V.“ 1.331, a... .7 ILLS}... THESlS nsmr ’uemms i llllllllllllllll‘llllllll‘ll llllllllll 3 1293 014172 This is to certify that the dissertation entitled Nonlocal Polarizability Densities and Molecular Softness: New Results for Electromagnetic Properties and Intermolecular Forces presented by Liu, Pao—Hua has been accepted towards fulfillment of the requirements for Ph.D degree in _C.b.emistr_y_ (L0 W Major professor Date July, 11, 1995 MSU is an Affirmative Action/E1; ual Opportunity Institution 0-12771 LIERAHY Michigan mate 4 University PLACE ll RETURN BOX to romovo this checkout from your record. TO AVOID FINES Mum on or More dot. duo. DATE DUE DATE DUE DATE DUE m 093531; 71"“! ' 6 nrrn MSU to An Affirmative Action/EM Opponmhy lm mm.- i——-————” 77 fi____ NONLOCAL POLARIZABILITY DENSITIES AND MOLECULAR SOFTNESS: NEW RESULTS FOR ELECTROMAGNETIC PROPERTIES AND INTERMOLBCULAR FORCES By Pao-I-lua Liu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1995 ABSTRACT NONLOCAL POLARIZABILITY DENSITIES AND MOLECULAR SOFTNESS: NEW RESULTS FOR ELECTROMAGNETIC PROPERTIES AND INTERMOLECULAR FORCES By Pao-Hua Liu By using nonlocal polarizability densities to characterize the changes in electronic charge density induced by molecular interactions, Dr. K. Hunt’s group has derived new results for dispersion, induction, and hyperpolarization forces. The first part of this thesis establishes that the nonlocal polarizability density theory meets the fundamental physical requirement for force balance between two interacting molecules A and B, order by order. Force relay plays an important role in this derivation, which stems from the application of Epstein’s force theorems: in a stationary state, the total force on the electrons is zero in fixed external fields. Thus when the electronic state adjusts adiabatically to a perturbation, the force of the external field on the nth order term in the electronic change density equals the force on the nuclei due to the (n+1)st order correction to the electronic charge density. The second part of this thesis rigorously relates electromagnetic properties and characteristics of molecular potential energy surfaces to the empirical concept of “soft- ness,” used to categorize Lewis acids and bases, and to summarize observed patterns of reactivity. New equations are derived that connect infrared absorption intensities, vibra- tional force constants, intermolecular forces at first order, and linear electric-field shield- ing tensors to softness kernels as defined in density functional theory. A generalization to nonlinear response--by introduction of the hypersoftness--leads to new equations in den- sity-functional terms for vibrational Raman band intensities, the cubic anharmonicities in molecular potential energy surfaces, intermolecular forces at first and second order, and nonlinear electric-field shielding tensors. The analysis employs relations of the softness and hypersoftness to nonlocal polarizability and hyperpolarizability densities that repre- sent the intramolecular distribution of response to applied electric fields. My dear God my parents, Hong-Ko and Chiao-Fong My brothers, Pao-Min and Pao-Kuo My friend Xiao deng-deng ACKNOWLEDGMENTS I would like to thank my research advisor Dr. Katharine Hunt for her guidance and encouragement throughout my graduate studies and during the work that led to this thesis. My thanks are also given to the members of my guidance committee, Dr. Robert Cukier, Dr. Daniel Nocera and Dr. James Jackson for their time and kindly help. My thanks are extended to the members of Hunt’s group and my officemates of Room 18 for their friendship. I wish I will have a husband to give my thanks to.... TABLE OF CONTENTS CONTENTS PAGE Chapter I. INTRODUCTION Chapter II. INTERMOLECULAR FORCES WITHIN NONLOCAL POLARIZABILITY DENSITY THEORY 2.1 Nonlocal Polarizability Densities 2.2 The Interaction Energy 2.3 Intermolecular Forces Obtained from Nonlocal Polarizability Densities 2.4 Adiabatic Approximation 2.5 Rigid Translation Chapter III. FORCE RELAY AND FORCE BALANCE 3.1 Force Relay 3.2 Force Relay within the Nonlocal Polarizability Density Theory 3.3 Force Relay and Force Balance a. Force Balance at First Order b. Force Balance for Second-Order Induction Forces c. Force Balance for Second-Order Dispersion Forces Chapter IV. MOLECULAR SOFTNESS FUNCTIONS 4.1 Introduction a. Density Functional Theory vi 21 23 30 3O 33 39 50 50 b. Chemical Hardness and Softness c. Density Functional Theory of Chemical Hardness and Softness (1. Local Quantities and Nonlocal Quantities 4.2 Relation between Molecular Softness and Nonlocal Polarizability Densities 60 4.3 Density of States and Nonlocal Polarizability Densities for Metals at Absolute Zero 71 CHAPTER V. MOLECULAR SOFT NESS, INFRARED ABSORPTION, AND VIBRATIONAL RAMAN SCATTERING: RELATIONS DERIVED FROM NONLOCAL POLARIZABILITY DEN SITIES 78 5.1 Vibrational Force Constants and Anharmonicities in Terms of Molecular Softness 78 5.2 Electric Field Shielding Tensors, Infrared and Vibrational Raman Band Intensities in Terms of Molecular Softness 85 5.3 Interaction Energies and Intermolecular Forces in Terms of Molecular Softness 88 vii Appendix A Appendix B Appendix C viii 96 98 101 CHAPTER I INTRODUCTION This thesis contains two major parts, the first is concerned with intermolecular forces and force relay, the second is concerned with electromagnetic properties such as force constants, infrared and vibrational Raman band intensities, and nonlinear electric- field shielding tensors. Both parts employ nonlocal polarizability density theory. As two interacting molecules approach each other, the distributions of charge and polarizability within molecules begin to affect the interaction energy before the electronic charge clouds overlap; hence representation of the molecules as point-polarizable multi- poles does not suffice. Distribution effects are expected to be particularly important for large planar or rodlike molecules, in configurations where the distances between nearby, but nonbonded nuclei in the two distinct molecules are smaller than the distances between many of the nonbonded pairs of nuclei in a single molecule. Hunt has developed a theory of molecular interactions that uses nonlocal polarizability densities [1-6] and nonlocal, nonlinear susceptibility densities [6] to incorporate the distribution effects. The nonlocal response tensors or (r, r’;—(:), (1)) and B (r, r’, r”;—0)o, to], (02) determine the intramo- lecular charge shifts induced by an applied electric field acting on a molecule with a fixed number of electrons N; their values reflect the distribution of polarizable matter within a molecule. Within linear response, or (r, r’;-(t), 0)) gives the m-frequency component of the polarization P (r, 0)) induced at point r in the molecule, due to an external field 86’“ (r’, (0) acting at point r’. The lowest-order hyperpolarizability density [3 (r, r’, r”;—(uo, (01, (1)2) gives the (no-frequency polarization induced at r by the con- certed action of the external field Sen (1", (01) at r’ and 8m (r”, (02) at r”, with 2 (00 = (i)1 + (.02. Thus, when an external field 86“ (r, 0)) is applied to a molecule, the electronic polarization PInd (r, (0) induced in the molecule satisfies P (130)) = P0(r, co) + Pind (r, (1)) ext = P0(l‘,(t)) +Jdr’a(r,r’;—(o,to) ~33 (r’,(i)) CXI (r II, w!) 1 oo +§I_wdw’Jdr’dr”B (r, r’, r”;—w, a) - (0’, (0') 23m“ (1", (D - (0') 3 +.... (1) The polarization Pind (r, (0) is related to pind (r, 0)) , the induced change in electronic charge density in the field Sc“ (r, (u) , by V - Pind (r, (o) = —pind (r, (0). (2) One purpose of the first part of this thesis, Chapters II and III, is to prove that the forces on interacting molecules A and B are equal and opposite, order by order, within the nonlocal polarizability density theory. An explicit proof is needed because of the differ- ences in the molecular properties that determine the forces on A and B, at each order in the interaction. For example, the first-order force on nuclei in A depends on the unperturbed charge density pg (r) of molecule B and on the first-order change pf’ A (r) in the elec- tronic charge density of molecule A [7]; thus it depends on the polarizability density of A. In contrast, the first-order force on nuclei in B depends on p3 (r) and p? B (r) , and hence on the polarizability density of B; but for distinct species A and B, there is no rela- tion between (1A (1', r’;—(i), to) and OLB (r, r’;—to, to) . As a second example, the disper- sion force on nuclei in A depends solely on the second-order, dispersion-induced change 3 pg’dA (r) in the electronic charge density of molecule A, while the force on nuclei in B depends on pg’dB (r) alone [8]. The quantity pg’dA (r) is determined by the hyperpolariz— ability density of molecule A; similarly for B [8], and the nonlinear susceptibilities of dis- tinct species are unrelated. Identical results for the forces at each order can be obtained by differentiation of the standard perturbation expression for interation energies, or by use of the Hellmann-Feynman theorem [9,10] for intermolecular forces [7, 8, 11]; therefore, the analysis given here has general applicability beyond the nonlocal response theory. Hunt and I [12] have found that force balance is derivable from “force relay,” a physical effect occurring when the electronic state of a molecule or group of molecules 6’“ (r) acts on a adjusts adiabatically to perturbations [13]. When a fixed external field 3 molecule, the force that 8w (r) exerts on the nth order term p: (r) in the electronic charge density is relayed in full to the nuclei by the (n+1)st order change p: + l (r) [13]. If the external field itself varies as the electronic charge density changes—for example, due ext n to molecular interactions—force relay takes a modified form. The change AF in the external force on the electrons is passed on to the nuclei by the (n+1)st order change in the electronic charge density; here ' n Ari“ = 2W1. lIdrpe(r)3m(r) I‘I’n_k ). (3) k = 0 In Eq. (3), ‘Pk is the kth order term in the normalized, perturbed wave function, including the field source, and be (r) and 3m (r) are operators. For fixed external fields, the force relay condition has been stated previously by Epstein [13]. Chapters II and III generalize the condition to cases in which the electronic charge density and the perturbing field are correlated. In Chapter II, the results from the nonlocal polarizability density theory for the first- and second-order forces on the nuclei in interacting molecules A and B [7, 8, 11] are summarized throughout Sections 2.1 to 2.3 and the need for an explicit proof of force bal- 4 ance at each order is shown. The key equations related to the adiabatic approximation are derived in Section 2.5 within the nonlocal polarizability density theory. In Chapter 1H, the force relay condition based on Epstein’s force theorem [13] is introduced in the first section. Section 3.2 prOveS that the force relay condition is satisfied within the nonlocal polarizability density theory, for fixed external fields. This is a conse- quence of interconnections among permanent charge densities, linear response, and non- linear response that Hunt’s group has derived earlier [14, 15]. Section 3.3 generalizes force relay to variable external fields and shows that force balance is a consequence, within the nonlocal response theory. Chapters IV and V link a set of molecular properties including infrared intensities, electric-field Shielding tensors, and harmonic force constants to softness [16] as defined in density functional theory [17, 18]. Expressions for these molecular properties have been derived previously [14, 19] in terms of nonlocal polarizability densities 0t (1', r’;—to, (o) . Here, the connection to molecular softness is established by relating a (r, r’;O, 0) to the softness kernel 5 (r, r’) [16]. A similar relation between the hyperpolarizability density and a “hypersoftness” o (r, r’, r”) is introduced in this work, in order to express vibra- tional Raman band intensities, nonlinear shielding tensors, and cubic anharrnonicity con- stants in density functional terms. New connections are also established between s (r, r’) , 0(1', 1", r”) , and long-range intermolecular forces. Within density functional theory, the softness kernel and the hypersoftness charac- terize the response of the electronic charge density to external perturbations. These func- tions quantify [16, 20-24] the concepts of chemical hardness and softness used by Pearson [25, 26] to categorize Lewis acids and bases. Empirically, numerous reactivity patterns are summarized by the statement that hard acids “prefer” to react with hard bases, both ther- modynamically and kinetically [26-27]. Typically, soft acids are large and highly polariz- able species, with low positive charges. Soft bases are highly polarizable, easily oxidized, and low in electronegativity. The opposite properties hold for hard acids and bases [25, 4 ance at each order is shown. The key equations related to the adiabatic approximation are derived in Section 2.5 within the nonlocal polarizability density theory. In Chapter III, the force relay condition based on Epstein’s force theorem [13] is introduced in the first section. Section 3.2 prOves that the force relay condition is satisfied within the nonlocal polarizability density theory, for fixed external fields. This is a conse- quence of interconnections among permanent charge densities, linear response, and non- linear response that Hunt’s group has derived earlier [14, 15]. Section 3.3 generalizes force relay to variable external fields and shows that force balance is a consequence, within the nonlocal response theory. Chapters IV and V link a set of molecular properties including infrared intensities, electric-field shielding tensors, and harmonic force constants to softness [16] as defined in density functional theory [17, 18]. Expressions for these molecular properties have been derived previously [14, 19] in terms of nonlocal polarizability densities (1(1', r’;—o), (n) . Here, the connection to molecular softness is established by relating or (r, r’;O, O) to the softness kernel 5 (r, r’) [16]. A similar relation between the hyperpolarizability density and a “hypersoftness” 0' (r, r’, r”) is introduced in this work, in order to express vibra- tional Raman band intensities, nonlinear shielding tensors, and cubic anharmonicity con- stants in density functional terms. New connections are also established between 5 (r, r’) , 0‘ (r, r’, r”) , and long-range intermolecular forces. Within density functional theory, the softness kernel and the hypersoftness charac- terize the response of the electronic charge density to external perturbations. These func- tions quantify [16, 20—24] the concepts of chemical hardness and softness used by Pearson [25, 26] to categorize Lewis acids and bases. Empirically, numerous reactivity patterns are summarized by the statement that hard acids “prefer” to react with hard bases, both ther- modynamically and kinetically [2627]. Typically, soft acids are large and highly polariz- able species, with low positive charges. Soft bases are highly polarizable, easily oxidized, and low in electronegativity. The opposite properties hold for hard acids and bases [25, 26]. The first section in Chapter IV is a general introduction of density functional the- ory and the concepts of chemical hardness and softness. In Section 4.2, the softness and hypersoftness kernels which also describe nonlocal response to external fields are related to the nonlocal susceptibilities at (r, r’;-(:), 0)) [1-5] and B(r, r’, r”;—(oo, (01, (1)2) [6]; the softness kernel is further generalized to a frequency-dependent form. The relation between harmonic force constants and the softness kernel 5 (r, r’) is derived in Section 5.1, which also shows that the cubic anharmonicity constants and the intensities of vibrational Raman bands depend on s (r, r’) and o (r, r’, r”) within the Placzek approximation [28]. In Section 5.2, the softness kernel 5 (r, r’) is related to the derivative of the molecular dipole with respect to a shift in nuclear coordinates; thus the softness determines both infrared absorption frequencies (within the harmonic approxima- tion) and infrared intensities; and the Stemheimer electric-field shielding tensors [9, 29- 33] are also related to the softness and hypersoftness. The Stemheimer shielding tensors give the difference between an external electric field 86’“ applied to a molecule and the effective field that acts at the nuclei, because these tensors account for the shielding or deshielding effects of the electronic redistribution induced by Sen. In Section 5.3, long-range intermolecular forces are treated in the density-func- tional framework. Earlier, Gézquez has expressed the second-order induction energy in terms of the softness kernel [34]; here it is proven that the force at first order depends on the softness kernel and the force at second order depends on the softness and hypersoft- ness. In Section 5.3, the dispersion energy is analyzed in terms of an imaginary-frequency dependent softness kernel, 5 (r, r’;-i(n, im) . References [1] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). [2] L. M. Hafkensheid and J. Vlieger, Physica 75, 57 (1974). [3] T. Keyes and B. M. Ladanyi, Mol. Phys. 33, 1271 (1977). [4] J. E. Sipe and J. Van Kranendonk, Mol. Phys. 35, 1579 (1978). [5] K. L.C. Hunt, J. Chem. Phys. 78, 6149 (1983). [6] K. L.C. Hunt, J. Chem. Phys. 80, 393 (1984). [7] K. L.C. Hunt and Y. Q. Liang, J. Chem. Phys. 95, 2549 (1991). [8] K. L.C. Hunt, J. Chem. Phys. 92, 1180 (1990). [9] R. P. Feynman, Phys. Rev. 56, 340 (1939). [10] H. Hellmann, Einfiihrung in die Quantenchemie (Deuticke, Leipzig, 1937), p. 285 [11] Y. Q. Liang and K. L.C. Hunt, J. Chem. Phys. 98, 4626 ( 1993). [12] P.-H. Liu and K. L. C. Hunt, J. Chem. Phys. 100, 2800 (1993). [13] S. T. Epstein, The Variation Method in Quantum Chemistry ( Academic, New York, 1974), Secs. 16, 18, and 19. [14] K. L.C. Hunt, J. Chem. Phys. 90, 4909 (1989). [15] K. L.C. Hunt, Y. Q. Liang and R. Nimalakirthi, J. Chem. Phys. 91, 5251 (1989). [16] M. Berkowitz and R. G. Parr, J. Chem. Phys. 88, 2554 (1988). [17] P. Hohenberg and W. Kohn, Phys. Rev. B 136: 864 (1964). [18] R. G. Parr, Ann. Rev. Phys. Chem. 34, 631 (1983). [19] K. L. C. Hunt, J. Chem. Phys.103, 0000 (1995). [20] R. G. Parr and R. G. Pearson, J. Am. Chem. Soc. 105, 7512 (1983). [21] R. F. Nalewajski, J. Am. Chem. Soc. 106, 944 (1984); J. Phys. Chem. 89, 2831 (1985); Int. J. Quant. Chem. 40, 265 (1991). [22] W. Yang and R. G. Parr, Proc. Natl. Acad. Sci. USA 82, 6723 (1985). 7 [23] RC. Parr and P. K. Chattaraj, J. Am. Chem. Soc. 13, 1854 (1991); P. K. Chattaraj, H. Lee, and R. G. Parr, J. Am. Chem. Soc. 113, 1855 (1991). [24] J. Cioslowski and M. Martinov, J. Chem. Phys. 101, 366 ( 1994). [25] R. G. Pearson, J. Am. Chem. Soc. 85, 3533 (1963). [26] R. G. Pearson, Science 151, 172 (1966). [27] For an example of applications in surface science, see L. M. Falicov and G. A. Somorjai, Proc. Natl. Acad. Sci. USA 82, 2207 (1985). [28] G. Placzek, Handbuch der Radiologie, edited by E. Marx (Akademische Verlagsge- sellschaft, Leipzig, 1934), Vol. 6, Chap. 2, p. 205. [29] R. M. Stemheimer, Phys. Rev. 96, 951 (1954). [30] P. Lazzeretti and R. Zanasi, Chem. Phys. Lett. 71, 529 (1980); J. Chem. Phys. 84, 3916 (1986); 87, 472 (1987). [31] S. T. Epstein, Theor. Chim. Acta 61, 303 ( 1982). [32] P. Lazzeretti, E. Rossi, and R. Zanasi, J. Chem. Phys. 79, 889 (1983); P. Lazzeretti, R. Zanasi, and P. W. Fowler, J. Chem. Phys. 88, 272 ( 1988). [33] P. W. Fowler and A. D. Buckingham, Chem. Phys. 98, 167 (1985). [34] I. L. Gézquez, in Chemical Hardness, Structure and Bonding, Vol. 80, K. D. Sen, Ed. (Springer-Verlag, Berlin, 1993) pp. 27-43. CHAPTER II INTERMOLECULAR FORCES WITHIN NONLOCAL POLARIZABILITY DENSITY THEORY 2.1 Nonlocal Polarizability Densities Maaskant and Oosterhoff introduced nonlocal polarizability densities in a study of optical rotation in condensed media [1]. They gave the nonlocal polarizability density in a sum-over—states form, with each matrix element itself given as an infinite series. Hunt [2] derived a simple form that permits practical calculations in cases when the field acting on a molecule is derivable from a scalar potential. When an external field 8”" (r, (1)) is applied to a molecule, the electronic polar- ization PInd (r, 0)) induced in the molecule satisfies P(r,a)) = P0(r,to) +Pi"d(r,(t)) ext (rl, (D) = P0(r,0)) +Idr’a(r,r’;—a),m) -S 1 °° I I II I II I I C“ I I 6’“ II I +-2-I_°°d(DIdl'dl' [3(r,r,r ;—(o,(o-0),u)):8 (r,(o—(i))3 (r,0)) +.... (I) The polarization Pind (r, (o) is related to pind (r, (1)) , the induced change in electronic charge density in the field 8”“ (r, 0)) , by V-Pi"d(r, (1)) = —pi"d(r, (1)), (2) and the same relationship holds for the polarization and charge density operators. From Eq. (1) one can see that the nonlocal polarizability density a(r, r’; —(i), (1)) determines the electronic polarization Pind (r, (1)) induced at point r in a molecule by an external electric field of frequency (1) acting at r’, within linear response. It is a fundamental molecular property which reflects the distribution of polarizable matter within the molecule. The polarizability density for a molecule in the ground state has the form (10,50, r’; —a), w) = < 0 I Pam C(co) P30“) 0) + < 0 | PB(r’) G(——(i)) Pa(r) lo ), (3) where 6(a)) is the reduced resolvent operator 6(a)) = (I — no) (H — 50 mm)“ 0— no). (4) and {J0 is the ground-state projection operator '0 ) ( O l. Hunt [3] has shown that or(r, r’; -(i), to) also determines the net field 3] acting on nucleus I of a molecule in a static, external field 8°“ (r) : SI : 31(0) +8ext(Rl) +Jdrdr’T(RI,r) o0t(r,r';0,0) -Sm(r’)+.... (5) SI (0) where is the field at nucleus I in the absence of the external perturbation, and T at} (R', r) is the dipole propagator, i.e., TaB(Rl,r) = VaV lR‘-rl"1 10 = [3 (Rl—r)a(Rl—r)B—8aBlRI—ri2]/iRl —ri5- 4355,3230 — R'). (6) The hyperpolarizability density [3(r, r’, r”; —0), 0) — 0)’, 0)’) represents the nonlinear re- sponse of lowest order; it gives the 0)-frequency polarization induced at r by the concerted o t I I t I I 1’ 0 action of the external fields Sex (r', 0) — 0) ) at r and Sex (r ’, 0) ) at r . Perturbation analysis gives the hyperpolarizability density in the form [3045103 r’, r”; —0)0, (01, (1)2) = 80371 <0 | Par) G(wo) Mr”) G(wl) Pg(r’) IO) + ( 0 | P70") G(—0)2) P“B(r’) G(— mo) Pa(r) I 0 > + < 0 I 1w") G(-w2) Imam G(wr) P30") |0 > 1. (7) where the operator 5037 denotes the sum of all terms obtained by perrnuting PB(r’) and PY(r”) and simultaneously perrnuting the associated frequencies ml and (02, in the expres- sion following the operator; and too = 0)] + $2. The operator P¢a(r) is defined by P¢a(r) = Pa(r) — < 0 l Pa(r) 10). The tensor densities 004303 r’; —0), 0)) and [SGML r’, r”; —0)0, 0)], 0)2) both repre- sent the distribution Of polarizable matter on the intramolecular scale. (10430, r’; —0), 0)) has applications in theories of local fields and light scattering in condensed media [4], and in approximations for dispersion energies [2], collision-induced dipoles, and collision- induced polarizabilities [2, 5] of molecules interacting at intermediate range. [304MB r’, r”; -0)O, 0)], (02) and the dipole propagator determine the derivatives of the polarizability density with respect to nuclear coordinates [3, 6], I I II I II damn, r ;—0), 0)) /8Ra =I dr [3375 (r, r , r ;—0), 0), 0) ll 1 ,, I XZ T8a(r,R). (8) where Z1 is the charge on nucleus 1, and RI is the nuclear position. An analogous equation also holds for any two adjacent-order polarizability densities. The higher-order nonlocal polarizability densities can be expressed in terms of a general nth order nonlocal susceptibility density x (n) defined by [2] X;7;2...an+l (r,r’, r(")’;-0);0)-0)’— ...m("' ”20)’, 0)”, 0)"' ") P r P r’ ...[P r(")’:l 1S 2 I: a,( )JOII[ 02( )]1112 a"+'( ) 1.,0 — 7 I II (ti-l), ("—1), ’ f1 1'12m,n(0)110—0)) (0)120—0) —0) —...0) )...(0),nO—...0) ) (9) In Eq. (9), the sum over intermediate states 11,12,...,ln runs over all electronic states including the ground state, and S represents the sum of all terms generated by pennuting P a: (r) , P (12 (r') , P +1 (rn’) and applying the same permutation to the frequencies a 1'1 —0);0)— 0)’ — ...0)("_ ”’,0)’, 0)”, ..., 0)"- 1’ in the expression given. 12 2.2 The Interaction Energy When molecules A and B interact, the net force FI on nucleus I in A is the sum of the force F 1(0) on I in the absence of molecule B and an interaction force AF 1. The interac- tion-induced force is related to the AB interaction energy AE by AF' = —aAE/aR‘a, (1) a where RI is the position of nucleus I. In this section I will show that the first-order and sec- ond-order interaction (energy separated into the induction energy and the dispersion energy) can be expressed in terms of nonlocal polarizability densities. At long range, perturbation techniques are suited for calculating the interaction energy order by order [7]. In Rayleigh-SchrOdinger theory the interaction energy AE can be expanded as a series in the perturbation Hamiltonian H' [2, 8-12]. For molecules inter- acting without appreciable charge-cloud overlap, H' is given by AA AB _l H’ = [p (r)p (r')lr—r'| drdr’, (2) AA AB where p (r) and p (r’) are the charge density operators for molecules A and B, respectively ; A . f) (r) = Ze5(r—rj)+2218(r—RI); (3) j I the sum over j runs over the electrons assigned to molecule A, with position operators rj, and the sum over I runs over nuclei in A with charges ZI and positions RI. The first-order interaction energy AB“) depends upon the permanent charge densi- 13 ties pAO (r) and p300“) of the unperturbed A and B molecules: 1 , AE( ’ = (‘I’OIH I‘I’O) A B I I ’1 I =Ip O(r)p O(r)Ir—r| drdr, (4) where ‘PO is the product of the unperturbed molecular wave functions of A and B, and PAoU') = <‘P,’,‘|pA mlwg‘) (5) The change of energy for the system at second order is A g (2) A B , A B A B , A B A AB = - Z 0113‘}!ng Wk \ngwk ‘1'ng I‘I’g ‘I’g )/(Ek —E ) katg A B , A B A B , A B B B — 2 (‘1’8 ‘1’ng Iwg ‘1’“)(‘I’g ‘I’ulH I‘vg ‘I’g )/(Eu —Eg) Uig A B , A B A B , A B A A B B — 2 (‘I’g‘I’ngI‘Pk‘I’u)(‘I’k‘I’uIHI‘I’g‘Pg)/[(Ek —Eg) +(Eu-Eg)] u,k¢g (6) The first term of the second—order perturbation energy corresponds to the permanent moments of B polarizing A. The induced moments of A then interact with the permanent moments of B. The second term corresponds to the permanent moments of A polarizing B. The third term corresponds to the second-order dispersion energy. The total interaction energy for molecules at long range is the sum of the induction energy Nimind (the first and second terms) and the dispersion energy AE(2)disp (the third term). The induction energy is determined entirely by the first—order, linear response of each molecule to the 14 field of its neighbor. The dispersion (van der Waals) energy results from dynamic reaction field effects, due to correlations of the spontaneous, quantum mechanical fluctuations in charge density on the interacting molecules [2, 8-12]. By using the definition of the nonlo- cal polarizability densities the first two terms can be rewritten as: AE(2). d = —1/2Jdrdr’aAaB(r,r’)380a(r)SBOB(r’) 1n -1/2jdrdr’aBaB(r,r')8A0a(r)8AOB(r'), (7) where SAOGU) is the OL component of the unperturbed field of molecule A and 0L0!B (r, r’) denotes the static susceptibility aaB (r, r’;0, 0) . The third term can be writ- ten as AEmdisp = —(h/41t2)Igdwjdrdr'dr'drmaAaB(r, r"';—i0), i0)) B H I._' ' I H H! xa 75(r ,r,10),10))Ta5(r,r)TYB(r ,r ). (8) Equation (8) is equivalent to that obtained from reaction-field theory [2, 13-15] and the fluctuation-dissipation theorem [8, 9, 16-23] 15 2.3 Intermolecular Forces Obtained from Nonlocal Polarizability Densities For interacting molecules with weak or negligible electronic overlap, within the Bom-Oppenheimer approximation, the forces acting on nuclei derived from the Hell- mann-Feynman theorem are equivalent order by order to those obtained from standard perturbation expressions for the interaction energy [see Section 2.2] by differentiating with respect to nuclear coordinates [24-26]. At first order, the force AF]l a on nucleus I in molecule A is determined by —aAE (”/aR'a AF'La a _ = —Jpa Aer( ——)RpBO(r’ ’)Ir—r’l ldrdl", (I) where pAO (r) and p30 (r’) are the expectation values of the charge density operators A B I) (r) and f) (r’) for the unperturbed molecules A and B [see Eq. (2, 2, 3)] ..A Ae,A p (r) =p (r)+22‘8(r—R'> l = ze6(r—rj)+2z'8(r—R'), (2) j e. A f) (r) is the electronic charge density operator for molecule A, the second term of the right-hand side of Eq. (2) is the nuclear point charge distribution, the sum over j runs over the electrons assigned to molecule A, with position operators r-, and the sum over I runs over nuclei in A with charges ZI and positions RI. The differential of the electronic charge density of molecule A is related to the nonlocal polarizability 0370, r’) [3] 16 ape,A0(r) -_ aVBPQBe’A (l‘) aR'a aR'a = —VBZIIdr’a€Y(r, r’) T70! (r’, R') , (3) where To”3 (RI, 1') is the dipole propagator: Tap (RI, r) = VaV IRl - rl_1. Greek sub- scripts denote Cartesian tensor components, and the Einstein convention of summation over repeated Greek subscripts is followed throughout. Substitution of Eqs. (2) and (3) into Eq. ( 1) shows that the first-order force is determined by the unperturbed charge den- sity pg (r) of molecule B, and by the first-order interaction-induced change in the elec- tronic polarization of molecule A, Pig (r) [24] -3 B AFLG = Z'J(RI—r)a|RI—rl p 0(r)dr +ZIJ‘TaB(Rl,r)Pig(r)dr; (4) where Pig”) = Jdr’aABy(r,r’)SgY(r’), (5) in terms of the field 83 Y (r’) due to the unperturbed charge distribution (electronic and nuclear) of molecule B. Similarly, the first-order force on nucleus J in molecule B depends on orgy”, r’) AFJ ZJJ(RJ—r)a|RJ—ri—3pA0(r)dr 1,0- +z’jTaB(R’,r)Pj:g(r)dr; (6) l7 and Pig”) = Idr’aBBy(r,r’)33Y(r’). (7) The net force on molecule A is obtained by summing overI and the net force on B by summing over I. Since the polarizability densities of A and B are unrelated (in general), force balance is not evident from a simple comparison of Eq. (4) with Eq. (6). At second order, the force is determined by differentiating the second-order inter- action energy which is the sum of the induction energy ABS: and the dispersion energy AB (2) [see Section 2.2], disp AF -8AE (2’ /aR'a 2,0 = -8(AEi(n2c: +AE335’p) /8Rla I I = AF 2, a,ind+AF 2,0t,disp' (8) The second-order induction force is A 8a (r r’) I , i B B , AF 2.a,ind =1/2jdrdr ‘37] 3 OB(r)Ss 07(r) an a A A a 3 (r)8 (r’) +1/2Jdrdr’aBBY(r,r’) [ DB 1 07 ].(9) an 0, Hunt et al. have shown that linear and nonlinear response are related by [3, 6] 8a (r r’) 57 ’ _ I II I II II I T _ zjdr emu; ,r )T8a(r ,R ), (10) a 18 and the differential of the field of the unperturbed molecule A is BSAOBU) rilapAOO’) aRI (I = —]dr'v Ir— (11) B {R (l Substitution of Eqs. (2) and (3) into Eq. (11), and Eqs. (10) and (11) into Eq. (9) shows that the second-order induction force AFIZ, a, ind is determined by P? g (r) and Pg’g ind(r) assuming that nucleus I is in molecule A; therefore it depends on BA (1., rl’ r”): 1 AF 2,0,,“ = z'jdrTaB(R',r) [szgm +P;:g,ind(r)], (12) where 9 I II III A I I II Fianna”) = Jdr dr dr aBY(r,r )TY5(r ,r ) B II III A III x083” ,r >304)” ) l +2ldr'd'"l3375(r. I", r")3807(r') 3805(r")- (13) The second-order induction force on nucleus J in molecule B has a similar form to Eq. (12). J AF Mind = z’jdr'raflmir) [Pj;g(r) +P;’,g’ind(r)]; (14) Eq. (14) depends on BB (r, r’, r”) , but Eq. (12) depends on BA (r, r’, r”) , so again force balance is not evident. The dispersion force on nuclei in A comes from the second-order, dispersion- induced change in the electronic polarization of A itself as can be proven by differentiat- l9 ing the second-order dispersion energy with respect to nuclear coordinates; (2) AE disp = —(h/41t2) Izdwjdrdr'dr'drmaAaB(r,r"';—i0),10)) B II I._‘ ' I II III x0t 75(r ,r ,10),10))Ta5(r,r)TYB(r ,r ). (15) By use of Eq. (10) generalized to the imaginary-frequency dependent polarizability den- sity, the dispersion force is obtained in the form 1 (2) 1 AF 2,0t,disp _ _aAEdisp/aR or Z'jdrTaBm‘, r)P;"3,disp(r). (16) e.A The polarization P2 B disp (r) is determined by the field from the fluctuating polarization of molecule B, and by the hyperpolarizability density of A, taken at imaginary frequencies [25] C,A _ 2 N I II III IV P2,B.disp(r) _ —(h/41t )jodwjdr dr dr dr x [39w (r ', r ", r;-i0), i0), 0) Tye (r ", r "’) B III . ' ° . ' XOL E:§(r ,r'v;—10),10))TC5(rw,r ). (17) The dispersion force on nuclei J in molecule B can be derived in the same way and has a form similar to Eq. (16), J _ (2) J 20 = zjjdr'rafim’,upggdispm. (18) The polarization P; g, disp (r) is determined by the field from the fluctuating polarization of molecule A, and by the hyperpolarizability density of B taken at imaginary frequencies. The hyperpolarizability density of B is unrelated to the hyperpolarizability density of A, in general. 21 2.4 Adiabatic Approximation In quantum mechanics an adiabatic process is a process for which the change of the Hamiltonian versus time is sufficiently slow to avoid quantum jumps from one eigen- state of the instantaneous Hamiltonian to another [27, 28]. If the external perturbing potential is small enough that perturbation theory can be used, it is possible to extend this treatment to a more general problem in which the perturbing potential undergoes a large change, but over a long period of time. Bohm [29] derived the condition that a process is adiabatic if the rate of change of the perturbation V is slow compared to the separation AB of the initial state from the neighboring states, in the sense that the expectation value of dV/dt must obey dV/dt « (AE) 2m. It is possible to perturb the electronic states of a molecule by slowly bringing another molecule near the first. Davydov [30] has defined an adiabatic collision as fol- lows: if the effective collision time is appreciably larger than the period 0)‘ lnm which char— acterizes the electronic energy spectrum, the collision is electronically adiabatic; in other words, in an adiabatic collision the inequality (0an A) » 1 is satisfied, where D is the distance of closest approach and 1) is the velocity of the approaching particle (approxi- mated as constant). In Section 2.3 the adiabatic energies of two molecules are discussed, based on the assumption that the nuclei move sufficiently slowly compared to the elec- trons that the force between the molecules can be calculated by differentiation of the adia- batic interaction energies [31]. Using the adiabatic approximation, one can assume that the wave function at any instant of time is nearly equal to that which would be obtained if dV/dt were zero, and V were equal to its instantaneous value. Two different approaches can be used to calculate the wavefunctions and the expectation values of forces. The first is based on time-inde- pendent perturbation theory with a fixed static external perturbation. The second is based 22 on time-dependent perturbation theory [32], but the nonadiabatic term in the expansion for the wavefunction is subtracted out. Both methods are consistent with the force relay con— dition [33]. To illustrate the second method, let us suppose that the perturbation V(t) acts only during some finite interval of time. Let the system be in the nth stationary state before the perturbation begins to act (or in the limit as t—) —oo). At any subsequent instant the state of the system will be determined by the function ‘I’ = zakn‘l’fio) , where, in the first- order time-dependent perturbation approximation, ._ (I) __ i : imnfi' , akn — akn — —%I_kane dt for k¢n, (1) _ (1)_ i r , an" —1+ann —1_1—’1 -..,oVnndt' For a perturbation V(t) that tends to zero as t —-> —oo and to a finite non-zero limit as t —-) +00, an integration by parts in Eq. (1) gives “.0 I . I kn m) r . r kn z z ' ane 8an e V ‘mkfl d I t k" it -~ *" fin) -°° at’ fin) kn _°° kn dt’. (2) The first term vanishes at the lower limit, while at the upper limit it is formally identical with the expansion coefficients in time-independent perturbation theory; the presence of an additional periodic factor exp(imk,,t) is merely due to the fact that “kn are the expansion coefficients of the complete wave function ‘1’. At any instant of time, the first term simply gives the change in the original wave function Tum) under the assumption that V is a con- stant and equal to its instantaneous value, while the second term describes transitions into other states. The magnitude of the second term depends on Ban/Bt’, which can be neglected if it is small enough. 23 2.5 Rigid Translation The forces on nuclei in a molecule can be obtained by using the Hellmann-Feyn- man theorem [34, 35] as well as from standard perturbation theory by differentiation of the interaction energy with respect to nuclear coordinates [24]. Because the interaction energy at different orders within the nonlocal polarizability density theory depends on electronic charge densities, nonlocal polarizability densities, hyperpolarizability densities, and higher order susceptibility densities, it is necessary to know the derivatives of nonlocal polarizability densities with respect to nuclear coordinates in order to prove force relay and force balance. It is also necessary to determine the effect on nonlocal polarizability densities of a simultaneous shift in the coordinates of all of the nuclei of the molecule in question. This section focuses on simultaneous shifts of all of the nuclei. If the electrons adjust adiabatically to the nuclear motion, the whole molecule moves uniformly: At an arbitrary but fixed nuclear configuration, let pe(r) denotes the expectation value of electronic charge density operator be (r) , and p°s(r) denote the elec- tronic charge density after an infinitesimal shift, and peo(r) denote the original electronic charge density, and let 5R denote the displacement. P 0 SR 965 r-8R 24 Then pes(r) = peo(r - 8R). Because SR is very small, a Taylor expansion can be used to connect pcs(r) and pe0(r): P:(r) = p§(r—5R) = 980‘) -5R-fo,(r); (I) also 8 e p°(r) P:(r) = potr)+6R-2 ° . (2) I ,3]: where I runs over the nuclei, and V denotes differentiation with respect to the spatial coor- dinate r. Comparison of Eqs. (1) and (2) shows that the effect on the electronic charge density of a simultaneous shift in the coordinates of all of the nuclei of a molecule is deter- mined by the gradient of the charge density 303(r) aR‘ = —Vpg (r) . (3) t The individual terms on that left-hand side of Eq. (3) are nucleus-dependent and the right- hand side is purely dependent on spatial coordinates for the electronic charge density. The rest of this section shows how to build up an analogous connection for each nonlocal polarizability density. Equation (3) can be related to the nonlocal polarizability density 0t(r, r’) by using the relations p (r) = —V - P(r) and P§(r) = Idr’a55(r, r’) f,“(r'). (4) Then the left-hand side of Eq. (3) becomes 3930') _ 9VaP§(r> Sf,“ (r’), (5) where E,“ (r’ ) is the static field which 1s derived from the scalar potential (Dex (r’) , via the relation SBX (r’ ) = —V’Be x(r’ ). The derivative of 0t(r, r’) with respect to nuclear coordinates in Eq. (5) can be derived from the definition aBB (r, r’) (‘i’OIP5(r)|‘Pk)(‘I’k|PB(r’) ‘I’ > = (1+C)2’|: E _E O], (6) k k 0 where C denotes complex conjugation, and P is polarization operator. The prime denotes the summation over all excited states k, excluding the ground state ‘I’O. By using integra- . . ext ext . tion by parts and the relat1ons 8a (r) = —Vad> (r), p (r) = -V - P(r) , the r1ght- hand side of Eq. (5) can be expressed as below: 801 (r,r ’) -2V8 Jdr I 5___B______ SEXI(rI) 3R] (1 90,,(1‘) Pkg“) =—Jdr’Z(I+C)2’aa[ R‘ E_E Ra k 0 (r) 8 ’ + pox pko” ) ]¢cxt(r,). (7) ER ‘ Eo 8R; In Eq. (7), the derivative of the energy difference a (Ek - E0) /8RL between excited state ‘I’k and ground state ‘I’O summed over nuclear coordinates is zero, Equation (3) also 26 A C P (r) hand side of Eq. (3) into Eq. (7) and integration by parts give holds for the expectation value p0k (r) = (‘I’O ‘Pk ). Substitution of the right- [Vapgkmjpiom —Idr’(l+C)2’[ E —E k k 0 C I e I 90ktr1V upkotr) + E E ]¢m(r’). k— 0 VBIdr’[VaaBB(r,r’) + Va’a5B(r,r’) 1 SE“ (r’). (8) Equation (8) that shows the sum of the derivatives of the nonlocal polarizability density 0t (r, r ’) with respect to nuclear coordinates is determined by the gradients of the polariz- ability density: BOLBB (r, r’) = --V a (r,r’) —V '0t (r,r'). (9) I (1 SB (1 SB 1 3R (1 Similar steps can be used to find the effect on the hyperpolarizability density [3 of a simultaneous shift in the nuclear coordinates, which is given by 861435 (r’, r”, r) , an' a = —V’aBYB5 (r’, r”, r) _VIIaBYB8 (r', rll, r) 27 —VGBYB5(r’,r”, r). (10) In general, the sum of the derivatives of the nth order nonlocal polarizability den- sity with respect to nuclear coordinates is (n) I II (n+1)! ax 76”.“)(1' ,r ,...,r ) l a Z , an (n+1), I (n) I II = —V ax YB...(o(r , r ’ 0.0, r ) II (n) I II (n+1); —V ax YBmm(r,r ,...,r )— (n+l), (n) , ” (n+1), _Va x YB.,.w(r , r , 00-, r )0 (1]) In Eq. (1 l), x(1)(r, t") denotes 0t(r, r'), x(2)(r, r', r") denotes [3(1', r', r"), and x(3)(r, r', r", r'") denotes y (3)(r, r', r", r'"). Eq. (1 1) holds not only for the static response tensors, but also for the imaginary-frequency susceptibility densities. 28 References [l] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). [2] K. L.C. Hunt, J. Chem. Phys. 78, 6149 (1983); 80, 393 (1984). [3] K. L.C. Hunt, J. Chem. Phys. 90, 4909 (1989). [4] T. Keyes and B. M. Ladanyi, Mol. Phys. 33, 1271 (1977). [5] K. L.C. Hunt and J. E. Bohr, J. Chem. Phys. 84, 6141 (1986). [6] K. L.C. Hunt, Y. Q. Liang and R. Nimalakirthi, J. Chem. Phys. 91, 5251 (1989). [7] J. N. Murrell and G. Shaw, J. Chem. Phys. 46, 1768 (1967). [8] H. C. Longuet-Higgins, Proc. R. Soc. London Ser. A235, 537 (1956); H. C. Longuet- Higgins and L. Salem, ibid. 259, 433 (1961); H. C. Longuet-Higgins, Discuss. Faraday Soc. 40, 7 (1965). [9] A. D. McLachlan, Proc. R. Soc. London, Ser. A 271, 387 (1963); 274, 80 (1963); A. D. McLachlan, R. D. Gregory and M. A. Ball, Mol. Phys. 7, 119 (1963). [10] J. Linderberg, Ark. Fys. 26, 323 (1964). [11] J. N. Murrell and G. Shaw, J. Chem. Phys. 49, 4731 (1968). [12] H. Kreek and W. J. Meath, J, Chem. Phys. 50, 2289 (1969). [13] B. Linder, Adv. Chem. Phys. 12, 225 (1967). [14] B. Linder and D. A. Rabenold, Adv. Quantum Chem. 6, 203 (1972). [15] D. Langbein, Theory of van der Waals Attraction (Springer, New York, 1974). [16] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951). [17] P. W. Langhoff, Chem. Phys. Lett. 20, 33 (1973). [18] N. Jacobi and Gy. Csanak, Chem. Phys. Lett. 30, 367 (1975). [19] A. Koide, J. Phys. B 9, 3173 (1976). [20] M. Krauss and D. B. Neumann, J. Chem. Phys. 71, 107 (1979); M. Krauss and D. B. Neumann, and W. J. Stevens, J. Chem. Phys. 66, 29 (1980); M. Krauss and W. J. Stevens, ibid. 85, 423 (1982). [21] B. Linder, K. F. Lee, P. Malinowski, A. C. Tanner, Chem. Phys. 52, 353 (1980); P. 29 Malinowski, A. C. Tanner, K. F. Lee, and B. Linder, ibid. 62, 423 (1981). [22] A. Koide, W. J. Meath, and A. R. Allnatt, Chem. Phys. 58, 105 (1981). [23] M. E. Rosenkrantz and M. Krauss, Phys. Rev. A32, 1402 (1985). [24] K. L.C. Hunt and Y. Q. Liang, J. Chem. Phys. 95, 2549 (1991). [25] K. L.C. Hunt, J. Chem. Phys. 92, 1180 ( 1990). [26] Y. Q. Liang and K. L.C. Hunt, J. Chem. Phys. 98, 4626 (1993). [27] P. W. Atkins, Quanta (Oxford University Press, 1991). [28] B. Boulil, M. Deumié, O. Henri-Rousseau, J. Chem. Educ. 64, 311 (1987). [29] D. Bohm, Quantum Theory (Prentice-Hall, Inc., Englewood Cliffs, N. J ., 1951) Chap. 20. [30] A. S. Davydov, Quantum Mechanics 2nd Ed. (Pergamon Press, Oxford, 1976) p. 388. [31] E. Merzbacher, Quantum Mechanics 2nd Ed. (John Wiley & Sons, Inc. N .Y.,1970). [32] L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon Press, Oxford, 1958). [33] P.-H. Liu and K. L. C. Hunt, J. Chem. Phys. 100, 2800 (1993). [34] R. P. Feynman, Phys. Rev. 56, 340 (1939). [35] H. Hellmann, Einfu'hrung in die Quantenchemie (Deuticke, Leipzig, 1937), p. 285. CHAPTER III FORCE RELAY AND FORCE BALANCE 3.1 Force Relay According to Epstein’s force theorem [see Appendix A] the force on any electron i vanishes if the electronic state adjusts adiabatically to an external perturbation. In effect, the force due to the external field passes through the electrons and acts on nuclei. This sec- tion shows how the force is relayed from the electrons to the nuclei. Let Fe be the total force operator on N electrons, Fe = F¢N+FCS (1) where F 6N is the force operator on the electrons due to the nuclei, and F c8 is the force operator on the electrons due to a static, nonuniform external field, 86’“ (r) . If we use the A e o o o 0 electronic charge density operator, p (r) wr1tten as a function of the contmuous vanable r, and a point charge distribution for nuclei, F 6N and F e8 can be expressed as follows: [see EC1-(2,2.3)] FCN .—. Idrpc(r)221(r-Rl)lr—Rl |—3 (2) I and Fes = jdrf) (r)8"“(r) (3) 3O 31 As the external perturbation is weak one can use time-independent perturbation theory to derive the force relay condition. The expectation value of the total force is zero in the perturbed ground state. That is, (Fe) =(‘1’0+‘1’1+‘1’2+...|Fe|‘1’0+‘1’l +‘P2+...) = 0, (4) and therefore (F63) = -‘”‘t (r') :| 81%;) 1 a 35 ex , 8[VY’P°. (r’)] = ;jdr'cb [(r) 31:: (I 893 (r’) 811‘ ' (I (8) = -2J‘dr’¢xt (r’) I The effect on the electronic charge density of a simultaneous shift in the coordinates of all of the nuclei [see Section 2.5] is given by 3133 (r’) 311' l a = -Va’pf, (r’) . (9) Substitution of Eq. (9) into Eq. (8) and an integration by parts yield saw) = jdr'pg(r’)8;’“(r’). (10) Thus, the force relay condition at lowest order (n = 0) has been proven within the nonlocal polarizability density theory: —3 —Zz‘jdrpj(r) (r—R')a|r-R'| = Jdr’p8(r’)86a“(r’). (11) I Next, I will prove the force condition in Eq. (1) for n 2 1. In Eq. (3) for Sa (n) , P:+1’B(r) is determined by the (n+1)st order polarizability density cast in terms of polarization operators, xv” ]) (r, r’, r”, ..., r (" +1)’) . Thus sum): EZ'IdrTaB(Rl,r)P:+BB(r) I 36 = 1/(n+1)IZZIJdrdr’dr”...dr("+l)’ 1 +1 I II +1, I I 1 ” XXézau ;(rar 9r 9...,r(n ) )3:X (1.)ng (r )... x82)XI(r(n+I)I)TaB(r,RI), (12) (n+1), where r denotes the spatial coordinate r with n+1 " primes." Hunt has shown that linear and nonlinear response are related by [1,3] Bays (r I, r II) _ I I II I 3R1 — ZIdl‘BY5B(l‘,r ’r)TBa(r9R ), (13) a and an analogous relation holds between x(") and x(" + 1’ rI I , 3x§§)Hw(1”...,r‘"+, )) 1 8Ron I 1 I II , = ZId'Xigtmis” ” ,...,r("+” ,r)TBa(r,R'). (14) Therefore Sa(") : 1/ (n + 1) IZJdrdr’dr”...dr (H 1)’ 1 () 1.! (+1), «3767;. m( l'”,...,r " ) 311‘ at ext , 8, (r ) xs;*‘(.-») ...szrwc”). (15> 37 Now the relation between the nth order polarizability density and its gradient is used [see Section 2.5] ”V;n+l)’x.;g?..w(r’ar”y---,r(n+l)’)- (I6) By use of the permutation symmetry of x (n) , Sam): —(1/n!)jdr’dr”...dr("+”’ XV I (n) r’, rll,...,r(n+l)l 8X! r! X ngt (r,r) Sext (l‘ (n+ 1);) (o 3 (17) and so Sa(n)= (l/n!)de'dr”...dr (n+I)I VY’VS”...V$H)’ (I!) I II (n+1), xxyamm(r,r ,...,r ) X [Va,¢ext (r,)](bext (r’,) ”.(bext (l' (n+ 1);) 38 —jdr'p: (r’) [Va,¢ext (r’)] jdr’p:(r’)8;’“(r’). (18) This completes the proof that the force relay condition is satisfied within the nonlocal polarizability density theory, for fixed external fields 8:“ (r) . 39 3.3 Force Relay Condition and Force Balance a. Force Balance at First Order The sum of the first-order forces on nuclei of molecule A is EAF‘W = zz’jm’ -— r)alRl — rl—3pBO(r)dr I I +22le (Rl r)Pe’A(r)dr (1) a8 ’ 1,13 ’ I where Pigm = Idr'aABY(r,r’)Sgy(r’), (2) in terms of the field 83 Y(r') due to the unperturbed charge distribution (electronic and nuclear) of molecule B. From the zeroth-order force relay condition, the second term of Eq. (2) can be replaced by a term containing the unperturbed electronic charge density of A and the unperturbed charge distribution of B, ZZlIdrTaB (R1, r) P??? (r) 1 = -221IdrPI'A(l') (l' 'RIMI" “R1 l-3 1 jdrp°”‘o(r)sga(r). (3) After substitution of Eq. (3) into Eq. (1 ), Eq. ( 1) can be rewritten as follows 2M“... = 2253,”) I l 4O +jdrpe’A0(r)Sga(r). (4) That is, the sum of the first-order forces on the nuclei of molecule A equals the force on the entire unperturbed charge distribution of A due to the entire unperturbed charged dis- tribution of B. Equation (4) can be decomposed further into four terms, giving the total force on nuclei in molecule A as XAFIM- = ZZIJpS’BU) (RI-r)alRI—r|—3dr I I +ZZIXZ’(R' — R’)a|R' —R’l_3 I J +zz’jp3’A (r) (r — RJ)alRJ — rl-Bdr .I +Jp3’A(r)p;‘B(r’) (r—r’)alr’—rl—3drdr’. (5) Similarly, for nuclei in molecule B, J J A J ZAFMJL = 22 80,010!) J J +Idrpe’Bo(r)88a(r); (6) and so 2M1... = EZ’JPS’Atr) (R’—r)a|R’-r|'3dr J .I +ZZ'ZZ’ (RJ — R‘) alR’ — R1 l—3 I J +zZlJp3’B (r) (r — RI) alR' — rl-Sdr 1 41 +jp§B(r)p§'A(r’) (r-r’)a|r’-r|—3drdr'. (7) Comparison of Eqs. (5) and (7) shows that the first-order forces on A and B are equal and opposite, within the nonlocal response theory. b. Force Balance for Second—Order Induction Forces The sum of the second-order induction forces is 2111,“, = XZ'JdrwR',r>1PT:E#:1311101], <8) I I where Fianna (r) = Idr'dr"dr"'agy(r, r')TY5(r', r") x (138 (r r'") 83.5 (r "') 1 +-2-jdr'dr"BB‘ys(1-, r ', r ") 330,0 ') 8305 (r "). (9) Substitution of Eq. (2) into the first term of Eq. (8) gives EZlIdrTaB (R', r) Pig (r) I = Zz‘jdrdr'TaBm',r)a‘§y(r,r’)83,(r’) I = 22' sfiam‘), (10) I where 42 SEQ(R') = Idrdr’TaB(Rl,r)0th(r,r’)83"y(r’). (11) Equation (10) gives the force on the nuclei of A, due to the first-order, interaction—induced field from molecule B, 8113, a (RI) , which depends on the change in charge density induced in B by the field from the entire unperturbed charge distribution of A. The second term of Eq. (8) contains two parts zz‘jdrTaB (R', r) 1’31;in (r), I = zz'jdrdr 'dr "dr ~er43 (R', r) 013,0, r ') T115 (r '. r "1 a3, (r r "') s; E (r "') I 1 +§2Z1Idrdr 'dr "TaB (R‘, r) [3&5 (r, r ', r ") 830,0 ') 8305 (r "). (12) 1 If one applies to Eq. (12) the same equations used to prove the force relay condition in Section 3.2, the first term in Eq. (12) becomes Zzlerdr Idr IIdr III'l‘aB (R1, r) agy(r, l.I) T75 (r I, r II) age (r II, I. III) 838 (r III) I 8P8’¢(r') B = 1 = ;Idr __8R:1 31.11“. ). (13) Then Tl = [drpe’Aou-mfiam, (14) where Eqs. (7)-(10) of Section 3.2 have been used to establish the equality of Eqs. (13) 43 and(l4), aPe’Au’) O, I -—Yl—- = z‘jdrTaB(R',r)a§Y(1-,r ), (15) and and 818,7(r’) : IdrIIdrIIIT‘Y6(rI’rII)ag€(rII,rIII)83£(rIII). ('6) Equation (14) gives the force on the unperturbed electronic charge distribution of A due to the first-order correction to the field from B. The second term in Eq. ( 12) can also be rear- ranged by using the same procedure used to prove the force relay condition within the nonlocal polarizability density theory [see Section 3.2]: I , ,, I II B I B ” EXz'jdmr dr TaB(RI,r)B§,5(r,r ,r )8 o,(r )3 050' > I A I II _ l I II 75(1. ,l' ) B I B II __ 2211.31. d 8R; 8 01/“ )8 05(r ). (17) Equation (17) is equal to S a (n = 1) from Section 3.2, and it has already been proven in Eqs. (12)-(18) of Section 3.2 that C Sa(n =1) = jdrpn'fl (”33a”). (18) Eq. (18) gives the force on the first-order correction to the electronic charge density of A, due to the unperturbed charge distribution of B. From Eqs. (10), (14) and (18), the sum of the second-order induction forces is I I B I zAF 2’ a, ind : 22 8 I, (I (R ) I I +Jdr[pe'A1(r)Sga(r) +pe’A0(r)S‘fia(r)]. (19) The first term in Eq. (19) is due to the direct force on the nuclei in A from the first-order, interaction-induced change in the charge density of B. The second term gives the induc- tion force on nuclei in A due to the response of electrons in A to the field from B, at sec- ond order in the interaction. The second term is also obtained by direct evaluation of the induction terms in [pe' A (1')ESB (r) J 1' Hence, the force relay condition is met for the second-order induction effects. To show that the induction forces balance in the theory, it is useful to designate the force terms as F [p?’ x —) pjo’ Y] according to their source S (electrons e or nuclei n) and molecule of origin X (A or B), and the object O in molecule Y, on which the source acts; i and j indicate the orders of the charge densities in the intermolecular interaction. Equation (19) implies ZAF'z. ma = F198 8 —> 91' A] + F193 B -> 01"] I +F[p§’B—>p3’A] +F|:pT'B—>p3’A]. (20) Similarly, for nuclei J in molecule B 2“]; ind = FIPS' A ‘9 Pi' B] + FIPS' A —’ Pi’ B] J +F[p‘;’A—>p3‘3]+F[p'i‘A—)p3’8]. (21) The terms in Eq. (21) correspond one-to-one with the terms in Eq. (20), taken in reverse 45 order. Then from the equalities F938 B—’Pn’=A] ‘FIPT'AHPS'V (22) F1913 ->1>3 AA]=-F[p$’ —>p, B] (23) Fpol: BTP1HI=TFIPI “’90 B] (24) and F1913 4133"] =-F[PS' 601B] (25) it follows that the induction forces balance within the nonlocal response theory. c. Force Balance for Second-Order Dispersion Forces The sum of the dispersion forces XIAFIZ' disp on the nuclei in A, due to the elec- trons in A, is obtained directly from Eqs. ( 16) and ( 17) in Section 2.3. To prove force bal- ance, it is easier to use the derivative of Eq. ( 15) in Section 2.3 with respect to nuclear coordinates, A . 8a By(r,r , 1011(1)) ZAF" 20Ldisp: (h/41t) j dtoXIdrdr'dr'or'" a 1 R I I 01 III I II B II III - - XTyfi” ,r )(Jt88 (r ,r ;-1o),1(o)TEB(r ,r). (26) Equation (9) in Section 2.5 holds not only for the static response tensors, but also for the imaginary-frequency polarizability density OLB7 (r, r ';-i(o, ico) [3]; that is 46 8am”, r';i(o) 3R1 = —Va0(BY(r, r';—i(1),i(i)) — Va aBY(r, r';—i(o, ico) , (27) l a Substitution of Eq. (27) into Eq. (26) shows that the sum of the dispersion forces on nuclei in A depends on the gradients of 0‘37 (r, 1' ';-im, im) : ZABJ’ALOLJ,isp = —(h/41t2) igdwldrdr'dr'arm 1 A y 0 - I A ' . a X[ I ;— ’ V ) ;_ I J VaaBYO' r 1(1) 1(1))+ a 0‘13?” r 1(1) 10)) I H B H ”I ' ' HI XTyB” ,r )oc68 (r ,r ;—1(1),10))TEB(r ,r).(28) Equation (28) is used next to prove that the force relay condition is satisfied for dispersion, within the nonlocal polarizability density theory. The first-order dispersion term in the product of the external field and the charge-density operator for molecule A is: Jdr[pe'A(l‘)SB(r)]1,disp e, B =-(1+C)jdr(\¥0l[b Ami! (r)]GA$B(0)VAB|‘PO>1 (29) where C denotes complex conjugation, ‘I’o is the ground state of the unperturbed AB pair, VAB is the perturbation due to the AB interaction, and GA 6 B is a particular term in the reduced resolvent for the AB pair (cf. Eq. (4) in Section 2.1): states of the AB pair give a nonzero contribution to GA 6 B only if both molecules A and B are excited. Eq. (29) trans- forms to 47 1dr 1p“ (r) 8" (011,,“p : _(] +C)J‘dl‘dl"dl‘ "drINT88(r, r')TB.Y(r", l.III) A A X2'( 111;; VaP5(r) ‘1'; X ‘11; PB(r")|~113) , B B , B B B B x2(‘1’0 Pen-)anwn Py(r)‘l’0) x [(E;_Eg)+(1a§_1ag)]“. (30) For simplicity, it is assumed that the states of A and B are real. Then by use of standard integral identities, the definition of the nonlocal polarizability density [see Eq. (3) in Sec- tion 2.1], index relabeling, and the symmetry of the dipole propagator T, the right-hand side of Eq. (28) is obtained. Since Eq. (28) follows both from the standard perturbation theory [see Eqs. (16) and (17) in Section 2.3] and from Eq. (29), the force relay condition holds for dispersion, at leading order. For force balance, Eq. (28) can be rewritten after an integration by parts zAFIhAlmdisp = (h/41t2) Igodconrdr'dr"dr"'[ag‘Y(r,r';—iw, im) I I II B II III, ° ' III xT75(r ,r )ase (r ,r ,—1(o,1w)VaTEB(r ,r) +01A (r r"—i0) i(())V' T (r' r")0tB (r" r""—i0) im) BY , I I (I 75 I 58 7 9 9 xTEfl(r"',r) 1. (31) 48 The same analysis, applied to nuclei J in molecule B gives the result on the right-hand of Eq. (31), but with VaTeB (r "', r) replaced by V"'mTEB (r "', r) and V’mTY5 (r’, 1'”) replaced by V”mT76 (r’, r”) . From the relations VaTefl (r "', r) = —V maTeB (r "', r) (r’, r”) = —V” T (r’, r”) , it is clear that dispersion forces balance in the andV’ T a Y5 (175 nonlocal response theory. 49 References [l] K. L. C. Hunt, J. Chem. Phys. 90, 4909 (1989). [2] M. Born, Optik (Springer, Berlin, 1933), p. 406. [3] K. L. C. Hunt, Y. Q. Liang, R. Nimalakirthi, and R. A. Harris, J. Chem. Phys. 91, 5251 (1989). CHAPTER IV MOLECULAR SOFTNESS FUNCTIONS 4.1 Introduction Over the years scientists have tried to categorize the behavior of molecules under different circumstances. From the earliest observations of some metal oxides and sulfides, scientists are able to classify metal ions into two groups, hard and soft, by comparing the differences between their cohesive energies of the metal oxides and sulfides [1, 2]. Pear- son [3] extended this work by adopting the generalized concept of acid and base intro- duced by G. N. Lewis to the entire range of chemistry. The principle of hard and soft acids and bases [4] (HSAB), and the principle of electronegativity equalization [5] provide a framework for simple physical interpretations of complex phenomena. Parr and Yang [6] related the parameters associated with these principles, hardness[3, 4, 7, 8] and softness [9], and electronegativity with fundamental variables of density functional theory [10]. Through their work, a solid theoretical basis has been provided for these concepts, this allows us to build a bridge between these con- cepts and wavefunction theory which provide an accurate description of the electronic structure of chemical systems, but otherwise far from providing a framework for simple interpretations. This also makes it possible to transform the relevant information contained in the wavefunction into an almost pictorial representation, ready to be analyzed with the principles mentioned above [8, 10-20]. The purpose of this chapter is to provide an over- view of the concepts of density functional theory needed for applications in Chapter V, and its relation with the nonlocal polarizability density theory. 50 51 a. Density Functional Theory Density functional theory originated with a 1964 paper by Hohenberg and Kohn [21], and its chief method of implementation is described in a 1965 paper by Kohn and Sham [22]. An earlier model which is regarded as the source of modern density functional theory is the Thomas-Fermi model [23-26], from which originated the idea of an "electron gas. " In this model all properties of a system are expressible in terms of the electron den- sity p (r) , the number of electrons per unit volume, as it varies through space. The Tho- mas-Fermi model fails to give an accurate description of electronic systems of chemical interest; for example, it cannot account for chemical binding [27]. But it is now possible to characterize the properties of any system in terms of its electron density via the density functional description. The Hohenberg-Kohn theorem ensures that the exact ground state density can be calculated from a variational principle involving only the electron density p, without solv- ing the Schrodinger equation exactly. For any system, the ground-state electronic energy is a functional of the density, given by the formula B1p1=[p(1)v(1)dr,+F1p1. (1) Here v is the external one-particle potential (for example, -Z/r for an isolated atom) and F[p] is the sum [21] Flo] = T1111 +V,,(p1, (2) where T [p] is the electronic kinetic energy and V e e [p] is the electron-electron repul- sion energy. Both T [p] and V e e [p] are well-defined, universal, but unknown function- als of the density. The spin-free density may be expressed in terms of the wavefunction, 52 p(l) = N]|\P(1,2,...,N)|2dm,dx2dx3...de, (3) where dxi = dmidti is a spin-space volume element, with dd)i the spin part. The number of electrons is given by the formula N=N[p] =[p(1)drl. (4) The quantity N, like E, is a functional of p. Suppose there is some p' from some approximation to the exact ground-state den- sity p, normalized to the proper number of electrons, N [p’] = N. Using the same defini- tion as Eq. (1), the energy associated with the electron density p' is given by E,1p’1 =[p'(1)v(1>dr.+F(p'1. (5) and Ev [p’] obeys the inequality Ev [P'] 2Ev [P]; (6) when p’ equals p, the equality holds and results in the true energy, E [p] . That is to say, the density p and energy E are determined from the stationary principle, 5{E,lp’] —uN[p’]} = 0, (7) where p. is a Lagrange multiplier. In Eq. (7) an arbitrary variation in p’ is allowed; the potential v is fixed. If the solution making E an absolute minimum is selected from all possible solutions, the associated value of u is characteristic of the system of interest [28] 53 and is called the chemical potential of the system. The chemical potential is related to the Mulliken definition of electronegativity [29, 30] (xM) through - a—P- (tar—p1 “ ' “Jd'aN ' Id” 5p 31: _ 8E) ~_I+A~_ - (3N v(r)= _ XM, (8) where v (r) , I and A are the external potential, the ionization potential and the electron affinity, respectively. From Eq. (7) and a theorem from the calculus of variations [3]] it follows that 5E[p] = — . 9 ll [(8p(1))v:|p=p[v] () The quantity 5E/8p is the functional derivative of the Hohenberg-Kohn functional with respect to the electron density; it is evaluated at the correct ground-state density at an arbi- trary point in space. The corresponding functional derivative with respect to the potential v may be determined from Eq. (1); it is (10) pH) = (5391—) P 8v(l) The total differential of E = E (N, v) accordingly is given by the fundamental equation dE=udN+Ip(l)dv(1)d‘cl. (11) This is the generalization, to include change in the number of particles, of a formula from first-order perturbation theory (the Hellmann-Feynman formula) [38]. 54 b. Chemical Hardness and Softness Chemical hardness is a helpful concept for describing a variety of acid-base reac- tions. This idea has been in the chemical vocabulary for almost four decades [3, 7]. A wide variety of chemical reactions can be encompassed by general reaction scheme, A + :B —> A: B (12) involving a Lewis acid (A) and a Lewis base (:B); electrons transfer from the base (:B) with an available pair of electrons to the generalized acid (A). Through study of the char- acteristics of acids and bases, both Lewis acids and bases have been divided into two cate- gories, called hard and soft [3] based on the following properties: a) Hard acid: high positive charge, low polarizability and small size, e.g., H+, Li+, Ca2+. b) Hard base: high electronegativity, difficult to oxidize and low polarizability, e.g., NH3. c) Soft acid: low positive charge, high polarizability, large size, e.g.,, Cu+, Ag+, Rs+,12. (1) Soft base: low electronegativity, easily oxidizable, higher polarizability, e.g., bases con- taining P, Se, 8, ml as donor atoms. From the time hardness was first defined within density functional theory [8], var- ious related concepts like softness, local hardness [32, 33] and local softness [9], hardness and softness kernels [34], and related hardness [35] have emerged to correlate and to ana- lyze experiment information on the interactions between different chemical species in many different situation. c. Density Functional Theory of Chemical Hardness and Softness Equation (11) gives the differential energy change if the external potential or the number of electrons or both are varied slightly, and the chemical system moves from one ground state to another: dE = udN+Ip(r)dv(r). (13) 55 In order to understand the energy change to higher order, one would need to know how the chemical potential behaves as the external potential or the number of electrons is changed differentially. The variation in u to first order is a d“ = (5%) 511 vmdN+j[8vm]Ndv(r)c111. (14) Both derivatives on the right hand side of Eq. (14) are chemically significant. The first derivative has been defined as chemical hardness 11 [8] 1 a 1 32E 1‘] : —(-a—u—) : —[_i] , (15) 2 N V(r) 2 8N v(r) and the second derivative is defined as the Fukui function f(r) [36] 5p 8p 1(1):] ] = [_] . (16) 5v (1') N 8N v (1.) With these definitions, Eq.( 14) becomes an = 2ndN+Jf(r)dv(r)dr. (17) Equation (17) connects three important molecular properties: the chemical potential (I; the hardness n , and the Fukui function f (r) , whose rigorous, quantitative definitions are supplied by density functional theory. The parameters [.1 and n are global quantities which do not depend on spatial coordinates, whereas f (r) is a local quantity varying from place to place in a molecule; f (r) is useful for explaining the frontier-orbital theory of chemical reactivity in molecules. Taking the finite difference approximation [11] for the curvature of the E vs. N curve one can obtain the following formula for hardness: 56 I — A = —. 18 n 2 ( 1 Comparison of Eq. (18) with Eq. (8) shows that both global quantities, the chemical poten- tial u and the hardness n of a molecule, can be determined from the ionization potential and the electron affinity of the molecule, just as can the electronegativity. From Eq. (15), another global quantity, softness, is defined in terms of the recipro- cal of the hardness 1 8N 2T] 311 v (r) Generalizing to treat a chemical system (atom, molecule or solid) in the grand canonical ensemble, the softness in Eq. (19) may be also defined in terms of number fluctuations as [9] a N l s = {-é—U = — (— 21. (20) u v, T kT where k is the Boltzmann constant and the brackets < > designate ensemble averages at constant T, v and u. (1. Local Quantities and Nonlocal Quantities In order to measure the chemical reactivity of a particular site in a molecule, differ- ent local variables are defined. The local softness is introduced by combining two impor- tant quantities, n and f (r) in Eq. (17): 5(r) = 53.51 = f(r)S = [3p(r):| (8N) : [M] . (2]) 2" v(r) v(r) v(r) BN 3;; Bu 57 Since the Fukui function is normalized (it integrates to unity), the local softness must yield the global softness on integration ]s(r)dr= s]f(r)dr = s. (22) A nonlocal quantity, the softness kernel [34] is defined by 5 s (r, r’) 5-5:((:,))1 (23) where u (r) is the difference between external and chemical potentials, i.e. u(r) = v(r) —p. (24) The softness kernel integrates to the local softness [9] ]s(r,r')dr' = s(r), (25) Combination of the definitions of the global softness S, the local softness s (r) , and the softness kernel 3 (r, I") gives the linear density response function [34], s(r)s(r’)_ , _ 6pm] 8 s(r,r) _ [5v(r’) N. (26) In more detail, from perturbation formula [33] ,< ‘11,] bml “W ‘1', bu) ‘11,) 5pm ] _ [8v(r’) N ' 21(13):: Ek-EO (27) Equations (26) and (27) provide a route to connect softness functions and nonlocal polar- 58 izability densities; this will be discussed in the next section. The fluctuation formulas for the local softness [9] and the softness kernel [37] are respectively given by 1 8(r) = fil--(p(r)>] - (29) The hardness kernel [32] 1] (r, r’) is defined in terms of functional the inverse of the softness kernel, _ 511m _ 52F[p] _ , 30 5P (r’) 5P (1059 (1") ( ) 2n (r. r’) E where F [p] is the universal functional of density functional theory [see Eq. (2)]. The hardness kernel integrates to the local hardness, 1] (r) , though not in the sense that the softness kernel integrates to local softness [Eq. (25)]. In this case 1 I I I u(r)sfiJn(r,r)p(r)dr, (31) and the local hardness integrates to the global hardness, n = Inmfmdr. (32) The reciprocal relations between local hardness and local softness, and between hardness and softness kernels [34] are 59 2J5 (r)n (r) dr = 1 (33) and 2J3 (r, r’) n (r’, 1'”) dr’ = 5 (r — r”) . (34) The local hardness involves the variation of p (r) at constant external potential; this situation makes the definition of the local hardness [32] ambiguous [37, 38]. [Interest- ingly, the hardness kernel and the local softness do not suffer from this drawback]. This ambiguity stems from the interdependence of v (r) and p (r) as they appear in density functional theory [21]. To avoid this ambiguity and take advantage of the simpler interre- lation of softness functions, it is easier to use the softness functions for the further deriva- tions in the next chapter. 60 4.2 Relation between Molecular Softness and Nonlocal Polarizability Densities Nonlocal polarizability densities and molecular softness functions are both related to the electronic charge density pe (r) . Nonlocal polarizability densities represent the dis- tribution of polarizable matter within a molecule; and softness functions represent the compressibility of the electronic charge cloud [33]. Within density functional theory, the nonlocality of molecular response to pertur- bations is expressed in terms of the softness kernel 5 (r, r’) [34] and the hypersoftness o (r, r’, r”) introduced in this section. The softness kernel s (r, r’) is a measure of the sensitivity of the electronic charge density p'3 (r) l to a change in the potential v (r’) rel- ative to the chemical potential [1. Here, the potential v (r) is the sum of the potential vn (r) due to the nuclei and the potential ve (r) due to an external perturbation. The chemical potential [I [11, 32, 33, 39, 40] is determined by the change in total energy E with a change in the number of electrons N, at constant v (r) : 11 = (BE/8N),; (l) Parr, Donnelly, Levy and Palke [11] have shown that the electronegativity equals ~11. It is convenient to define a modified potential u (r’) by [34] u(r’) = v(r’) —u. (2) The softness and hypersoftness are derivatives of the electronic charge density with respect to the modified potential u(r). As shown by Berkowitz and Parr [34], u(r) deter- ]. The sign changes if the number density p (r) is used instead of electronic charge density, in the defini- tion of the softness kernel. 61 mines all properties, subject to the assumption that E is a convex function of N: Since v(r) vanishes as r —> co, the long-range behavior of u(r) yields )1; then for finite r, v(r) follows by subtraction. Given [1 and the convexity property of the energy, N is determined. Then the softness kernel is a functional derivative [34], s(r, r’) = 5pe(r)/8u(r’). (3) Integrating the softness kernel 3 (r, r’) over all space with respect to r’ gives the local softness s (r) , [8, 9, 34, Eq. (24) in Section 4.1] s (r) = Is (r, r’) (11"; (4) where the local softness, s (r) , is defined by [8, 9, 34] _ __ 3126(1)] 31:1 __[Bpe(r)] s(r)—f(r)S— [ 8N v(au)v— a” v. (5) In Eq. (5), S is the global softness; f (r) is the Fukui function [36], a normalized local softness, which is useful for explaining the frontier-orbital theory of chemical reactivity in molecules. A Maxwell relation yields [see Eqs. (27) and (28), below] __awuq _[5u] f”) ' [ aN ,7 5v(r) N' (6) Since the Fukui function is normalized (it integrates to unity), the local softness must yield the global softness on integration [8, 9, 34] Is(r)dr = SIf(r)dr = S. (7) 62 The global softness, S, is reciprocally related to the hardness, n , as defined by Parr and Pearson [8] 1 N s z _ = (L). (8) 211 911 v The molecular softness determines the change in the electronic charge density at r due to a change in the external potential ve (r’) , for a system with a fixed number of electrons N, via 8 [M] = so, r’) —s(r>s(r’) S". (9) five (r’) N as proven by Berkowitz and Parr [34] (see also Handler and March [41]). By definition, the derivative [5pe (r) /5ve ( r’) ] N is related to the static charge- density susceptibility x (r, r’;0, O) by [42] (t‘1p"(r)/2‘>v‘i(r')1N = x(r,r’;0.0). (10) Equation (9) permits a connection of the softness kernel and the local softness to the nonlocal polarizability density, since the relation between the charge-density suscepti- bility x (k, k’;0, O) in Fourier space and the longitudinal component 0LL (k, k’;O, O) of the nonlocal polarizability density is known. In terms of the polarization operators Pa(r) and PB(r ), the aB-component of (x (r, r’;—(r), (0) satisfies [43-47, Eq. (3) in Section 2.1] 010430. r ’; -(.0, (1)) = ( O I Pa(r) G(co) Pfl(r’) IO) + ( 0 l PB(r’) G*(—(o) Pa(r) I 0 ), (1 1) 63 where G(m) is the reduced resolvent operator G(w)=(1- poxH-Eo-hm)“ <1— (00), (12) 500 is the ground-state projection operator [0 ) (O I, H is the unperturbed molecular Hamil- tonian, and E0 is the unperturbed ground-state energy. After Fourier transformation into k-space, the longitudinal component of the static nonlocal polarizability density (1" (k, k’;O, O) is related to the charge-density susceptibil- ity [47, 48]. Specifically, with the convention a (k, k’;0, 0) = Jdrdr’exp (—ik - r) exp (ik’ - r’) on (r, r’;0, 0). (l3) and V~P(r,(1o) = -—Ap(r, (11)), (14) one obtains [47, 48] (1"(k, k’;0, 0) = E12101, k’;0, O)/kk’. (15) Then from Eqs. (9), (10) and (13), one can obtain 01“ (k, k’;O, 0) = iEiE'[s (k, k’) — s (-k) s (k’) 5"] (kk’) ". (16) The longitudinal component of the polarizability density 011‘ (k, k’;0, O) suffices to deter- mine all of the physical properties considered in the next chapter. In r space Eq. (16) is VmV’B awn, r’;0, 0) = s(r, r’) —s (r) s(r’) S". (17) Equation (16) gives an exact formulation of the observed qualitative correlation between polarizability and softness; it is consistent with the result of Vela and Gézquez for the total polarizability [49]: aaB = Idrdr’[s(r,r’)—s(r)s(r’)S_]]rar’B. (18) Vela and Gazquez have used Eq. (18) and a local approximation to the softness kernel to correlate the isotropically averaged polarizability a and the global softness S, in an approximate fashion [49]. The total polarizability can be obtained from Eq. (16) in the _ as(-ki) as(k'j) _. [ ak. H Bk’. ]3 ’ “9) k=1(’=0 ‘ 11:0 1 k’=O in terms of the derivatives of the softness kernel with respect to k and k’. In Eq. (19), i form a _ 62$ (k1, k’j) 11' Bkiak’j and j denote unit vectors in the directions of the i and j axes. Further, Eq. (16) suggests a generalization of softness to include frequency depen- dence, with s (r, r’;—(i), (1)) determined by the (n-frequency component of the change in electronic charge density p‘3 (r, 0)) induced by the (1)-frequency component of the modi- fied potential 11 (r’, 0)) , within linear response: 51)" (r, (0) (Eu (r’, (1)) ° (20) s (r, r’;—(n, (1)) = To extend the density-functional analysis to nonlinear response, Fuentealba and Parr [50] have used higher-order derivatives of the molecular hardness n (r) [8]. Here, we introduce a hypersoftness o (r, r’, r”) , which permits a more direct relation to the hyperpolarizability density. The hypersoftness o (r, r’, r”) is defined by (21) Szpe (r) ] = 55 (r, r') 0(r’r’r) : [8u(r’)8u(r”) 5u(r”) ' 65 From Eq. (21) and the symmetry of s (r, r’) , o (r, r’, r”) is symmetric in all three vari- ables r, r’ and r”. The analysis below requires contracted versions of the hypersoftness, obtained by integration with respect to one or more of the spatial variables: G(r, r’) = Jdr”o(r, r’, r”) = Jdr”o(r’, r”, r) = Jdr”o(r”, r, r’) , (22) and o (r) = Idr’dr’b’ (r, r’, r”) = Jdr’dr’b (r’, r”, r) = Idr’dr’b (r”, r, r’) . (23) When the external potential ve (r) changes while the number of electrons N is held constant, the change Ape (r) in the electronic charge density at point r satisfies C C l C Ap (r) = 8p (r) +582p (r) +... Idr’x (r, r’) five (r’) + ;Idr’dr”x(2) (r, r’, r”) 8ve (r’) (3ve (r”) + ...(24) where x (2) (r, r’, r”) is the nonlinear charge—density susceptibility (at lowest order). For comparison, within the density functional framework, Apc (r) is obtained from [see Appendix B] e __ I 6pe(r)] I Ap (r) — [dr [5u(l") 8u(r) 2 .1. I II 6 pc(r) ’ ” +zj'dr dr [on (r,) on (r”) Jou (r )5u (r ) 1 .8601] 2 . +zjdr[8u(r,) 5u(r)+... (25) C . . . to second order. When v (r) 1S changed but N 18 held constant, there is a non-zero second 66 variation in u(r), due to the second variation in u. The first variation in u at constant N is specified by the Fukui function f(r): C 811N = jdr f(r)v (r), (26) where [34, 5]] f(r) ____[§£_(r_)] = 8” , (27) 3N v 5v"’(1-)N The equality of the two derivatives in Eq. (27) follows from a Maxwell relation [52] for the mixed second partial derivatives of the energy E, with N and v'3 (r) taken as the inde- pendent variables: d5 = udN -]d1-p° (r) v6 (r). (28) The second variation in u stems from nonlinear response to the external potential: 5f(r) five (r’) azuN = [drdr’[ ] 8ve(r)5vc(r’). (29) N From Eqs. (2) and (3), within the density functional framework the change in charge den- sity is Ape(r) = Jdr’s(r, r’) [5ve(r’) -5u] +-;-Jdr’dr”o (r, r’, r”) [five (1") - 51.1] [5Ve (r”) — 511] —-;—Idr’s (r, r’) 52p + (30) 67 Then from Eqs. (26) and (29) Ape(r) = [dr'sng r')5v"(r') —]dr"s(r, r")]dr'f(r')5ve(r') +%Jdr'dr"o (r, r', r") five (r') five (r") -%]dr'dr"'o (r, r', r'") five (r') jdr"f(r") 5vc (r") -%]dr"dr"'o (r, r'", r") 5ve (r") Jdr'f(r') 8ve (r') 1 m iv ,., iv , , e , ,, n e n +§Idr dr o(r,r ,r )Idrf(r)5v (r)]dr f(r )8v (r) 8f(r3 5ve(r") —%J’dr"'s(r, r"')]dr'dr"[ ]8v°(r') 5ve(r") + (31) From Eqs. (2), (22), and (23), Eq. (31) becomes Ape(r) = Idr’[s(r, r’) -s(r)f(r’)]5ve(r’) 1 +§Idr’dr”{o(r, r’, r”) -6(r, r’)f(r”) -G(I'1 r”)fU') 8f (r’) 6v°(r”) +o(r)f(r’)f(r”) —s(r)[ :l}5ve(r’)5ve(r”) +... (32) The Fukui function f(r) is related to the local softness s(r) via the global softness S, f(r) = s(r)S_]. (33) 68 Then Ape(r) = Idr’[s(r, r’) -s(r)s(r’) S—1]5ve(r’) 1 _ _ +§Idr’dr”{o(r, r’, r”) -6(r, r’)S(r”) S 1-00‘, 1'”) $075 I +o(r)s(r’)s(r”)S—2—s(r)[ 8f(r’) :l } ave (r”) x 811" (r’) 5ve(r”) + (34) The final term in Eq. (34) can be further simplified as [see Appendix C] l: 6f(r’) ] : G(r’, r”) S-I_O_(rl)s(rn) S-‘Z five (r”) N — o (r”) s (r’) S“2 + OS (r’) s (r”) 8-3. (35) Substitution of Eq. (35) into Eq. (34) gives Ape(r) = Jdr'[s(r, r’) —s(r)s(r’)S-]] 8ve(r’) +%J‘dr’dr” {0' (r, r’, r”) — [0' (r, r’) s (r”) + o (r, r”) s (r’) + 0' (r’, r”) s (r) ] S" + [o(r)s(r’)s(r”) +O'(r’)s(r)s(r”) +O'(r”)s(r)s(r’)]S_2 —o s(r) s(r’)s(r”) 8‘3 }8vc(r’) 5ve(r”) + (36) 69 Then by comparison with Eqs. (24) and (36), the lowest order nonlinear charge—density susceptibility x (2) (r, r’, r”) in terms of molecular softness and hypersoftness is x (2) (r. r’, r”) = O'(l‘, r’, r”) — [G(l‘, r’) s (r”) + o (r, r”) s (r’) + G(r’, r”) s (r) ] S-1 + [o(r)s(r’)s(r”) +o(r’)s(r)s(r”) +O'(r”)s(r)s(r’) ] s‘2 —o s (r) s (r’) s (r”) 5'3 (37) = C (r, r’, r”) [1/3 c (r, r’, r”) — o (r, r’) s (r”) S"1 +o(r) s (r’) s (r”) 84-1/3 0' s (r) s(r’) s (r”) S_3 J, (38) where 0' is obtained by integrating over 1' in Eq. (23), and C (r, r’, r”) denotes the oper- ator that sums the cyclic permutations of r, r’ and r” in the expression that follows it. The symmetries of xm (r, r’, r”) and o (r, r’, r”) with respect to r, r’ and r” are evi- dentin Eq. (38). The lowest order nonlinear charge-density susceptibility xm (r, r’, r”) is related to the longitudinal component of the B hyperpolarizability density in k-space, via (3“ (k, k’, k”) = 112123?) (2’ (k, k’, k”) (kk’k”) ". (39) Thus, the relation between BL (k, k’, k”) and the hypersoftness is 70 BL (k, kl, k”) A = ikk’k”C(k, k’, k”) [1/3 G(k, k’, k”) —o(k,k’)s(k”)S—' +o(k)s(k’)s(k”)S_2—1/3 o s(k)s(k’)s(k”)S—3 ] (kk’k”)-'. (40) For comparison, a perturbation analysis gives the hyperpolarizability density in the form [48, 52] Bafigr, r’, r”; —(r)o, (1)1, (02) = (0(1) 1 < 0 | Pam G(coo) 111).") G(m1) Pam 10> + ( 0 11w") G(-(1)2) Wfiu’) G(—(11)O) Pa(r) IO) + <0 | P70") G(—w2) Wan) G(w11 Pam I 0 > 1. (41) where the operator 5037 denotes the sum of all terms obtained by permuting PB(r’) and P70”) and simultaneously permuting the associated frequencies (01 and (02, in the expres- sion following the operator; and (00 = (1)] + (1)2. The operator Walk) is defined by Wan) = Pa(r) — ( o l Pa(r) Io ). 71 4.3 Density of States and Nonlocal Polarizability Densities for Metals at Absolute Zero In the Kohn-Sham [53] formulation of finite-temperature density functional the- ory, one has the self-consistent equations l [— §V2+veff(r) ‘11]‘l’i : ‘3in (1) p(r) = 2|w,(r)|21(e,—11). (2) ' 5F [1)] _ ,P (1') xc veff(r) — v(r) +Idr Ir—r’] +—5p(r) , (3) where f (t:i — It) is the Fermi function, 1 f(ei—u) = (4) 1+exp(B(e,-11)1 and the “’1 (r) are the normalized Kohn-Sham orbitals. FxC [n] is the exchange-correla- tion free-energy functional. Equations ( l) to (4) hold for molecules or solids, for the specific case of solids, N =[d1-p(r) = Zuei—u) 2V = _3]dk mm _,11], (5) (2n) where the discrete sum has been replaced by an integral; a sum over the band index is assumed here and in the formulas below. Alternatively [9, 54, 55], 72 N = Jde g(8)f[e-ul. where g (e) , the density of states at energy 8, is given by g (e) = 2561—8) 2V (2 )3]dk6[e(k)—e]. 1E At T = O, u is equal to the Fermi energy 8F and l €i<].1 f(Ei-Ll) = I O ei>u At T = 0 one therefore has _ u N - [Ode g(e) andhence EN) 1 —— = — : S : g(8 ), (Bu 1w 2n F (6) (7) (8) (9) (10) where the volume V and the lattice structure, remain fixed through all the differentiations here and later. For a metal at absolute zero, the global softness is the density of states at the Fermi level. The local density of states g (e, r) is defined by g(8.r) = Elvi(r)|28(ei-e) 73 2V (211) 3 [dk|1yk(r)|28[e(k)—e]. (H) In analogy with Eq.(9), the electron density is given at absolute zero temperature by p(r) = [Ede g(e, r). (12) Consequently, using Eq. (10) and Eq. (16) in Section 4.1, one can connect the Fukui func- tion [36] with the density of states g (e) and the local density states g (e, r) : _ 8pm _ 8pm] 93 f(r) _[ aN ]T,v'[ an T,V(BN)T’V g (2,, r) g (2F) ° (13) 211s (8p, r) = The Fukui function is the normalized local density of states at the Fermi level; the normal- ization [f(r) dr = 1 corresponds to Ig (8F, r) dr = g (8F) . Equations (10) and (13) give the product of the global softness S and Fukui func- tion, that is the definition of the local softness s (r) [see Eq. (21) in Section 4.1] s(r) 2 (8p (1.)) = Sf(r) = g(8F,r). (14) T,v(r) Equations (10) and (14) extend the application of the molecular softness functions to met- als at T = O: S = g (8F) . (15) s(r) = s(epr). (16) The integration relation also holds for Eqs. (15) and (16), thus 74 S = Idrs(r). (17) Chemisorption and catalytic reactions on metals can be regarded as soft-soft chemical reactions. Falicov and Somorjai [54] have pointed out that f (r) or g (e, r) appears to determines site selectivity for metals in Chemisorption and catalysis. In general metals are soft [4], with large g (e) [9], and transition metals are particular active because of their high g (e). 75 References [l] G. Schwarzenbach, Experentia. Suppl. 5, 163 (1956). [2] S. Ahrland, J. Chatt, and N. Davies, Quart. Revs. (London) 12, 265 (1958). [3] R. G. Pearson, J. Am. Chem. Soc. 85, 3533 (1963). [4] R. G. Pearson, Hard and soft acids and bases (Dowden, Hutchinson and Ross Strouds- ville, PA 1973). [5] R. T. Sanderson, Chemical Bonds and Bond Energy, 2nd Ed. (Academic, New York 1976). [6] R. G. Parr and W. Yang, Density functional theory of atoms and molecules (Oxford University Press, 1989). [7] R. S. Mulliken, J. Am. Chem. Soc. 64, 811 (1952). [8] W. Yang and R. G. Parr, Proc. Natl. Acad. Sci. USA 82, 6723 (1985). [9] W. Yang and R. G. Parr, Proc. Natl. Acad. Sci. USA 82, 6723 (1985). [10] R. G. Parr and W. Yang, Density functional theory of atoms and molecules (Oxford University Press, 1989) [l 1] R. G. Parr, R. A. Donnelly, M. Levy and W. E. Palke, J. Chem. Phys. 68, 3801 (1978). [12] R. G. Pearson, J. Am. Chem. Soc. 107, 6801 (1985). [13] R. G. Pearson, Inorg. Chem. 27, 734 (1988). [14] C. Lee, W. Yang and R. G. Parr, J. Mol. Struc. (Theochem) 163, 305 (1988). [15] F. Méndez and M. Galvan in: J. K. Labanowski, J. W. Andzelm (Eds.) Density func- tional methods in chemistry (Springer, Berlin, 1991) p. 387. [16] R. F. Nalewajski, J. Korchowiec and Z. Zhou, Int. J. Quantum Chem. 822, 349 (1988). [17] R. F. Nalewajski, J. Phys. Chem. 93, 2658 (1989). [18] R. F. Nalewajski and J. Korchowiec, J. Mol. Catal. 54, 324 (1989). 76 [19] W. Yang and W. J. Mortier, J. Am. Chem. Soc. 108, 5708 (1986). [20] W. J. Mortier, S. K. Ghosh and S. Shankar, J. Am. Chem. Soc. 108, 5708 (1986). [21] P. Hohenberg and W. Kohn, Phys. Rev. B 136: 864 (1964). [22] W. Kohn and L. Sham J. Phys. Rev. A140, 1133 (1965). [23] N. H. March, Adv. Phys. 6, 21 (1957). [24] S. Lundqvist and N. H. March Eds. Theory of the Inhomogeneous Electron Gas (New York, Plenum, 1983) pp. 1-77. [25] N. H. March, Self-consistent Fields in Atoms (London, Pergamon, 1975). [26] E. H. Lieb, Rev. Mod. Phys. 53, 603 (1981). [27] N. L. Balazs, Phys. Rev. 156, 42 ( 1967). [28] The smallest It ordinarily will go with the smallest E, but this may not always be the case. [29] R. S. Mulliken, J. Chem. Phys. 2, 782 (1934). [30] K. D. Sen and C. K. Jorgensen, Electronegativity (Springer-Verlag, Berlin, 1987). [31] I. M. Gelfand and S. V. Fomin, Calculus of Variations (Prentice-Hall, Englewood Cliffs, NJ, 1963). [32] S. K. Ghosh and M. Berkowitz, J. Chem. Phys. 83, 2976 (1985). [33] M. Berkowitz, S. K. Ghosh and R. G. Parr, J. Am. Chem. Soc. 107, 6811 (1985). [34] M. Berkowitz and R. G. Parr, J. Chem. Phys. 88, 2554 (1988). [35] Z. Zhou and R. G. Parr, J. Am. Chem. Soc. 111, 7371 (1989). [36] R. G. Parr and W. Yang, J. Am. Chem. Soc. 106, 4049 (1984). [37] M. K. Harbola, P. K. Chattaraj and R. G. Parr, Israel J. Chem. 31, 395 (1991). [38] S. K. Ghosh, Chem. Phys. Lett. 172, 77 (1990). [39] R. P. Iczkowski and J. L. Margrave, J. Am. Chem. Soc. 83, 3547 (1961). [40] J. P. Perdew, R. G. Parr, M. Levy and J. L. Balduz, Jr., J. Phys. Rev. Lett. 49, 1691 (1982). [41] G. S. Handler and N. H. March, J. Chem. Phys. 63, 438 (1975). 77 [42] B. Linder, Adv. Chem. Phys. 12, 225 (1967). [43] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phy. 8, 319 (1964). [44] L. M. Hafl } 3RY L¢l a[(1— 511) z‘z’TaB (R‘, 18)] K 3RY (7) Using the same assumptions as for Eq. (3), one needs the derivatives of the polarizability density with respect to nuclear coordinates, which are determined by the hyperpolarizabil- ity density and the dipole propagator [9], 3013, (r, r’) 311' or = ZIIdrBBYS(r,r’,r”) T5a(r”,R1). (8) 82 to lead to an analytic expression for the third derivatives of V({R}), and hence the anhar- monic force constants, 83V({ R})/BRLBREBR: . These quantities depend on the first and . . 8 . . second derivatives of p (r) With respect to the nuclear coordlnates: 83v BRLBR$BR$ = —Z‘Z’ZKIdrdr’dr”B&§ (r, r', r”) T0,,5 (r, R‘) T55 (r’, R’) TY: (r”. RK) + C 6,,z‘z"jdrdr'ol88 (r, r’) Tags (r, R1) "rye (r’, RK) IJK + 511511 and excited states I k ). [sA (r, r ';—i0), iw) - sA (r;iu)) SA (r’;i(u) /SA (iw)] = (4n2/h)2wko(mio+w2)"<0|p(r)| k> Fe = zzzafl— 28(rs) EFeN-theES’ (Al) —r s=l a a s s=l also Fe=—2(aH/Brs) = i[H,P] (A2) 3 where P is the total electronic momentum operator, H is the Hamiltonian operator. Za is the electron charge of the nucleus a. In Eq. (Al) the double sum yielding the force on the electrons is due to the nuclei and the single sum is the force due to an applied electric field, 3 (if any). The electron-electron force cancels. From Eq. (A2) if the hypervirial theorem for P is satisfied, i.e., if ([H, P] ) = 0, then the average of F e vanishes. As an interesting application of the force theorem, consider the average total force on all the nuclei. Since the operator for this total force is N (rs-Ra) F" = Z 220T+Zzasmp EFNe-i-FNS, (A3) 5:] a I S— 0| a and since FN. - -F.~, (A4) if follows that 1. From S. T. Epstein, The Variation Method in Quantum Chemistry (Academic Press, NY, 1974) 96 97 (Fm) = ' (FeN>+ (FNS>' However, if the force theorems are satisfied, then 0=+ = = (ilH,Pl), and Eq. (A5) can be rewritten as <17") = - (FeN>+ (FA/3) :