation operators acting on the vacuum state. The creation operator product also is placed in a conventional order. lnm may ' 'nnlmfl> = (a1(lsa))m”(al(lsg))msfi---(al(nlrnp))nm| Z) (1) The wavefirnction is written in terms of n with the understanding that n is either 0 or 1. Thus, Equation (1) can be rewritten neglecting creation operators raised to the zeroth power. In,” my ° °nnhnfi> = a*(lsu)a*(ls;,)' ' °a*(nlmn)l E) (2) The electrons in the multi-electron wavefimction must be indistinguishable. Therefore, when the electrons are placed in a different order into a multi-electron state, the differently ordered state is still a valid wavefunction, though the sign of the function may change depending on how and how many electrons were exchanged. The anticommutation rules for electron operators allow for the proper accounting of the sign of the multi-electron state when electrons are exchanged. 148 {alual}: aInan + anam= 0 {amaan} =aman+anam=0 (3) {alnan} = afuan + again = 6m Specific examination of the anticommutation relations for the creation operators yields the proper sign of the wavefimction under electron exchange. a3, a}; = — a1 a3, (4) “1 Therefore, in the occupation number formalism, the relationship becomes aman|®)= —a;arnlg>=>lnmnn>=-Innnm>° (5) Thus every time the order of a pair of creation operators is reversed, a sign change results. 9; When a creation operator operates on a multi-electron wavefunction, the number of particle exchanges required to place the operator in standard order must be determined so that the correct sign of the function can be found. Consider the operator aI operating on the wavefunction Inm nu). (In the examples, standard order is alphabetical order.) allnmnn)= aramanl®>= (- 1)ama,an|@) =(- 1)(-1)aLaI.aII@> = Inmnnnr) (6) The a: operator needed to be exchanged twice to place it in conventional order. Thus, a factor of (-1)2 is included in the final wavefunction. The two following examples further illustrate the concept. a: l nm nu nr>= 31 am an ar Tl Q): (’ l)31’am an ara ang): —| Ilm Ill] 11: m) (7) aLInmnnnx)= apamanarlg>= (-1)2 amanaparlg) = )Inmnnnpnr> The factor of -1 can be written using the notation (—1)S’ where Sp indicates the number of operator exchanges, i. e. particle exchanges, needed to put the operator a; into conventional order. 149 A similar situation arises when an annihilation operator is applied to a multi-electron state. The anticommutation relations yield anal. = -a.I..a.. m ¢ n (8) anan=1- anan 111:1]. Consider the effect of the annihilation operator a. on the examples in Equation (7), a.|n...n.. mu.) a.a...a..a.aslg> (-1)aLa.ana. aslg) (-1)2 amana.a.a.I@) If} =(-l)2 aman(1—aral’)a$Ig>: amanas|®>+(-1)3amanararaslg) (9) =amana3|$>+(-1) a...a..a.a.a.|@)= a...a..a.|®>+0=lnmn..n.) a.n...n..n,.n.>= aramanapa.I@)= (- l)ama,anapa,|®)= (-1)2 amana.a,.a.IQ> =(-1)3 amanaPaYaTI®)= (__31) amanap(1-arar)lg> (10) = -a...a..aplg> + 0 = -In...n..np> where a,|®> = 0. Therefore, the same set of operator exchanges is needed when determining the sign of the multi-electron wavefimction when an annihilation operator is applied. The effect of a creation operator upon an arbitrary occupation number wavefunction can be written as aXInrnzmm‘“) = (...1)Sllmm...m +1...) m = 0 (11) The restriction on the occupation number m = 0 should be noted since the application of the operator gives zero when m = 1, i. e., the complete destruction of the multi-electron state. The effect of an annihilation operator upon an arbitrary occupation number wavefunction can be written as a‘lmnz...n‘...> =(-1)S"|n1n2'°-n..-1'°°) n‘ =1 (12) Similarly when 11,, = O , the annihilation operator gives zero. 150 The matrix element of an arbitrary single-electron number-conserving operator can be calculated using Equations (11) and (12). (n'.n'2°"n'x'°'n'i'“Ialaxlmnzmmmm'°') = (—1)S“S"(n'.n'2'"n'..°°'n'i'“Innnzmnx-1'°'m+1'"> (13) _ SA’SK — (-1) 8n'1-m 8n'z-nz ' ' .8n".n.¢—l . . 'Sn'xmrl ' . ' Note that (—1)S“S" in Equation (13) has been written identically as (—1)S*'S" . 151 APPENDIX D: RELATION OF POLARIZATION PROPAGATOR DERIVATIVE TO POLARIZABILITY DENSITY DERIVATIVE The derivation relating the derivative of the polarization propagator to the first-order nonlinear polarization propagator is suggested by the relationship found by Hunt et a1.” between the nuclear-coordinate derivative of the polarizability and the hyperpolarizability density, aaw(f;f',m)/6Rf = Idf"Bw5(f;f',co,f",0)ZKT51(f”,RK). (1) The hyperpolarizability density can be expressed in terms of the quadratic polarization propagator. amazemwuw")=[pat]:[w’).]:[p(r'),]:11:22:20»um"). <2) There are no linear polarization propagators in Equation (2). The purpose of this appendix is to demonstrate that the linear polarization propagators'that arise from taking the derivative of the polarizability density cancel each other to yield an expression with only the quadratic polarization propagator. The polarizability density written in terms of the polarization propagator is. aus(ffr":w) = -A§,‘§,<%|P(f).li<>r1¢:fi3(w) (3) 152 The derivative of the polarizability density with respect to any arbitrary parameter is WWW) ( K %p(f).lx>n:::1(w)+among““WVWIW \ ' K, (4) 2 ;. +0»!p(f).lx>‘a%'v(f').lx'>ntzii(w)+Ifiii3(‘°) an - . -. ,6rli:fi3(w) K+<>~|p(r),lx)<>~ |p(r )..|K >—--——a,1 ) Upon substitution of the polarization propagator derivative, the polarizability density becomes aaaafifi'w) an 2 2 {%p(f>.lx><>~'lp(f').lx'>r1256»)+.%">n:::1(w) LN K.K' +<>~lp(f).lx>an:::i(w)+<>~'Ip(f').a—an9n::£(w)} <5) -,2,20.1p.lx><>ulp(r').lx'> 2[23,131n2;2:;::(w,0)+§,c....n2::(w)+,2,c...n22(m) . + .5... C2. Ht? (0)) + 3.. Ca 113:5 (0)) The derivative of the single electron wavefunctions in Equation (5) can be expressed as a sum-over-states. %= m‘kcmlno (6) 153 Substitution of Equation (6) into Equation (5) yields 6a....(f.f';w) ',2,c;m ““"(‘°) _+n§,,cx'm<>~’lp(f’).lm>J ‘1 2%] 11;:22 :22 (a) 0) :2. 2.2122222wax)x 2+ 2 21222122 2 c.2222) ~— . + 2 C;.n.’::'t(w)+ 2:car1t::(w) (7) J. V o After simple rearrangements of coefficients and summations, Equation (7) becomes aaaagrf'm) = 322:2 ‘2 31Cg(m|p(r)a|x)(w|p(r')B|x')2(a)) -12,‘2,tn2xcm<>~lp(f),lm>rI.t 2(a)) —AgggycivmwP(f).|K> > —l§.'x.zx'm§flcrm<>~'|p(r'..)K>[%n‘9]:n:3:1222(20) (8) -,§,,§,m§,cm<2'Ip(r'). lK' >112? (2)) 32,2, 3,,cm<2|p(f).lK><2'lp(2 ).IK >n2:2(w) -A213,,,§,c22<>~'|p(f').|K'>n::t'(w) — 2 2 2 Chmp(f).|K><>~'|2(2'),|K'>nt:21(w)- 1,1’ K,K' km: 154 Since A, N, K, K', m and k are dummy indices, they can be reassigned different labels. In the sixth term, let m be interchanged with 7. (m <—> 72). In the seventh term, In (-> N. In theeighthterm,k—>K’andx'—)m. Intheninthterm,k—>xandr<—>m. 2222:2222)=-3333,E2Ec;..(m|2(2).)K><2'Ip(2').IK> 2122(2) 2(2) -3,2,2,cm<2|p(2).lm><>»'|p(f').lK'>n2 .2 —2,2,2,2.2212(2).)K>(m(p(2'),IK>n2 2(2) 1 i, -,§,,§,,,§,,Cm<2lp(f),lK)<2'l2(2'),lm>n2.23(w) -,§,,2,,,2,,<2|p(f).lK><2'|p(2’),lK')[-al‘9]:r1:221:21(2 o) (2) EE -,2,,2,,2,c2(m)p(2).K | 22 |p(r ),IKn 2(2) >122 -,2,,2,,_2,,c...(2)p(2)K l K>n2 2(2) > mu): > - 2 2 2 CI...)(7~lp(f) IK (7V |P(r'mrl)..| 323(0)) - 2 2 2 Ec...<>»lp(r) lm><>~' 19(2')..|K' 1125(0))- Common quantities can be factored from terms (1 and 6), (2 and 9.), (3 and 7) and (4 and 8) in Equation (9). 22222.2) =— 2 2 2 [C;m+C...](mlp(f),lK)(?~'|P(?').|K')Hizfi3(w) 322ng >3 E[c...+c;..](llp(r),|m m><72 '|P(f').|'<')flfi:23(°3) _ 2 2 2 [c;.m+c,,.]().|p(r),| x (m|p(r'),|x')n:;::(m) (10) > -2 2 2E[me+C;2]<’~|P(r),K|><1|P(r).|m> 1125(0)) -,2 ,2, ,2, (2(2) l K><2'Ip(f').lK {2,22 ]: r22 :2 (2 0) 155 Because of the orthonormality condition placed upon the perturbed wavefunctions Cu, + CL; = O , the sum of coefficients in the [] brackets is zero.3 Therefore, 6222222“) = -,2,,§,,2,<2I2(2).IK>(2I2(22).IK'>[%] n22;2:(2,o) (11) Thus, the derivative of the polarizability has no dependence on linear polarization propagators. ‘ K. L. C. Hunt, J. Chem. Phys. 90, 4909 (1989). 2 K. L. C. Hunt, Y. Q. Liang, R. Nimalakirthi, and R. A. Harris, J. Chem. Phys. 91, 5251 (1989) 3 Chapter 2, pg. 32. 156 "IT'EHEIHMMT'L'"l“lflifl'lfl'flmEET