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AAAAAA‘" AAAAAAAAAAAAAAAAAA 'AAAAAA AAAAAA'AAAA 3129 3900 A This is to certify that the dissertation entitled Energies, polarizabilitles, and forces of interact- Ang molecules at long or.in+ermediate range presentedby YAng Q- Liang has been accepted towards fulfillment of the requirements for Ph.D degreein Chemical PhYSlCS (flew Major professor Date MSU is an Affirmative Action/Equal Opportunity Institution 0-12771 LIBRARY Mlchlgan State Unlverslty PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE A AL— J ‘5‘” L___ 1—1 ? A__:A "lAfl—A MSU Is An Affirmative ActioNEquol Opportunlty Institution amp-nu.- —_- ENERGIES, POLARIZABILITIES, AND FORCES OF INTERACTING MOLECULES AT LONG OR INTERMEDIATE RANGE By Ying Q. Liang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements ' for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1992 ABSTRACT ENERGIES, POLARIZABILITIES, AND FORCES OF INTERACTING MOLECULES AT LONG OR INTERMEDIATE RANGE By Ying Q. Liang Collision-induced molecular phenomena are widely researched subjects. In com- plement with other research, deriving new results to understand the nature and effects of the interaction between molecules at long or intermediate range is the main goal of this thesis. By applying Rayleigh-Schrodinger perturbation theory, we have obtained the molecular interaction energy to second order in terms of nonlocal polarizability densities. The derivation also includes the effects of an applied field. The nonlocal polarizability density o.(r; r’, cu) plays a central role in this research. The polarizability density is a linear-response tensor that determines the electronic polar- ization induced at point r in a molecule, by an external electric field of frequency to, acting at r’. When a nuclear position in the molecule shifts infinitesimally, we find that the change in 0t(r; r’, (o) is connected to the same hyperpolarizability BM" r’, 0), r”, 0) that describes the electronic charge distribution’s response to external fields, i.e.: acwmyakla = I dr dr’ dr” 5515‘" r’, m, r”, 0) 21 “rage", R1). This is a generalization of the relationship between BaBY(O)/8R1a and [3375“; r’, r”). Due to establishment of the relationships between BGIMBRIG and [3575’ we have obtained new analytical results for the forces acting on nuclei in a molecule. For the first time, we have proven the equivalence of forces from interaction energy calculations and those obtained via the Hellmann-Feynman theorem, order by order. We are also able to separate forces on nuclei in one of the interacting molecules (A) into those due to its “own” electrons vs. forces due to the charge distribution of the collision partrrer, B. By taking the long range limit of the new analytical results for forces acting on nuclei in a molecule, we express the electrical shielding effects in interacting molecules through nonlocal polarizability and hyperpolarizability densities. Intermolecular fields are screened via the same tensors that describe shielding of external fields. An explicit expression for the momentum distribution of a particle in a one- dimensional box is also included in this thesis. It is a result from my work as a teaching assistant for the graduate course in quantum chemistry for several terms. The result cor- rects misconceptions about the momentum distribution in several quantum chemistry text- books. To All People Who Support, Direct and Cooperate with Me in Scientific Research and Education, Among Them the Industrious Family of Mine iv ACKNOWLEDGMENTS I offer my most grateful thanks to Dr. Katharine L. C. Hunt, whose guidance during the theoretical exploration led to the results of this research. She always shares her research experience and knowledge with me as well as her encouragement to smooth my transition in a research direction from pure physics to a combination of physics and chem- istry. This also opened the door for my further research work. My thanks are extended to the members of my guidance committee, Dr. Robert Cukier (who served as second reader), Dr. Subhendra Mahanti, Dr.Thomas A. Kaplan, and Dr. Jack Hetherington for their generous donation of time and assistance. My thanks also go to Dr. K. M. Chen for his invitation for me to be a visiting researcher in his group for two years, where I obtained training in theoretical research and computational work. My gratitude must also be expressed to Dr. Henry Blosser and Dr. Sam Austin for introducing me to the physics graduate program in the National Supercon- ducting Cyclotron Laboratory, from which I leaned much since they are exemplary scien- tists. I thank Dr. George Leroi, Dr. Paul Hunt, and Dr. Themas Atkinson for their administrative and computational support for my research. I gratefully acknowledge financial support for this research from the National Science Foundation through grants to Dr. K. I... C. Hunt, and I am grateful for the award of a Summer Alumni Fellowship from Department of Chemistry, Michigan State University. I sincerely thank my family members for their understanding and encouragement, especially my wife Ruiqun Xuan, whose consrant love and great support are very truly appreciated. She also deserves as much credit as I for the completion of this dissertation. I also express my thanks to all of my friends, whose help and assistance are very much appreciated. vi Table of Contents LIST OF FIGURES .................................................................................................... I. INTRODUCTION ................................................................................................. II. CHANGES OF ENERGY FOR INTERACTING MOLECULES AT INTERMEDIATE RANGE IN AN APPLIED ELECTROSTATIC FIELD ....... 2.1 The Perturbation Energy .............................................................................. 2.2 The Nonlocal Polarizability Densities .......................................................... 2.3 Analytical Results for the Perturbation Energy in terms of Nonlocal Polarizability Densities ................................................................. III. CHANGES IN ELECTRONIC POLARIZABILITY DENSITIES DUE TO SHIFTS IN NUCLEAR POSITIONS, AND A NEW INTERPRETATION FOR INTEGRATED INTENSITIES OF VIBRATIONAL RAMAN BANDS. 3.1 Introduction .................................................................................................... 3.2 N onlocal Polarizability Densities and Polarization Induced by External Fields ........................................................................................... 3.3 Change in Polarizability Density due to an Infinitesimal Shift in Nuclear Position ........................................................................................ 3.4 Discussion ..................................................................................................... IV. FORCES ON NUCLEI IN INTERACTING MOLECULES ............................ 4.1 Introduction .................................................................................................... 4.2 Forces on Nuclei in Interacting Molecules .................................................. 4.3 The Long-Range Limit, and New Sum Rules for Polarizability and Hyperpolarizability Densities ...................................................................... 4.4 Discussion and Summary ............................................................................. V. ELECTRIC FIELD SHIELDING EFFECT S IN INTERACTING MOLECULES .................................................................................................. 5.1 Chemical Shift and Electric Field Shielding at Nuclei ............................... vii ix 1 4 4 6 7 11 12 14 16 18 22 24 31 39 48 56 56 5.2 Relationships Between Electric Field Shielding Tensors, Dipole Derivatives, Polarizability Derivatives, and Nonlocal Polarizability Densities ...................................................................................................... 57 5.3 Electric Field Shielding Effects in Interacting Molecules ......................... 59 VI. QUANTUM THEORY OF MOMENTUM DISTRIBUTION FOR A PARTICLE IN A ONE-DIMENSIONAL BOX ................................. 65 viii List of Figures 6.1 The position probability densities of a particle in a one-dimensional box; it indicates the energy level of the particle (not to scale) .............................. 75 6.2 The momentum probability densities of a particle in a one-dimensional box; n indicates the energy level of the particle (not to scale) .............................. 76 CHAPTER I INTRODUCTION This thesis is written so that each chapter is independent. However, there do exist internal logical relations between these chapters. We consider two molecules interacting at long or intermediate range in an applied electro-static field. We assume that the intermolecular separation is sufficiently large that the overlap of molecular wave functions is weak and we neglect exchange of electrons be- tween molecules. We focus our attention on the changes of energy and polarization of the system, the forces acting on nuclei in each molecule, and the electrical shielding effects between them. By applying Rayleigh-Schrodinger perturbation theory [1] to the system in Chap— ~ ter 11, we obtain the changes of energy to second order in terms of nonlocal polarizability densities. 0t(r; r’, co) is a linear-response tensor that determines the electronic polarization induced at point r in a molecule, by an external electric field of frequency to, acting at r’, which was introduced by Maaskant and Oosterhoff in a study of optical rotation in con- densed media [2]. Hunt derived a simpler form suited for practical calculations in cases when the field acting on a molecule is derivable from a scalar potential [3]. The nonlocal polarizability density is discussed in Section 2.2. The results for energy changes in terms of the nonlocal polarizability densities are summarized in Section 2.3. Besides the inter action between applied field and each molecule. and the molecular pair interaction, there are trinary interactions among the applied field and the molecules, associated with the collision-induced dipole. In Chapter 111, it is shown that the change in frequency-dependent electronic polar- izability densities due to shifts in nuclear positions depends upon the hyperpolarizability density. This generalizes the relation initially established by Hunt [4] in the case when ex- ternal field is static. By generalizing the relationship to the frequency-dependent case, we are able to give a new interpretation for integrated intensities of vibrational Ramarr bands as well as new analytical results for van der Waals’ forces acting on nuclei in interacting molecules [5]. Chapter IV gives analytical results for the induced forces acting on nuclei in inter- acting molecules, in terms of nonlocal polarizability densities. We calculate these forces both by direct differentiation of the interaction energy and by use of the Hellmann- Feynman theorem [6]. By proving the equivalence order by order, we unify the two approaches. Chapter V discusses the electrical shielding effects in interacting molecules at long range. By using the relationships initially established by Hunt [4] and generalized in Chapter III, and taking the long range limit of the forces acting on nuclei in interacting molecules obtained in Chapter IV, we prove that the same electrical shielding tensors de- scribe not only the response to an external field, but also the response to local intermolec- ular fields. Lastly, Chapter VI discusses a simple but general problem in quantum mechanics: the momentum distributions of a particle in a one-dimensional box. A simplified and ex- plicit formula for this case is'obtained and used to correct misconceptions in some quan- tum chemistry textbooks. References [1] See, for example, E. Merzbacher, Quantwn Mechanics (Wiley, New York,1970), Chap. 17. [2] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). [3] K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983); 80, 393 (1984). [4] K. L. C. Hunt, J. Chem. Phys. 90, 4909 (1989). [5] .K. L. C. Hunt, J. Chem. Phys. 92, 1180 (1990). [6] H. Hellmann, Eirgft'ihrung in die Quantenchemie (Franz Deuticke, Leipzig, 1937), p. 285. R. P. Feynman, Phys. Rev. 56, 340 (1939). CHAPTER II CHANGES OF ENERGY FOR INTERACTING MOLECULES AT INTERMEDIATE RANGE IN AN APPLIED ELECTROSTATIC FIELD 2.1The Perturbation Energy The perturbation Hamiltonian for molecules interacting at intermediate or long range in an applied electrostatic field is H' = A pA(r) me T( r - r3 dr dr' + l er> Am dr + A me ¢ dr .(1) where pA(r) and pB(r) are the charge density operators for molecules A and B, respective- ly; T( r - r’) = I r - r’ I"; and ¢(r) is the scalar potential of the applied electrostatic field. The external field Emu) is 13%) = —V ¢(r), (2) Assuming that the Rayleigh-Schrodinger perturbation theory [1] can be applied to this case, we obtain the change of energy for the system to first order as AB“) = l pAo(r) T( r - r) 9300') dr dr’ + A pAo(r) ¢dr+ 1 p30 (r) ¢ dr. (3) where pAo(r) and p30 (r) are the unperturbed charge densities of A and B, respectively. The first term in Eq. 3 is the electrostatic interaction energy of the unperturbed charge distributions of the pair, while the second and third terms are the electrostatic interaction energies of the unperturbed charge distributions of molecules A and B with the external field. The change of energy for the system to second order is A50): _ 2 (gAgB |H'|kAgB)(kAgB IH’lgAgB)/(EkA’EgA) katg ‘1}:‘(gAgB IH’lgAJ’BHgAJB AH'ISAEBV‘EJB'EzB) m —2 (gAgB IH'IkAjBXkAjB IH’lgAgB)/[(EkA-E3A)+(Ej3-EgB)] j.k¢8 =-Idrdr’dr ’"dr ”2 (gA lpA(r) lkAXRA IPA“ 7 'gAWEk '5 3A) xpBo(r’)|r- r"B’|1po(r ’”’)|r -r’ ”'l1 —l drdr’dr”dr”’.22’/[(E.A-EAA)+(EjB-EAB)A ><|r-r’l'1xlr”-r’”|'l —l drdr’dr”2’(gA lpA(r) Mm:A lpA(r”) lgA)/ (E.I AEA“) x ¢(r)p30(r’) I r” - r’ I'1 + complex conjugate — A dr dr'dr" 2' < gB lme IjB > (AB lpB/ (E,-B - ES“) 1' x ¢(r) pA0(r’) I r” - r’ I'1 + complex conjugate -ldrdr'2'(gA IpA(r) IrcA‘\)(1cA lpA(I") lgM/(EkA-Eg") k x ¢(r)¢(r’) -Adrcnr'>:' 40)] < o I Pa(r) G(m) PB(r’) lo ), (5) when the frequency a) is off-resonance with molecular transition frequencies. C((o —9 -(r)) designates the operator for complex conjugation and replacement of to by -(0, and G(m)=(1- 600) (H - Bo - Am»-1 (1 - too). (6) where 500 is the ground-state projection operator I 0 ) ( 0 l. The electronic polarization Pind(r, (0) induced in a molecule by an external field F(r, (0) depends on the polarizability density a(r; r’, to), the hyperpolarizability density B(r; r’, (0’, r”, 0)”) and higher-order nonlinear response tensors Pi“d(r, to) = I dr’ a(r; r’, (1))° F(r’, to) +1/2 Loo” dm’ l dr’dr” an; r’, m- (0’, r”, (0’) : F(r’, (o- (0’) F(r”, to’) +... . = Pi“d(r, (0)“) + Pind(r, (0)“) + . . . . (7) where PM“, (0)“) =I dr’ a(r; r’, (u) - F(r’, to). Ph‘d(r, (0)“) gives the electronic polar- ization in a molecule by the external field to first order, and Pind(r, (0)0) = 1/2 Loo” dto’ I dr’ tit” 50': r’, m - (0’, r”, m’) : F(r’, co - w’) F(r”, m’). Pind(r, (0)“) gives the second order term. B(r; r’, O) - m’, r”, (0’) is the hyperpolarizability density; when (0’ = O, Bamm r’, to, r”, 0) = [1 + C((o —> -m)] x {( o | Pam 0(a)) [Pytro - Pymtro] Gtw) P30") l0) + < o l Pa(r) 0(0)) [pan-3 - 93mm] G(O) Pym l0) + (o I Pym 0(0) [ram - Pam(r)] can) pan-3 I 0)} . (3) 2.3 Analytical Results for the Perturbation Energy in terms of Nonlocal Polarizabil- ity Densities We now rewrite the perturbation energy AEQ) in terms of the nonlocal polarizabil- ity densities and obtain: AE(2)=—2 (gA gB IH'IkAgBHkA g3 IH’lgAgBHaEkA-ESA) -k.2alw(g"gB IH’lgAJ’BMgAJ’B III’lg“‘g‘3>/(E,-B 4533) J¢8 —2: (gAgB IH’IkAJ'i>= —1/2 I dr Mm“) 730m— 1/2 1 dr Page“) 7" onto _ 0° , n u, A 0,. B n ,- h/41c2l0 dmldrdrdr dr a “Boar ,rm)a16(r ,r,rw) x Task, r’)TpY(r”, r’") —A dr PAa(r) Fe"t am - l dr PBa(r)<1>Fwa(r) —1/2! dr PAexta(r)(l> Fm an) .12! dr PBexta(r)(1) Fe" cl(r) (10) where PAa(r)(1)= l dr’ “Aug“, r’) rBoBm (11) is the first order polarization of molecule A induced by the field due to the unperturbed charge distribution of molecule B, while ch‘a(r)(l) = l dr’ awn, r’) Fext Ba) (12) is the first order polarization of molecule A or B induced by the applied field. From Eq. 9, we know that when two molecules interact with each other, there is not only the electrostatic interaction between their unperturbed charge distributions (in first order), but also the interaction between the induced charge density of one and the un- perturbed charge distribution of the other (the first and second terms in Eq. 9, induction energy). Furthermore, there is a dispersion interaction between the two molecules, which comes from the fluctuations of the charge distributions of A and B, a purely quantum effect (the third term in Eq. 9, dispersion energy). When the molecules interact with an applied field, the external field affects the electrostatic and induction interactions between them, via the same polarizability density (the sixth and seventh terms in Eq. 9). The applied field acts on the interacting molecules not only directly with each, but also indi- rectly on one through the other. An alternative point of view is: each of the interacting molecules not only directly responds to the external field, but also does so indirecly through the other. For example, the field produced by the unperturbed charge distribution of B induces in A a polarization which interacts with the applied field (the fourth term in Eq. 9). An alternative point of view is: the applied field induces in A a polarization which interacts with the field produced by the unperturbed charge density of B. The same also happens for B (the fifth term in Eq. 9). We call this kind of interaction trinary. 10 References [1] See, for example, E. Merzbacher, Quantum Mechanics (Wiley, New York,1970), Chap. 17. [2] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). [3] K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983); 80, 393 (1984). CHAPTER III CHANGES IN ELECTRONIC POLARIZABILITY DENSITIES DUE TO SHIFTS IN NUCLEAR POSITIONS, AND A NEW INTERPRETATION FOR INTEGRATED INTENSITIES OF VIBRATIONAL RAMAN BANDS Abstract: The nonlocal polarizability density or(r; r’, to) is a linear—response tensor that determines the electronic polarization induced at point r in a molecule, by an external electric field of frequency to, acting at r’. This work focuses on the change in a(r; r’, (1)) when a nuclear position shifts infinitesimally. We prove directly that the electronic charge distribution responds to the change in Coulomb field due to the nucleus via the same hy- perpolarizability density that describes its response to external fields. This generalizes a result established previously for the static (or = 0) polarizability density. The work also provides a new interpretation for the integrated intensities of vibrational Raman bands: it proves that the intensities depend on the hyperpolarizability densities and the dipole prop- agator. ll 12 3. 1. Introduction The nonlocal polarizability density a(r; r’, (1)) gives the (1)-frequency component of the polarization induced at point r in a molecule by an external electric field F(r’, 0)) acting at the point r’, within linear response [1-5]. This property reflects the distribution of po- larizable matter within the molecule; it represents the full response to external fields de- rived fiom scalar potentials of arbitrary spatial variation. Thus a.(r; r’, (1)) is a fundamental molecular property. It has applications in theories of local fields and light scattering in con densed media [3,6] , and in approximations for dispersion energies [4], collision-induced di- poles, and collision-induced polarizabilities [5,7] of molecules interacting at intermediate range. Recently, Hunt [8] has shown that or(r; r’, 0) also determines the net field FI acting on nucleus I of a molecule in a static, external field F‘(r): F1 = FRO) + F°(RI) +l dr dr’ T(RI, r) - a(r; r’, 0) - F°(r’) + , (1) ' where FRO) is the field at nucleus 1 in the absence of the external perturbation, and Taflml, r) is the dipole propagator, i.e., Taflml, r) = V a VB ( I RI - r H). Specializing Eq. (1) to the case of a uniform external field leads to an expression for the linear electric field shielding tensor 71 [9- 15] in terms of a(r; r ’, 0). Further, the nonlocal polarizability density determines the derivative of the molecular dipole moment with respect to the position of each of the nuclei: If ZI is the charge on the Ith nucleus, then [8] aha/6R1 2.215043 +21 ldrdr am; r;r ,1‘0) (r, R1). (2) Th1s woiic focuses on the changes 1n the frequency ependent molecular polariz- ability density when a nucleus shifts infinitesimally. The results are important because of the roles of the polarizability density noted above. In addition, the theory yields the deriv- atives of th polarizability (1043(0)) with respect to the normal mode coordinates qv, which l3 determine the integrated intensities of vibrational Raman bands, within the Placzek approx- imation [l6]. Earlier Hunt [8] has shown that the derivative of the static polarizability BOLCLB(O)/E)RI7 is related to the nonlinear response tensor 50'; r’, 0, r”, 0). This accounts , for the connection between the polarizability derivative and the quadratic electric field shielding tensor (cf. Ref. 15). The purpose of this work is to prove that the relation between Baas“; r’, 0)]8RI7 ' and the nonlinear response tensors generalizes to the frequency-dependent case. The anal- ysis in Ref. 8 employs the electrostatic Hellmann-Feynman theorem [9,17], and therefore does not apply to a(r; r’, m) with a) at O. A new approach is needed to prove the generali- zation. In this work, we have used direct differentiation to evaluate Bataan; r’, (ID/31111. For the derivative of the total polarizability (1(0)), we obtain BaBY(m)/3R‘a = I dr dr’ dr” awe; r’, (o, r”, 0) 2I 754"", RI). (3) This result yields physical insight into the change in polarizability (at frequency to) that results from an infinitesimal shift in the position of nucleus 1. The molecule responds to the change in the Coulomb field of the nucleus via its hyperpolarizability density B(r; r’, (u, r”, 0). All of the quantum mechanical influences are contained within B, and the remainder of the calculation is classical. Eq. (3) also provides the basis for a new in terpretation of integrated Raman band intensities, without requiring that (105(0)) be approximated by its zero-frequency limit. 14 3.2 Nonlocal Polarizability Densities and Polarization Induced by External Fields The electronic polarization Pind(r, (1)) induced in a molecule by an external field F(r, (1)) depends On the polarizability density a(r; r’, to), the hyperpolarizability density [3(r; r’, 03’, r”, to”) and higher-order nonlinear response tensors: Pind(r, (1)) = I dr’ a(r; r’, (1)) - F(r’, to) + 1/2 Loo” dw’ I dr’ dr” [3(r; r’, (o - 0)’, r”, (0’) : F(r’, a) - (0’) F(r”, m’) + . (4) The polarization Pind(r, to) is related to pind(r, to), the induced change in electronic charge density in the field F(r, to), by V - Pindtr. m) = - pimtr. co). ' <5) and the same relationship holds for the polarization and charge density operators, P(r) and p(r) respectively. The polarizability density for a molecule in the ground state has the form “an“; r’, to) = [1 + C((o -+ -(o)] ( o lPa(r) 6(a)) PB(r’) lo ),(6) when the frequency a) is off-resonance with molecular transition frequencies. C(to -9 ~03) designates the operator for complex conjugation and replacement of (.0 by -m, and 6(0)) = (1 - too) (H - 150- mo)" (1 - too). (7) where 500 is the ground-state projection operator I 0 ) ( 0 I . It should be noted that the nonlocal polarizability density completely determines the electronic charge redistribution linear in a perturbing field F(r, 0.)), and not simply the dipolar component. Integration of 0t(r; r’, to) over all space with respect to r and r’ gives the dipole polarizability (1(a)); but or.(r; r’, to) also determines all of the higher-multipole, linear-response tensors [5]. 15 The hyperpolarizability density B(r; r’, (o’, r”, or”) gives the polarization induced at r bythe lowest-order nonlinear response to a field of frequency 0)’ acting at r’ and a field 4 of frequency to” acting at r”. Integrating Bah“; r’, (u’, r”, to”) with respect to r, r’, and r” over all space yields [3 ((1)’, to”), while moment integrals of Bap-(m r’, w’, r”, to”) a 7 give all of the third-order higher multipole susceptibilities. For the proof to be given here, we require the hyperpolarizability density Barfly“; r", to, r”, 0), which has the form Bony“; r’, a), r”, 0) = [1 + C(tu —+ -0))] x {( o | Pa(r) 6(a)) [Pym - P700(r”)] 6(a)) PB(r’) l0) + ( o l Pa(r) 6(0)) (950-) - PBOOh- )1 0(0) P70“) | o ) + ( o I P7(r”) (3(0) [Pa(r) - Pawn» 6(a)) Pfl(r’) I o )} ’. (3) Eq. (8) is derived by analogy with Eq. (43b) in Ref. 18. For compactness, we have used the notation Puma) = ( 0 I Pa(r) I 0 ), and similarly for Pfloo(r’) and P700(r”). Damping has been neglected in Eq. (8). From Eq. (4), if a molecule is placed in a static external field Fs(r), its reaction to an additional external field F(r, to) [19] can be characterized by the effective polarizability density a°(r; r’, to; F5), given by ae(r; r’, 0); F) = a(r; r’, to; F3 = 0) + I dr” B(r; r’, (u, r”, 0) . F5(r”) + . (9) The permutation symmetry of the B hyperpolarizability density has been employed to obtain this result. 16 3.3 Change in Polarizability Density due to an Infinitesimal Shift in Nuclear Position A shift 5RI in the position of nucleus I in a molecule changes the nuclear Coulomb field acting on the electrons. In this section, we prove directly that the resulting change in polarizability density is determined by the same hyperpolarizability density Ba Y(r; r’, 0)’, r”, m”) that fixes the response to external fields. Specifically, we show acme; r’, cu)/aR1a = I dr 5518‘" r’, to, r”, 0) 2I The", R1), (10) where ZI is the charge on nucleus I and T5a(r”, R1) is the dipole propagator. The proof in this section is based on direct differentiation of the polarizability den- sity “0130.; r’, (u) with respect to RI? . From Eq. (6), Bataan; r’, (tn/8R17 = [l + C(co —-> -o))] x [ ( 30/3187 Irma) G(to) PB(r’) Io ) + (0 Iran) ileum/31?}1r 95m I o ) + (o IPa(r)G((o) PB(r’) 130181117) 1. (11) To convert Eq. (11) into Eq. (10), we first take the derivative of the ground state with respec to an arbitrary parameter 1] in the Hamiltonian, lac/an) =-o(0) aH/an IO). (12) We also require the derivative of the operator 6(a)): 86(w)/8n = - G(w) 3(H - 130an 6(0)) + pearl/an (3(0) G(to) + G(tu) 0(0) art/an too . (13) Specializing Eqs. (12) and (13) to the case n = R17 gives the derivatives in Eq. (11). The change in the Hamiltonian due to the shift 8R17 is given by art/aidY = I dr zI VIY Ir - R1 l-1 p(r). (14) where V17 denotes B/BRIT Eq. (14) for arr/3R1? can be rewritten in terms of the polariza- 17 tion operator P(r) by using Eq. (5) for P(r) and p(r), integrating by parts with respect to r, andusing I -l _ I I -l Valr-Rl --VaIr-RI . . (15) Thisgives 1 _ _ n I . u 1 arr/aRY- I dr 2 P5(r')Tfi(r ,R). (16) Together, Eqs. (8), (11)-(13) and (16) prove Eq. (10) for 8am“; r’, (ID/3R1“. Equivalently, the polarizability density “6043“; r’, to) for the molecule perturbed by an infinitesimal shift of nucleus I satisfies Eq. (9), with Fs(r”) replaced by 8f I r”), the infin- itesimal change in the Coulomb field of nucleus 1, due to its displacement by 8R1. This shows that the molecule responds via to the change in the Coulomb field of nucleus I via the same hyperpolarizability density that governs its response to external fields. Integrating Eq. (10) over all space with respect to r and r’ gives the Eq. (3) for the derivative of the electronic polarizability (157(0)) with respect to Rla. Clearly the derivative of the polarizability (1043(0)) with respect to the normal mode coordinate qv is aaafl((r))/aqv = 2 BaaB((o)/3RIY only /aqv. ' (17) 18 3.4 Discussion This work has shown that the derivatives of the polarizability density with respect to nuclear coordinates depend upon the dipole propagator and the hyperpolarizability den- sity B(r; r’, (0’, r”, 0); the density B yields the lowest-order nonlinear response to imposed fields, on integration. Thus, we have generalized the relationship between static linear and nonlinear response tensors [8] to the frequency-dependent case. Our work also gives a new expression for the second derivative of the dipole with respect to nuclear coordinates, for a molecule in any nuclear configuration. From Eq. (2) for auB/aala and Eq. (10), day/812103319B = zI zJ I dr dr’ dr” awn: r’, o, r”, 0) rash-'3 R’) T5a(r’, R1) + 218,, I dr dr’ case; r’, 0) TJMu-z R1), (18) where Time, R1) = - Va VB VY( l r - R1 H). (r tensors of odd orders are odd in the difference between the two arguments r and R’). A shift in the position of nucleus I from RI to RI + 5RI changes the field at point r, due to nucleus 1, from fla(r) = - VaZ‘( Ir - R1 I") fto f1a(r) + 8f1a(r) + 1/2 82f1a(r) + = - Va 21 ( Ir - R1 l-1)+ 21 TaB(r’ R1) 8111‘3 + 1/2 zI Timur, RI) 8R‘B 6R1? + . (19) Thus the first term on the right in Eq. (18) represents the nonlinear response (via B) to the changes in the nuclear Coulomb field at r’ and r”, while the second term represents the linear response (via a) to the second variation in the field at r’. Extensions of this analysis to find higher derivatives of the dipole and to find second and higher derivatives of the sus- l9 ceptibilities are straightforward. Immediate uses of the results from this work are concep- tual rather than computational. Applied to sin gle-molecule polarizabilities, our work pro- vides a new physical interpretation for integrated intensities of vibrational Raman bands, by showing that the band intensity depends on the response of the molecule to the change in Coulomb fields of the nuclei, via the B hyperpolarizability density. For interacting mol- ecules with nonoverlapping or weakly overlapping charge distributions, induction and dis- persion energies, collision-induced dipoles and collision-induced polarizabilities are all related to the sin gle-molecule polarizability densities. Thus, Eq. (10) determines in part the nuclear-coordinate dependence of these properties. For computational purposes, methods of finding the required components of a(r; r’, 0) are known (see Refs. 5 and 7, and references therein), and methods of approxi- mating B(r; r’, a), r”, 0) are currently under development. With information on B(r; r’, to, r”, 0), it should be possible to identify the regions of the electronic charge dis tribution that make the principal contributions to the vibrational Raman band intensities for isolated molecules; and this would facilitate tests of atom- or group-additivity approx- imations. The dipole propagator tensors appearing in dragon/3R1“ weight the regions nearest to nucleus 1. This tends to support additive approximations, provided that B(r; r’, to, r”, 0) is largest for small Ir - r' I and Ir - r” I. References [1] W. J. A. Maaskant and L. J. Oosterhoff, Mol. Phys. 8, 319 (1964). [2] L. M. Hafltensheid and J. Vlieger, Physica 75, 57 (1974). [3] T. Keyes and B. M. Ladanyi, Mol. Phys. 33, 1271 (1977). [4] K. L. C. Hunt, J. Chem. Phys. 78, 6149 (1983). [5] K. L. C. Hunt, J. Chem. Phys. 80, 393 (1984). [6] Additionally, optical rotation in condensed media can be treated using a generalized linear response tensor (see Refs. 1 and 2). [7] K. L. C. Hunt and J. E. Bohr, J. Chem. Phys. 84, 6141 (1986). [8] K. L. C. Hunt, J. Chem. Phys. 90, 0000 (1989). [9] R. P. Feynman, Phys. Rev. 56, 340 (1939). [10] R. M. Stemheimer, Phys. Rev. 96, 951 (1954). [11] H. Sambe, J. Chem. Phys. 58, 4779 (1973). [12] P. Lazzeretti and R. Zanasi, Chem. Phys. Lett. 71, 529 (1980); J. Chem. Phys. 84, 3916 (1986); 87, 472 (1987). [13] S. T. Epstein, Theor. Chim. Acta 61, 303 (1982). [14] P. Lazzeretti and R. Zanasi, Chem. Phys. Lett. 112, 103 (1984). [15] P. W. Fowler and A. D. Buckingham, Chem. Phys. 98, 167 (1985). [16] See, e.g., J. Tang and A. C. Albrecht, in Raman Spectroscopy, Vol. 2, H. A. Szyman- ski, ed. (Plenum, New York, 1970), pp. 33-68; and Vibrational Intensities in Infiared and Raman Spectroscopy. W. B. Person and G. Zerbi, eds. (Elsevier, Amsterdam, 1982). [17] H. Hellmann, Einflihrung in die Quantenchemie (Franz Deuticke, Leipzig, 1937), p. 285. [18] B. J. Orr and J. F. Ward, Mol. Phys. 20, 513 (1971). 21 [19] Related results for the effective frequency-dependent polarizability of a molecule in an external field have been given by A. D. Buckingham, Proc. Roy. Soc. London, Ser. A 267, 271 (1962); and D. M. Bishop, Mol. Phys. 42, 1219 (1981); J. Chem. Phys. 86, 5613 (1987) CHAPTER IV FORCES ON NUCLEI IN INTERACFING MOLECULES Abstract: When the charge overlap between interacting molecules or ions A and B is weak or negligible, the first-order interaction energy depends only upon the molecular positions, orientations, and the unperturbed charge distributions of the molecules. In contrast, the first-order force on a nucleus in molecule A as computed from the Hellmann-Feynman theorem depends not only on the unperturbed charge distribution of molecule B, but also on the electronic polarization induced in A by the field from B. At second order, the inter- action energy depends on the first-order, linear response of each molecule to its neighbor, while the Hellmann-Feynman force on a nucleus in A depends on second-order and non- linear responses to B. One purpose of this work is to unify the physical interpretations of interaction energies and Hellmann-Feynman forces at each order, using nonlocal polariz- ability densities and connections that we have recently established among permanent mo merits, linear response, and nonlinear response tensors. Our theory also yields new information on the origin of terms in the long-range forces on molecules, through second order in the interaction. One set of terms in the force on molecule A is produced by the field due to the unperturbed charge distribution of B and by the static reaction field from B, acting on the nuclear moments of A. This set origi- nates in the direct interactions between the nuclei in A and the charge distribution of B. A second set of terms results from the permanent field and the reaction field of B acting on the permanent electronic moments of A. 22 This set results from the attraction of nuclei in A to the electronic charge in A itself, polar- ized by linear response to B. Finally, there are terms in the force on A due to the perturba- tion of B by the static reaction field from A; these terms stem from the attraction of nuclei in A to the electronic charge in A, hyperpolarized by the field from B. For neutral, dipolar molecules A and B at long range, the forces on individual nuclei vary as R'3 in the intermolecular separation R, at long range; but when the forces are summed over all of the nuclei, the vector sum varies as R4. This result, an analogous conversion at second order (from R‘6 forces on individual nuclei to an R'7 force when summed over the nuclei), and the longrange limiting forces on ions are all derived fiom new sum rules obtained in this work. 24 4.1 Introduction When molecules A and B interact, the net force F1011 nucleus I in molecule A is the sum of the force FRO) on I in the absence of molecule B and an interaction-induced force AFI. The interaction-induced force is related to the AB interaction energy AE by AFICt = - aAE/aala , (1) where RI is the position of nucleus 1. Throughout this work, we use the Born-Oppenheimer approximation: we determine the forces on the nuclei as functions of the nuclear coordi- nates, fixed within individual calculations but not resn'icted to the equilibrium configura- tion. The electronic state is the fully perturbed ground state of the AB pair in the specified nuclear configuration. For molecules with weak or negligible charge overlap, the first-order interaction energy, denoted by AB“), is determined completely by the molecular positions, the orien- tations, and the unperturbed charge distributions of molecules A and B. In contrast, the first-order interaction-induced force AFIm on nucleus I in molecule A, obtained directly from Eq. (1), depends not only on the interaction of that nucleus with the unperturbed charge distribution of B, but also on interaction-induced changes in the electronic charge distribution of A. At second order in the A-B interaction, the induction energy AE(2)ind is determined entirely by the first-order, linear response of each molecule to the field of its neighbor. Yet the associated induction force AFI(2)ind on nucleus I in molecule A does not originate solely in the first-order perturbed charge distributions of A and B. Instead, Mimi“ also depends on the hyperpolarization of the electronic charge in A by the field from B, and on the second-order change in the electronic charge density of A due to linear response to the perturbed charge distribution of B. The results stated above appear counter-intuitive, although they are fully consis- tent with the Hellmann-Feynman theorem [1, 2] applied to compute forces on nuclei in interacting molecules, given the electronic charge distributions. One purpose of this paper is to connect the physical interpretations of interaction energies and Hellmann—Feynman forces, order by order. We allow for the possibility that A and B may be ionic. An impor- tant component in the analysis is the inter-relation that we have recently established among permanent moments, linear response, and nonlinear response tensors [3, 4]. In Sec. 4.2, we use nonlocal polarizability densities to find AFIU), AEmind, and Alfie)“. The polarizability density tensors give the polarization produced at one point in a molecule due to the application of an external field at other points [3-12], and thus repre- sent the distribution of polarizable matter throughout the interacting molecules. Quantum mechanical definitions for these tensors are given in Sec. 4.2. Earlier, it has been estab- lished that the nonlocal polarizability density determines the derivatives of the molecular dipole with respect to nuclear coordinates [3], and that the first hyperpolarizability density determines the derivatives of the polarizability with respect to nuclear coordinates [3, 4]. In each case, the molecule responds to the change in Coulomb field 'due to an infinitesimal shift in nuclear position via the same susceptibility density that determines its response to external fields. Our approach, based on nonlocal polarizability densities, holds even when low- order, point-multipole models break down, provided that overlap and exchange between molecules A and B are minimal; for example, our analytical results apply to planar mole- cules in "sandwich" configurations and to long, chain-like molecules in configurations 26 where appreciable charge overlap could be produced by rotation of either molecule. Our . approach includes the direct effects of overlap on the electrostatic and inductive interac- tions, but it does not account for modifications of the classical interactions due to electron exchange or charge transfer between A and B. Overlap damping effects on dispersion energies have been studied extensively at this level of approximation [9, 12-27]. In Sec. 4.3, we take the long-range limits of the forces AFIO) and Mimi“, and express the results in terms of the field and field gradients due to molecule B, together with the screening tensors that represent the effects of electronic redistribution in mole- cule A [28-44]. We then sum over all nuclei in molecule A in order to find the total force on molecule A. In the process, we resolve a problem connected with the Hellmann- Feynman interpretation of the long-range forces. For specificity, we focus on the long- range forces on neutraldipolar molecules A and B: The lowest-order, long-range force on an individual nucleus in A varies as R'3 in the separation R between A and B--but when this force is summed over all of the nuclei in A, the R’3 component must drop out, leaving an R' 4 force on the entire molecule, to leading order. Since the summation runs over nuclei only, elimination of the R'3 component is not a simple charge cancellation effect. At second order, the leading term in the force AFIGQ)“ on an individual nucleus in A depends in part on the attraction of the nucleus to the second-order perturbed elec- tronic charge distribution of molecule A, and thus it varies as R'6--but when the force is summed over all nuclei in molecule A, the result must vary as R7. The elimination of the R‘3 and R’6 terms in the forces on molecule A follows fi'om new sum rules that we derive in this work. The sum rules apply to integrals involving polarizability densities and dipole propagators from points in the electronic charge distribution to the nuclei, summed over the nuclei. In Sec. 4.3, we also analyze the long-range, interaction-induced forces on molecule A into components originating in the interaction with the perturbed electronic charge dis- tribution of A, or with the charge distribution of B, and we obtain new results at both first and second order. At first order, the interacrion-induced force can be written as a sum of two sets of terms. One involves the net charge on all of the nuclei in A, and the nuclear con- tributions to the dipole, quadrupole, and higher moments, while the other involves the net electronic charge and the electronic contributions to the permanent charge moments. We ' show that all of the terms in AFAU)“ containing the nuclear charge or nuclear moments of A result from the direct interaction between the nuclei in A and the unperturbed charge dis- tribution of molecule B; the terms in AFAO)‘ll containing permanent electronic moments of A result from the attraction of the nuclei to the electronic charge on A, perturbed to first order by interaction with B. At second order, the induction energy AEmind is determined by the static "reaction fiel " acting on molecules A and B. To lowest order, the static reaction field at A is the field resulting from the polarization of B by the permanent charge and moments of A (similarly for the reaction field at B). We show that interactions of A nuclei with the polarized charge distribution of B appear directly in the force, as terms involving reaction field effects on the nuclear moments of A. We also show that force terms involving reac- tion field effects on the electronic moments of A stem from the attraction of the A nuclei to the second-order, linear change in the electronic charge density of A itself. Finally, there are terms in the interaction-induced force on molecule A that are associated with the reaction field at B. These contain linear response tensors on A. We show that these terms 28 stem from the attraction of nuclei in A to the electronic charge of A, hyperpolarized by the field from B. At second order, the total interaction energy for molecules at long range is the sum of the induction energy AEmind discussed above and the dispersion energy AE(2)disp' The dispersion (van der Waals) energy results from dynamic reaction field effects, due to cor- relations of the spontaneous, quantum mechanical fluctuations in charge density on the interacting molecules. Previously, we have analyzed dispersion forces using nonlocal polarizability densities [45], and the results are summarized briefly below. Through second order in the molecular interaction, the total interaction-induced force is obtained by adding the forces AF“) and M10)“, determined in this work to the dispersion force found earlier. We have shown that the dispersion force on molecule A results entirely from the attraction of nuclei in A to the dispersion-induced change in the electronic charge distribu- tion on A [45]. In Ref. 45, a direct perturbative approach is used to find the dispersion terms in the charge densities of molecules A and B, through second order in the interaction. The polarization of A due to dispersion depends on the frequency-dependent hyperpolarizabil- ity density of A and the polarizability density of B, taken at imaginary frequencies. Sepa- rately, the nonlocal polarizability density theory gives the dispersion energy: Spontaneous fluctuations in the polarization of molecule A produce a field that polarizes B nonlocally. The induced polarization of B gives rise to a reaction field at A, with a resultant energy shift that, depends on correlations of the fluctuating polarization of A at two points. Via the flue- tuation-dissipation theorem, the correlations are connected to the imaginary part of the non- local polarizability density of A. The total dispersion energy is obtained by adding the energy shifts due to the reaction field effects at A and B, and then it is cast as an integral 29 (over imaginary frequencies) of the product of the polarizability densities of the two inter acting molecules. Comparison of the dispersion force on a nucleus in A evaluated by differentiating the dispersion energy vs. that calculated from the dispersion-induced change in the polarization of A establishes the origin of the dispersion force. Hunt [45] proved a conjecture by Feynnian about the origin of forces between atom in S states [1], and generalized it to molecules of arbitrary symmetry. Feynman originally suggested that [1]: "The Schrodinger perturbation theory for two interacting atoms at a separation R, large compared to the radii of the atoms, leads to the result that the charge distribution of each is distorted from central symmetry, a dipole moment of order l/R7 being induced in each atom. The negative charge distribution of each atom has its center of gravity moved slightly toward the other. It is not the interac- tion of these dipoles which leads to van der Waals’ force, but rather the attraction of each nucleus for the distorted char e distribution of its own electrons that gives the attractive 1/R force." Prior to Hunt’s work, this conjecture had been proven by Hirschfelder and Eliason [46; see also 47], for the particular case of two hydrogen atoms, both in the 1s state. Hunt [45] pro- vided the first explicit, general proof, and resolved two problems associated with the con- jecture. First, the dispersion-induced change in charge density and the dispersion dipole both depend on nonlinear response tensors [10, 48-51] for molecules interacting at long range, while the dispersion energy and thus the dispersion forces depend on linear response [9, 12-27, 52], to leading order. The required connection between linear and nonlinear response is provided by our recent proof that the hyperpolarizability density determines the changes in the polarizability when nuclei shift [3, 4]. Second, while the dispersion dipole varies as R'7 for distinct, nonoverlapping atoms A and B in 8 states, the dispersion-induced change in charge density actually varies as R'6. Therefore Feynman’s electrostatic inter- pretation would predict an R’6 dispersion force in the absence of additional constraints. 30 A sum rule on the frequency-dependent hyperpolarizability density B(r, r’, r”; in), 0) [45] eliminates the net attraction of the nuclei to the R'6 component of the electronic charge dis- tribution. This result is particularly striking for noncentrosymmetlic molecules [45]. For these species, the long-range dispersion dipole varies as R'6, while the net dispersion force varies as R'7, as for heteroatoms. The rationale behind Feynman’s conjecture failsuyet the electrostatic interpretation of the dispersion forces still holds, because of the sum rule on B(r, r’, r”; in), 0). This work is related to an electrostatic force theory based on the Hellmann—Feynman theorem, which has been deve10ped by Nakatsuji and Koga [53; see also 54, 55] and applied to the special case of interactions between two atoms. Within this theory, a density matrix analysis is used to decompose the forces on nuclei into distinct terms, and only two forces--the atomic dipole (AD) force and the extended gross charge (EGC) force--act between atoms at long range. The atomic dipole force on the nucleus of atom A results from the polarization of A induced by interaction with B [53]. It corre- sponds to the attraction of the nucleus of A to the electronic polarization PAa(r)(“) (with n > 1), in our approach (see Sec. 4.2). The extended gross charge force results from electro- static interactions of the nucleus in A with the electrons and nucleus of atom B [53]; it corresponds to the force due to the charge distribution pBo(r) and the polarization PBa(r)(“) (with n > 1). We find the force on a nucleus in A in terms of PAa(r)(1) and p30(r) at first order in the A-B interaction, and in terms of PAa(r)(2) and P3a(r)(1) at second order. This is consistent with the electrostatic force theory. Our approach differs from that in Ref. 53, however, since we use nonlocal polarizability densities to derive PAa(r)(“) and PBa(r)(“) and thus to deduce the forces, while Nakatsuji and Koga have given the AD and EGC forces in terms of density matrix elements that are not further specified, in general. 31 Based on the density-matrix analysis, Nakatsuji and Koga [53] have concluded that the electrostatic force theory of long-range interactions is "quite difi’erent from the traditional energetic theories in both theoretical and interpretative views." This conclusion also con- trasts with our work. By use of relations we have recently derived among permanent moments, polarizability densities, and hyperpolarizability densities [3, 4], our work unifies the theories.This work is related in an indirect way to the incorporation of induction effects into density functional theory, carried out by Harris [56]. Our approach yields the mean- field interaction energy in terms of the unperturbed charge densities and the induced polar- ization, computed from the polarizability densities. 4.2 Forces on Nuclei in Interacting Molecules In this section, we find the interaction-induced forces on nuclei in a pair of mole- cules A and B from the perturbation series for the interaction energy AE, and we establish the physical interpretation of the forces. The forces are determined for fixed nuclear con- figurations. The unperturbed electronic states of the AB system are taken as direct prod nets of states on A with states on B, under the assumption that overlap between the charge distributions of molecules A and B is weak or negligible. AB is expanded as a series in the perturbation vAB [9, 12-27], given by vAB = I pA(r) pB(r’) Ir-r'l-l drdr’ , (2) where pA(r) and pB(r’) are the molecular charge density operators: pA(t-) =2: e 8(r - rj) +2: zI 8(1- - R1) ;(3) j 1 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. 32 The interaction energy AB“), taken to first order in V“, depends upon the perma- nent charge densities pA0(r) and pBO(r’) of the unperturbed A and B molecules: AB“) = I pAo(r) pBO(r’) Ir - r’ I’1 dr dr’ . (4) For a nucleus I in molecule A, the force AFIQ“) derived from AB“) has two components: the first results from the change in the nuclear charge density of A when nucleus I shifts, while the second is due to the change in the permanent electronic charge density p°A0(r) of molecule A (unperturbed by interactions with B) due to an infinitesimal shift of nucleus 1: Apia“) = - I 3p°A0(r)/3Rla pBo(t-') | r - r’ 1'1 dr dr’ — zI I 85(r - RIVER!“ pBo(t-') Ir - r’ 1'1 dr dr’ . (5) The derivative of p°A0(r) with respect to RIm satisfies ap°A0(r)/8Rla = I i dr” 21 Vla Ir” - R1 I-1 x fut. Ipwr) loom IpWr') let) + (gA Ip°A(r") IkAHkA Ip°A(r) ISA) 1/(Eg-Ek). (6) where VI0L denotes differentiation with respect to Rlm and it operates only on I r” - RI I '1. The prime on the summation indicates that the sum runs over the excited electronic states I k A ) of the unperturbed A molecule, omitting the ground state I g A ). The energies of the ground and excited states are E8 and Bk; and p‘A(r) is the electronic charge density operator for A. Eq. (6) shows that the derivatives of the permanent charge density with respect to nuclear coordinates depend upon a molecular susceptibility. This observation is significant for the subsequent analysis. The electronic charge density operator p°(r) is connected to the electronic polarization operator P°(r) used in defining the nonlocal polarizability density by 33 V - Pe(r) = - p°(r) . (7) In sum-over-states form [9, 10], the nonlocal polarizability density mafia, r’) satisfies aag(r.r’) = (1+ 60045)}:(8 |P°a(r) |k>//(Ek_Eg) = - I at ifi"dr"'£’ < gA IpAm I1:A > < kA I We) lgA > / (B, - 15,) x p30(r'k) Ir - r’ I ’1 p300”) I r” - r’” I ‘1 . (14) With Eq. (8) for the nonlocal polarizability density, successive integrations by parts [57] yield the first term in Eq. (13), and the second term is obtained by interchanging the roles of molecules A and B. The induction force on nucleus K in molecule A is determined by the derivative of the polarizability density aAaBO‘, r’) and by the derivative of the field ona(r) with respect 36 to the coordinates of K. In Refs. 3 and 4, we have shown that the derivative of the polariz- ability density with respect to RK depends on the hyperpolarizability density Bum“, r’, r”), a nonlinear response tensor that gives the polarization Pa(r) induced at r by the concerted action of static electric fields YB“) and 9'7“”). Explicitly, the static hyperpolarizability density is given by [58, 59] timer-2m = 5'2 omits lPa(r) |m>1-8m..=2’[(g lpa |k>(k lqg, lg)-(g 'qu |k>(k Iva Ian R = 21 info; IIPa.Ho] |k>/(E,-E,) (37) where Irn denotes the imaginary component of the expression that follows. Matrix ele- ments for the commutator between p aj for the jth electron and the unperturbed Hamiltonian H0 are determined by the force on the jth electron; and N (gllp,.Hol |k>=ir- (g In, Im>l -(g|ug|g>(g|pa|n). (54) the analogous relation for( g I [pm tfifi] In), the fact that ( g I [1)“. LLB] I n ) vanishes for any n ¢ g, and (assuming for simplicity that the molecular states are real) (mlpa|n>=-(n|pa|m>. (55) Together, these results yield A): 2K I dr dr’dr” Te¢(RK, r”) Mme", r, r’) Ta rm (r'A - RAA) (r’K - RAA) = i/(3rzlra'rmpzl'1ug lipstml |n>n = I...” ‘I’n(x) x ‘I’n(x) dx = 2 la I0 ‘1 rt sin2(mtx/a) dx = a / 2. (4) The average value of x2 is n = Log” ‘I‘n(x) x2 ‘Pn(x) dx =2 la lo a x2 sin2(n1tx/a) dx = (al2n1t)2(4n21r2/3 -2). (5) Thus the root-mean-square deviation of the position for the particle is Ax =V:xz>n- 112 =(a/2nrt)\/(n21t2/3 -2). (6) 67 The momentum distribution of a particle in a box is also very different from classical expectations. From a classical point of view, the particle should have a definite value of the momentum due to conservation of energy. Since E = p2/2m, therefore p = flim—E. Classi- cally, the particle moves forward and backward in the box with constant speed Quantum mechanically, the energy eigenstate ‘I’n(x) = 3/27a sin (ntrx/a) is not an eigenstate of the momentum operator, because MEI—a sin(mcx/a) = -i(h/2n)d/dx 14271 sin (nu/an =-i(h/21t) (nu/a) \IZ-li-l cos (mat/a) it p0 4273 sin (nrtx/a). (7) This means that the outcome from a momentum measurement for the particle cannot be pre- dicte with certainty. From the expression ‘I’n(x) = v27; sin (nrtx/a) = -w2£[exp(innx/a)exp(-mrtx/a)] (0 S x S a) (3) it can be shown that the wavefunction inside the box is a superposition of traveling waves with equal and opposite momentum. However, it would be incorrect to jump to the con- clusion that the state ‘I’n(x) consists of two eigenstates of momentum operator with the same weighting factor, and therefore, the particle moves forward and backward uniformly in the box with a definite absolute value of the momentum lpl = pH = nh/Za [9-11]. This is identical to the classical point of view about the motion of a particle in a box. 68 The function exp(:ti21tpx/h) is an eigenstate of momentum operator when it is valid for all x, without spatial limitation. Only when a particle moves in free space can its momentum take on a definite value. This is easy to show [4]: If a particle moves with a definite momentum p0 , its wave function in the momentum representation is ¢(P) = 5(P'Po); (9) Then by Fourier transformation, the wave function in the position representation is ‘P(x) = 1/‘15 I...” 5(P-Po) CXP(21tiPx/h) dp = lNh exp(21tipox/h). (10) The probability density to find the particle within the infinitesimal range dx about x is p(x) =I ‘I’(x)| 2: l/h = constant. (11) This means that the particle can be found anywhere in space with the same probability. That is to say, if the momentum of a particle is fixed, its position is totally unknown. In the opposite case, if a particle is fixed in space at x = x0. its wave function in the position representation is ‘I’(x) = 8(x-xo); (12) Then by Fourier transformation, in the momentum representation the wave function is ¢(p) = 1N1: Loo” 8(x-xo)exp(-2uipx/h) dx = l/‘Jli exp(-21tipx0/h). (13) Hence the probability density to find the particle with a momentum in the infinitesimal range dp about p is P(P) =4 (p)l 2= 1/h = constant. (14) 69 This means that the particle has equal probability to be observed with any momentum value (relativistic effects are neglected). In other words, when the position of the particle is fixed, its momentum is totally unknown. Between these two extremes, a particle in a one-dimensional box is spatially con strained (in the interval 0 S xS a). The function ‘P(x) = exp (:tinrtx/a) for 0 S x S a, but ‘I’(x) = 0 otherwise, is not an eigenstate of momentum operator. The actual momentum dis- tribution is found by transforming the energy eigenstate from the position representation to the momentum representation by Fourier transformation [6,15]: ¢,,(p) =(1NB') I...” ‘I’n(X) exp (zaps/h) dx = (INK) (,3 («Iz/a) siren/a) expt-znipx/h) dx = (Win) Ioa [CXP(in1tx/a)-CXP(-in1txla)] eXp(-21tipx/h) dx = (W) Pn/(Pnz-PZ) [1-(-1)“eXP(-21tiPa/h)] (W) pn 2 i sin(Pafi/h) ¢XP(-i1tpa/h)/(Pn2'P2) (11 even) - (W) Pn 2008(Pau/h) cXP(-i1tpa/h)/(Pn2-P2) (n odd) where pn=nhl2a, n=l,2,3, .. . (15) Equation 15 gives an explicit, simplified expression for the wave function in the momentum representation, from which it is easily seen that the wave functions are amplitude-modulated and n-dependent also. The corresponding probability densities to observe the momentum in the infinites- imal range dp about p are 021((P)=(2h/3JC2) P 22ksin2(pna/h)/(p22k-p2)2, (n even, n = 2k) p2k-l(p) = (2h/a1t2) p22k_lcos2(prta/h)/(p22k_l-p2)2. (n odd, n = zit-1) (16) where k =1, 2, 3, ‘70 Figure 2 shows the momentum probability densities for the particle in the first several energy eigenstates. This figure and Eq. 16 show clearly that there is a nonzero probability density to obtain many values other than tab/28 in a measurement of momen- tum for the particle in state 11. This is very difl'erent from the classical behavior. C. Cohen-Tannoudji, B. Diu, and F. Laloe, have analyzed the momentum probabi- lity density of a particle in a box in their textbook Quantum Mechanics [1]. In their expres- sions for the momentum wave functions and probability densities, they keep two terms sep- arate as “diffraction functions” and this leads to a valuable physical interpretation of the momentum probability density distributions. But they did not simplify the expressions to the more explicit forms of our Eq. 15 and 16. From Eq. 16, one can easily find all the maxima and minima of the momentum distribution. From dP(P) ldp = 0, the conditions for the maxima are cot (pan/he -2 (uh/an) / (p2 as?) (n even. p... p2,), tanrpart/h)=2 (Pb/310K132 zit-1112) (nodd. P‘leell (17) These equations can be solved numerically (see remark 3, below.) i The minima of the momentum distributions occur when p(p) = 0, i.e. when sin(part/h) .= 0, (11 even, p1: pa) and cos(patt/h) = 0. (n odd, pit pn). (18) The separation between typical minima in the momentum distribution is obviously n-independent and equals h/a. We make several observations: 71 l. The momentum distribution of a particle in a box gives a definite probability for observing values of p other than those corresponding to the eigenenergies of the particle. It is very interesting to note that, in analogy with the nodes in the position distribution of the particle (points in space where the particle has zero probability density to be found), the momentum distributions also have some zeroes at special values of the momentum. In even n states (n=2k), the particle cannot have the momentmn values of p = l h/ a, where 1 it k, while in odd n states (n= 2k-1), momentum values of p = (2 l-l)h/2a (l at k) cannot be mea sured. We could also regard the momentum wave function as a standing wave set up in the momentum space but it is amplitude-modulated. 2. The momentum of the particle in an eigenstate averages to zero due to the sym- metry of momentum distribution: p(p) is an even function of p, so < p >n = I_°°°°p pn (p) dp = I...” ‘11,, (x)(-ih/Zrt d/dx) \Pnot) dx=0. (19) The probability densities at zero momentum are Farm): 0, (n = 2k) p2,.1(0)=8a/ (2k-l)2 hltz. (n = 2k-1) (20) In even n states, one cannot observe the particle with zero momentum (the probability is zero), while in odd n states, one does observe zero momentum of the particle with a certain probability. Surprisingly, in state ‘I',(x) (n=l), the most probable momentum is zero, rather than p1: h/Za. When odd n becomes larger, the probability of finding the particle with zero momentum decreases. The mean value of p2 is given by < p2 >n = I_.,°°p2 pn (p) dp = L,” ‘Pn(x)(-ihf21t curls)2 and) dx = (uh/2a)2 (21) Thus the root-mean-square deviation of the momentum is Ap= @n-¢>n2=m (22) Combining Eq. 22 with the root-mean-square deviation of the position for the particle, one obtains a result consistent with the uncertainty principle: Ap Ax = (mam/zoom) 2 h/4n. (23) 3. The probability density at p = :l: Pn is a constant (n-independent): lim P(P)= am! (24) P—Wn We note that the most probable momentum is not :1: Pn when the particle in state ‘I’,(x). Instead, by numerical calculations (Eq. 17) we find the following values for the most prob- able momentum pm in different states: 3 pm (one) 1 0.000 2 1.675 3 2.790 4 3.845 5 4.950 "lo 9.985 From Fig. 2, it can be seen vividly that the distribution of the momentum for the particle bifurcates from a single peak (in the n=l state) to two separate peaks near p = :1: Pn (for all states 11 2 2). The most probable value of the momentum for the particle is n dependent. Only when 11 becomes large, does the probability density p(p) reach its maximum at pm 5 :1: pa. This can be proven by taking the limit for (19(1)) /dp when p—rpn: limdp(p) /dp .. Mn. (25) an . We note that p(ipn) is n independent. This matches the classical description. Acknowledgments This work grew out of problem sets in (EM 991, Selected Topics in Quantum Chemistry taught by Prof. K. L. C. Hunt. The authors thank Prof. K. L. C. Hunt and Dr. V. Sethura- man for interesting discussions. YQL is grateful for a research assistantship, supported through NSF Grant CHE-9021912 to KLCH, and for a Summer Alumni Research Fellow- ship, from the Department of Chemistry, Michigan State University. 74 References 1. Cohen-Tannoudji, C., Diu, B., Laloe, F., Quantum Mechanics, John Wiley & Sons, 1977. 2. Merzbacher, E., Quantum Mechanics, John Wiley & Sons, 1970. 3. Landau, L. D. and Lifshitz, E. M. Quantum Mechanics (3rd ed.), Pergamon Press Inc., 1977. ' 4. Dirac, P. A. M., The Principles of Quantum Mechanics (4th ed.), Oxford University Press, 1958. 5. Kramers, H. A., Quantum Mechanics, Interscience Publishers, Inc., 1957. 6. Feynman, R. P.and Hibbs, A. R., Quantum Mechanics and Path Integrals, McGraw-Hill, 1965. 7. Schiff, L. 1., Quantum Mechanics (3rd ed.), McGraw—Hill, 1968. 8. Shamkar, R., Principles of Quantum Mechanics, Plenum Press, 1980. 9. McQuarrie, D. A., Quantum Chemistry, University Science Books, 1983. 10. Atkins, P. W., Molecular Quantum Mechanics, Oxford University Press, 1970. 11. Hanna, M. W., Quantum Chemistry (3rd ed.), University Science Books, 1981. 12. Lowe, J. P., Quantum Chemistry (student ed.), Academic Press, 1978. 13. Levine, I. N ., Quantum Chemistry (3rd ed.), Allen & Bacon, Inc., 1983. 14. For example, see Volkamer, K. and Lerom, M. W., J. Chem. Educ. 1992, 69, 100; El-Issa, B. D., J. Chem. Educ. 1986, 63, 761; Miller, G. R., J. Chem. Educ. 1979, 56, 709. 15. Arfken, G. Mathematical Methods for Physicists (3rd ed.), Academic Press, Inc., 1985. 75 "=10 n: N- A X v C .9: II + 3'? V: "=2 O. ”=1 A l A l n l n l 0.0 0.2 0.4 0.0 0.0 X(a) 1.0 Figure 6.1 The position probability densities of a particle in a one-dimensional box; n indicates the energy levels of the particle (not to scale). 76 n=10 M ~— A a. V: 9. ll L ’5 V n=2 C O. r I n l a 1 a l n l n .15 .10 s o s 10 ‘5 P(h/za) Figure 6.2 The momentum probability densities of a particle in a one-dimensional box; 11 indicates the energy levels of the particle (not to scale). Figure 6.1 The position probability densities of a particle in a one-dimensional box; n indicates the energy levels of the particle (not to scale). Figure 6.2 The momentum probability densities of a particle in a one-dimensional box; 11 indicates the energy levels of the particle (not to scale). IIllillllllllllli