Preface	7
	Contents	9
	1 Introduction	13
	1.1 Sum Frequency Generation	13
	1.2 Visible-Infrared SFG Vibrational Spectroscopy	16
	1.3 Solutions to Problems	19
	1.3.1 Inversion Symmetry of ?(2)	19
	1.3.2 Time and Frequency Domains	19
	1.3.3 Red Shift of O-H Frequency	21
	Bibliography	22
	2 Electrodynamics at Interface	24
	2.1 Electromagnetic Fields at Interface	25
	2.1.1 Maxwell Equations	25
	2.1.2 Boundary Conditions at Interface	26
	2.1.3 SFG Signal Emitted from Interface	29
	2.1.4 Fresnel Factor	31
	2.2 Response to Incident Lights	33
	2.3 Summary of Factors in SFG Spectra	35
	2.4 Solutions to Problems	36
	2.4.1 Boundary Condition at Interface (1)	36
	2.4.2 Boundary Condition at Interface (2)	38
	2.4.3 Boundary Condition at Interface (3)	39
	2.4.4 Electric Field and Interfacial Polarization	41
	Appendix	45
	A.1 Singularity of Source Polarization	45
	A.2 Fresnel Factors for Two-Layer Model	47
	A.3 Fresnel Factors for Three-Layer Model	50
	A.3.1 Fields and Wavevectors	50
	A.3.2 Boundary Conditions	52
	(1) Medium i and Interface	52
	(2) Interface and Medium j	53
	(3) Media i and j	54
	A.3.3 Solution of Boundary Equations	55
	Bibliography	56
	3 Microscopic Expressions of Nonlinear Polarization	58
	3.1 Density Matrix	58
	3.1.1 Definition	59
	3.1.2 Features and Advantages	60
	(i) Ensemble of States	60
	(ii) Partial System in Bath	62
	3.2 Perturbation Forms of Susceptibilities	64
	3.2.1 Perturbation Expansion of Density Matrix	64
	3.2.2 First-Order Susceptibility	65
	3.2.3 Second-Order Susceptibility	67
	3.3 Properties of ?(2)	69
	3.3.1 Vibrational Resonance	69
	3.3.2 Relation to Molecular Orientation	73
	Two Factors: Density and Orientation	74
	Rotational Matrix	74
	3.3.3 Tensor Elements of ?(2) and Polarization	77
	3.4 Solutions to Problems	79
	3.4.1 Formulas for Mixed States	79
	3.4.2 Pure and Mixed States	80
	3.4.3 Derivation of ?(2)	82
	3.4.4 Effective ?(2) Formula	83
	Appendix	85
	A.1 Off-Diagonal Elements of Density Matrix	85
	A.2 Interaction Energy of Nonmagnetic Materials	87
	A.3 Polarizability Approximation for Raman Tensor	88
	Bibliography	90
	4 Two Computational Schemes of ?(2)	91
	4.1 Energy Representation	91
	4.2 Examples of ?(2) Tensor and Orientation	94
	4.2.1 O-H Stretching	95
	4.2.2 C-H Stretching	97
	4.3 Time-Dependent Representation	100
	4.3.1 Time Correlation Function	100
	4.3.2 Classical Analogue	102
	4.4 Motional Effect on ?(2)	106
	4.4.1 Relation of Two ?(2) Models	107
	4.4.2 Slow Limit and Fast Limit	109
	4.5 Solutions to Problems	110
	4.5.1 Polarization Ratios	110
	4.5.2 Canonical Time Correlation Function	112
	Bibliography	112
	5 Molecular Theory of Local Field	114
	5.1 Local Field Correction Factor	115
	5.2 Local Field Correction for ?(2)	121
	5.3 Interfacial Dielectric Constant	124
	5.4 Solutions to Problems	129
	5.4.1 Interfacial Dielectric Constant	129
	Bibliography	130
	6 Charge Response Kernel for Electronic Polarization	132
	6.1 Charge Response Kernel (CRK)	133
	6.2 Electronic Structure Theory of CRK	134
	6.3 Polarizable Model with CRK	140
	6.4 ?(2) Formula with CRK Model	143
	6.5 Solutions to Problems	145
	6.5.1 ESP Charge	145
	6.5.2 Charge Response Kernel	146
	6.5.3 Energy and Force with Polarizable Model	148
	6.5.4 Polarizability	149
	Appendix	150
	A.1 Derivation of CRK from CPHF Equation	150
	A.1.1 Wavefunction Under External Field	150
	A.1.2 Derivative of Wavefunction	152
	A.1.3 Formula of CRK	155
	A.2 Reorganization Energy of Electronic Polarization	155
	A.2.1 Derivation of Reorganization Energy	156
	A.2.2 Variational Principle of Polarization	157
	Bibliography	158
	7 Quadrupole Contributions from Interface and Bulk	160
	7.1 Beyond the Three-Layer Model	161
	7.2 Extended Nonlinear Susceptibility	166
	7.2.1 Extended Source Polarization	166
	7.2.2 Effective Polarization and Susceptibility	168
	7.2.3 Interface Contribution	169
	7.2.4 Bulk Contribution	171
	7.2.5 Expression of Bulk Term ?B	175
	7.2.6 Summary of Derivation	178
	7.3 Microscopic Formulas of Quadrupolar Susceptibilities	181
	7.3.1 Perturbation Expressions	181
	7.3.2 Time-Dependent Expressions	187
	7.4 Invariance to Molecular Origin	191
	7.5 Summary	195
	7.6 Solutions to Problems	195
	7.6.1 Isotropic Tensor Components	195
	7.6.2 Bulk Term ?B	198
	7.6.3 Transformation of Quadrupolar Susceptibility	199
	7.6.4 Levi-Civita Tensor	202
	Appendix	203
	A.1 Physical Meaning of ?IQB	203
	A.2 Levi-Civita Antisymmetric Tensor	204
	A.3 Definition of Bulk Polarization	207
	Bibliography	208
	8 Other Topics	210
	8.1 ?(3) Effect at Charged Interfaces	211
	8.1.1 Properties of ?(3) Tensor	211
	8.1.2 Role of ?(3) in Electrolyte Solution	214
	8.1.3 Calibrating ?(3) Effect in SFG Spectra	218
	8.2 Chiral Elements of ?(2)	219
	8.2.1 Symmetry	220
	8.2.2 Intensity	222
	8.2.3 Future Development	223
	8.3 Solutions to Problems	224
	8.3.1 Guoy-Chapman Theory	224
	8.3.2 Chiral ?(2) Components	226
	Bibliography	226
	9 Applications: Aqueous Interfaces	228
	9.1 Water Surface	229
	9.2 Ice Surface	232
	9.3 Electrolyte Solution Surfaces	234
	9.3.1 Halide Ions: Surface Segregation	236
	9.3.2 Buried Ions: F-, SO42-	239
	9.3.3 Acid	241
	9.3.4 Base	243
	9.4 Oil/Water Interfaces	246
	9.5 Water at Monolayers	248
	Bibliography	250
	10 Applications: Organic Interfaces	256
	10.1 C–H Bands of Alkyl Groups	257
	10.1.1 C–H Modes	257
	10.1.2 Modeling of C–H	258
	10.1.3 Methanol C–H Vibrations	259
	10.1.4 Ethanol C–H Vibrations	261
	10.2 Benzene: SFG from Centrosymmetric Molecules	262
	10.3 Molecular Orientation and Polarization Analysis	263
	10.3.1 Methanol	264
	10.3.2 Acetonitrile	266
	Bibliography	268
	11 Summary	270
	11.1 Outline of Theory	270
	11.2 Future Directions	271
	Bibliography	272