PHYSICS OF MASS
WHICH THE EXTERNAL FIELD ET MANAGES TO SET THE CHARGES IN MOTION IS, TO FIRST ORDER, EXPRESSED IN TERMS OF THE LINEAR ELECTRIC POLARIZABILITY IN GENERAL, THE RESPONSE OF THE CHARGES DEPENDS ON THE FREQUENCY OF THE ELECTRIC FIELD, AND THE POLARIZABILITY IS THEREFORE FREQUENCY DEPENDENT AS WE SHALL SEE LATER THE TIME-DEPENDENT POLARIZATION BECOMES (2) T = 0 +ET WHERE 0 IS THE PERMANENT ELECTRIC DIPOLE MOMENT OF THE MOLECULE WE ARE CONCERNED WITH OPTICAL FIELDS, SO THE WAVELENGTH OF ET IS AT LEAST IN THE ORDER OF A FEW HUNDRED NANOMETERS WHEREAS THE SIZE OF THE MOLECULE IS NO GREATER THAN A FEW NANOMETERS IN MOST CASES, THE ELECTRIC FIELD IS THEREFORE TAKEN TO BE UNIFORM OVER THE MOLECULE NOW THERE SEEMS TO BE ONE IMMEDIATE AND IMPORTANT QUESTION TO BE ADDRESSED: IF THE PARTICLE AND WAVE PICTURES OF LIGHT ARE TWO VERSIONS OF THE SAME THING, HOW CAN WE UNDERSTAND NONLINEAR LIGHT–MATTER INTERACTIONS WHEN THE ELECTRIC FIELD IS REPRESENTED BY A PLANE WAVE? THE ANSWER LIES IN A GENERALIZATION OF EQ (2) WHICH LETS THE POLARIZATION BE EXPRESSED AS A TAYLOR SERIES IN THE ELECTRIC FIELD STRENGTH (3) T = 0 +ET+ 2 E2T+ E3T+ THIS EQUATION INTRODUCES THE FIRST-ORDER (NONLINEAR) HYPERPOLARIZABILITY , THE SECOND-ORDER HYPERPOLARIZABILITY , AND SO FORTH JUST AS THE LINEAR POLARIZABILITY, THE NONLINEAR COUPLING CONSTANTS DEPEND ON THE FREQUENCY OF THE APPLIED FIELD IT IS CLEAR THAT THE TIME-DEPENDENT POLARIZATION CAN HAVE FREQUENCY COMPONENTS SEPARATE FROM THOSE OF THE EXTERNAL FIELD DUE TO ITS POWER DEPENDENCE ON THE ELECTRIC FIELD STRENGTH, AND THE MOLECULE CAN THUS EMIT SUM-FREQUENCY-GENERATED RADIATION IN CORRESPON- DENCE WITH FOR EXAMPLE FIG 2 IN OPTICS, EQ (3) PROVIDES THE FUNDAMENTAL ORIGIN OF NONLINEARITIES, AND, AT THE MICROSCOPIC LEVEL, IT IS THE EXPANSION COEFFICIENTS IN THIS EQUATION, OR THE HYPERPOLARIZABILITIES, THAT GOVERN THE NONLINEAR OPTICAL PERFOR- MANCE OF THE MATERIAL HENCE, THEORETICAL MODELING OF NONLINEAR OPTICAL PROPERTIES IS CONCERNED WITH THE DETERMINATION OF THESE QUANTITIES GIVEN THE STRUCTURE OF THE SYSTEM LET US NOW EXAMINE EQ (3) IN SOME DETAIL THE ELECTRIC FIELD IS VECTORIAL AND GENERALLY CONSIDERED TO BE COMPOSED OF A STATIC COMPONENT AND ONE OR MORE TIME- OSCILLATING COMPONENTS ACCORDING TO (4) ET = E E?IT WHERE E ARE THE FOURIER AMPLITUDES OF THE ELECTRIC FIELD ALONG THE MOLECULAR AXIS WE NOTE THAT THE USE OF GREEK SUBSCRIPTS FOR TENSORS IN NONLINEAR OPTICS FOLLOWS THE ORIGINAL NOTATION OF BUCKINGHAM [19] THE SUMMATION INCLUDES BOTH POSITIVE AND NEGATIVE FREQUENCIES, AND, SINCE THE EXTERNAL FIELD IS REAL, WE HAVE E = E?? FURTHERMORE, A VECTORIAL ELECTRIC FIELD IMPLIES THAT THE LINEAR POLARIZABILITY IS A PHYSICS OF MASS SECOND-RANK TENSOR, THE FIRST-ORDER HYPERPOLARIZABILITY IS A THIRD-RANK TENSOR, AND SO FORTH WE INSERT THE EXPRESSION FOR THE TIME-DEPENDENT ELECTRIC FIELD IN EQ (4) INTO EQ (3) TO OBTAIN T = + ? E E?IT (5) + E?IT + E?IT + WHERE DENOTES THE SUM OF OPTICAL FREQUENCIES: FOR TERMS INVOLVING THEN = 1 +2 AND FOR TERMS INVOLVING THEN = 1 +2 +3 THE EINSTEIN SUMMATION CONVENTION FOR REPEATED SUBSCRIPTS IS ASSUMED HERE AND ELSEWHERE WE NOTE THAT ANY PAIRWISE INTERCHANGE OF THE INDICES AND FREQUENCIES 1, 2, AND 3 CAN BE MADE WITHOUT ALTERING THE PHYSICALLY OBSERVABLE POLARIZATION T IT IS THEREFORE CUSTOMARY, BUT NOT NECESSARY, TO DEMAND THAT THE INDIVIDUAL TENSOR ELEMENTS ARE INTRINSICALLY SYMMETRIC ? 12 = ? (6) ? 123 = ? (7) = ? 213 = ? = ? 321 = ? FURTHERMORE, SINCE THE MOLECULAR POLARIZATION T AS WELL AS THE ELECTRIC FIELD ET ARE REAL, WE HAVE ? = ? ? (8) ?1?2 = ? 12? (9) ?1?2?3 = ? 123? (10) THE FREQUENCY IS THAT OF THE GENERATED MOLECULAR POLARIZATION, AND, SINCE THE SUMMATIONS IN EQ (5) RUN OVER BOTH POSITIVE AND NEGATIVE FREQUENCY COMPO- NENTS, THE NONLINEAR HYPERPOLARIZABILITIES WILL CREATE BOTH SUM-FREQUENCY AS WELL AS DIFFERENCE-FREQUENCY GENERATED POLARIZATION IT IS ILLUSTRATIVE TO CONSIDER A FEW SPECIFIC EXAMPLES FOR INSTANCE, LET TWO LASERS A AND B, WHICH OPERATE AT FREQUENCIES A AND B, RESPECTIVELY, INTERACT THE EXTERNAL ELECTRIC FIELD EXPERIENCED BY THE MOLECULAR SYSTEM WILL IN THIS CASE BECOME (11) ET = EA COSAT+EB COSBT PHYSICS OF MASS FIGURE 3 FREQUENCY DECOMPOSITION OF TWO INTERFERING LASER FIELDS WITH A FREQUENCY DECOMPOSITION THAT APPEARS AS IN FIG 3 ACCORDING TO EQ (5) AND INCLUDING TERMS UP TO SECOND-ORDER IN THE FIELD, THE TIME-DEPENDENT POLARIZATION WILL BE T = + EA COSAT+EB COSBT (12) + EA COSAT+EB COSBTEA COSAT+EB COSBT WITH THE USE OF THE TRIGONOMETRIC IDENTITY (13) COSUCOSV = 2COSU+V+COSU?V WE CAN REWRITE THE POLARIZATION AS T = + =AB ? E COST (14) + ?E E + E E COS2T + ?A +B ABEA EB COSA +BT + ?A ?B A?BEA EB COSA ?BT THE FREQUENCY DECOMPOSITION OF THE POLARIZATION IS ILLUSTRATED IN FIG 4, AND, IN THIS FIGURE, WE RECOGNIZE THE LINEAR POLARIZATION AT THE FREQUENCIES OF THE EXTERNAL FIELD AND THE SECOND-HARMONIC GENERATION FREQUENCIES AS A RESULT FROM TWO-PHOTON ABSORP- TION (SEE ALSO FIG 2) IN ADDITION TO THESE FREQUENCIES, WE SEE THAT THE INDUCED MOLEC- ULAR POLARIZATION WILL CONTAIN THE FREQUENCIES A +B (SUM-FREQUENCY GENERATION) AND A ?B (DIFFERENCE-FREQUENCY GENERATION) AS WELL AS A STATIC COMPONENT = 0 SUM-FREQUENCY GENERATION IS SIMILAR TO SECOND-HARMONIC GENERATION IN THAT ?B 2?B 2?A ?A –2?A –(?A + ?B) (?A + ?B) –(?A – ?B) (?A – ?B) –2?B –?A –?B FIGURE 4 FREQUENCY DECOMPOSITION OF THE MOLECULAR POLARIZATION PHYSICS OF MASS +E COST INCLUDING TERMS UP TO THIRD ORDER IN THE ELECTRIC FIELD, THE POLARIZATION IN THIS CASE BECOME T = +E COST (16) + +E COST + COS3T PHYSICS OF MASS (18) + (19) + E E (20) + E E E (21) AN ALTERNATIVE SUMMARY OF THIS RESULT IS SOMETIMES EXPRESSED AS AN EXPANSION OF THE POLARIZATION AMPLITUDES IN TERMS OF THE ELECTRIC FIELD AMPLITUDES ˜ = ? 1E1 + (22) + ? 123E1E2E3 + WHERE THE FACTORS KN ARE REQUIRED FOR THE POLARIZATION RELATED TO THE MOLECULAR RESPONSE OF ORDER N TO HAVE THE SAME STATIC LIMIT BY A DIRECT COMPARISON WITH EQS (17)–(21), WE ARE ABLE TO IDENTIFY THESE FACTORS FOR SOME COMMON NONLINEAR OPTICAL PROCESSES, SEE TABLE 1 WE HAVE SEEN HOW THE MOLECULAR PROPERTIES IN NONLINEAR OPTICS ARE DEFINED BY THE EXPANSION OF THE MOLECULAR POLARIZATION IN ORDERS OF THE EXTERNAL ELECTRIC FIELD, SEE EQ (5); BEYOND THE LINEAR POLARIZATION THIS DEFINITION INTRODUCES THE SO-CALLED NONLINEAR HYPERPOLARIZABILITIES AS COUPLING COEFFICIENTS BETWEEN THE TWO QUANTITIES THE SAME EQUATION ALSO EXPRESSES AN EXPANSION IN TERMS OF THE NUMBER OF PHOTONS INVOLVED IN SIMULTANEOUS QUANTUM-MECHANICAL PROCESSES: , , , AND SO ON INVOLVE EMISSION OR ABSORPTION OF TWO, THREE, FOUR, ETC PHOTONS THE CROSS SECTION FOR MULTI- PHOTON ABSORPTION OR EMISSION, WHICH TAKES PLACE IN NONLINEAR OPTICAL PROCESSES, IS IN TYPICAL CASES RELATIVELY SMALL AND A HIGH DENSITY OF PHOTONS IS REQUIRED FOR THESE TO OCCUR PHYSICS OF MASS STATIC 0; 0, EOPEA ? STATIC 0; 0, 0, EOKEC ? A ELECTRO-OPTICAL POCKELS EFFECT B SECOND-HARMONIC GENERATION C ELECTRO-OPTICAL KERR EFFECT D INTENSITY-DEPENDENT REFRACTIVE INDEX E ELECTRIC FIELD-INDUCED SECOND HARMONIC GENERATION F THIRD-HARMONIC GENERATION 3 TIME-DEPENDENT PERTURBATION THEORY IN THIS SECTION WE SHALL DERIVE EXPLICIT EXPRESSIONS FOR THE RESPONSE FUNCTIONS THAT DESCRIBE THE INTERACTION OF A QUANTUM MECHANICAL SYSTEM AND AN EXTERNAL ELECTRIC FIELD THE FIRST ORDER RESPONSE FUNCTION WAS IN THE PREVIOUS SECTION REFERRED TO AS THE MOLECULAR POLARIZABILITY AND HIGHER-ORDER RESPONSE FUNCTIONS AS THE MOLECULAR HYPERPOLARIZABILITIES WE WILL USE BOTH THESE TERMINOLOGIES AS IF THEY WERE SYNONY- MOUS ALTHOUGH THE NOTION OF RESPONSE FUNCTIONS IS MORE GENERAL AND CAN BE USED IN OTHER APPLICATIONS THAN NONLINEAR OPTICS AS TOOL WE WILL USE TIME-DEPENDENT PERTURBATION THEORY AND ASSUME THAT THE SOLUTIONS TO THE EIGENVALUE PROBLEM OF THE UNPERTURBED SYSTEM ARE KNOWN IN PRACTICE WE ARE OF COURSE NOT ABLE TO OBTAIN EXACT SOLUTIONS EVEN TO THE UNPERTURBED SYSTEM, BUT OUR ANALYSIS WILL NEVERTHELESS HIGHLIGHT THE DEPENDENCE OF THE POLARIZABILITIES ON OTHER MOLECULAR PARAMETERS SUCH AS EXCITATION ENERGIES AND TRANSITION MOMENTS A THOROUGH INTRODUCTION TO VARIOUS ELECTRONIC STRUCTURE METHODS FOR DETERMINING UNPERTURBED REFERENCE STATES HAVE BEEN GIVEN IN [31] THE TIME-EVOLUTION OF THE STATE VECTOR IS GIVEN BY THE TIME-DEPENDENT SCHRÖDINGER EQUATION (23) I? TT = ˆHT WHERE ˆH IS THE HAMILTONIAN WHICH MAY BE TIME-INDEPENDENT OR TIME-DEPENDENT IN THE FORMER CASE, THE PROBLEM OF SOLVING EQ (23) REDUCES TO SOLVING THE EIGENVALUE EQUATION FOR THE HAMILTONIAN, BUT IN THE LATTER CASE SUCH SIMPLIFICATIONS ARE NOT POSSIBLE HOWEVER, WHEN THE TIME-DEPENDENT PART OF ˆH IS A SMALL PERTURBATION, WE PHYSICS OF MASS N , ETC BY STRAIGHTFORWARD TIME-INTEGRATION IN ACCORDANCE WITH THE EXPANSION OF THE COEFFICIENTS DN IN ORDERS OF THE PERTURBATION ˆV, IT IS CUSTOMARY TO WRITE THE WAVE FUNCTION AS (30) T = 0T+1T+2T+ PHYSICS OF MASS I? T ? N M ˆ E?I1T?ET?NEIMNT?0NDT? (38) = M ? ARE ZERO DUE TO THE POSITIVE INFINITESIMAL, THE RESULT IS CONSISTENT WITH THE MOLECULE BEING IN ITS GROUND STATE AT T = ? PHYSICS OF MASS E?I1T?ET?NEIMNT? (39) × N0 ?2 ?IEIN0?2T?ET? DT? = E?I1T?ET?NEIMNT? (40) × N0 ?2 ?3 ?I2P0 ?3 ?IEIN0?2?3T?E2T?DT? = (41) = 0EIE0T/?ˆ N N0 ?1 ?IEIN0?1TETE?IENT/?N + N ? N 0 ˆNN ˆ E?I1TET + ?EI1TET = E?I1TET PHYSICS OF MASS IN THE LAST STEP WE HAVE USED THAT E1?= E?1 AND THE FACT THAT 1 RUNS OVER BOTH POSITIVE AND NEGATIVE FREQUENCIES BY A DIRECT COMPARISON TO EQ (5), WE ARE ABLE TO IDENTIFY THE RESULTING QUANTUM MECHANICAL FORMULA FOR THE LINEAR POLARIZABILITY AS (42) ? = N0 ++I IT IS CLEAR THAT THIS FORMULA CAN BE USED DIRECTLY AS IT STANDS FOR PRACTICAL COMPUTATIONS OF THE OPTICAL POLARIZATION ONCE THE EXCITATION ENERGIES AND TRANSITION MOMENTS OF THE SYSTEM ARE KNOWN MORE IMPORTANTLY, HOWEVER, IT ELUCIDATES THE DEPENDENCE OF THE LINEAR POLARIZABILITY ON THE QUANTUM MECHANICAL PROPERTIES OF THE MOLECULE FOR THE TWO TERMS IN THE SUM-OVER-STATES EXPRESSION IN EQ (42) THAT INVOLVE THE GROUND STATE N = 0, THE TRANSITION FREQUENCY N0 IS ZERO THE TWO TERMS ARE OF OPPOSITE SIGN AND WILL THEREFORE CANCEL, AND IT IS COMMON PRACTICE TO EXCLUDE THE GROUND STATE FROM THE SUMMATION AND TO USE A PRIMED SUMMATION SYMBOL ? N FOR THE SUM OVER EXCITED STATES WE HAVE MADE A POINT OF CARRYING ALONG THE POSITIVE INFINITESIMAL IN THE PERTURBATION NOT ONLY TO AVOID SINGULARITIES OR DIVERGENCIES IN THE TIME-INTEGRATION STEP OF THE EXPANSION COEFFICIENTS DN ABOVE BUT ALSO TO AVOID DIVERGENCES IN THE LINEAR RESPONSE FUNCTION ITSELF WITH A REASONABLE LASER DETUNING, HOWEVER, THE IMAGINARY TERM IN THE DENOMINATOR OF EQ (42) CAN SAFELY BE NEGLECTED IN THE CALCULATION OF ? , AND, SINCE THE SET OF EIGENSTATES N CAN BE CHOSEN AS REAL, WITHOUT LOSS OF GENERALITY, THE LINEAR RESPONSE FUNCTION IS REAL IN THE NONRESONANT REGION THE WORKING FORMULA WILL IN THIS CASE TAKE THE FORM (43) ? = ? N ? 0 ˆNN ˆ N0 + WE NOTE THAT AFTER HAVING EXCLUDED THE GROUND STATE FROM THE SUMMATION, THE POLARIZABILITY IS CONVERGENT IN THE LIMIT OF NONOSCILLATING FIELDS ALSO WITHOUT THE IMAGINARY TERM IN THE DENOMINATOR THE SINGULARITIES THAT OCCUR FOR STATIC FIELDS WITH = 0 AND INCLUSION OF THE GROUND STATE IN THE SUMMATION ARE CALLED SECULAR DIVERGENCES, SINCE THEY CAN BE REMOVED BY MATHEMATICAL MANIPULATIONS OF THE SUM-OVER-STATES EXPRESSION WE WILL LATER SEE THAT SECULAR DIVERGENCES APPEAR FOR THE HYPERPOLARIZABILITIES AS WELL HOWEVER, WITH = 0, THE LINEAR POLARIZABILITY, OR THE LINEAR POLARIZATION PROPAGATOR AS IT IS SOMETIMES DENOTED, WILL ALSO BE DIVERGENT FOR PERTURBATION FREQUENCIES EQUAL TO THE TRANSITION FREQUENCIES OF THE SYSTEM THESE SINGULARITIES ARE TRUE DIVERGENCES AND ARE KNOWN AS RESONANCES OF THE SYSTEM THIS FACT IS FREQUENTLY UTILIZED IN APPROXIMATIVE CALCULATIONS IN ORDER TO FIND ESTIMATES OF THE MOLECULAR EXCITATION ENERGIES AS POLES OF THE POLARIZATION PROPAGATOR THE RESIDUES OF THE PROPAGATOR CAN BE USED TO IDENTIFY THE ABSOLUTE VALUE OF THE TRANSITION MOMENTS BETWEEN THE GROUND AND EXCITED STATES AND THEREBY DESCRIBE THE INTENSITIES PHYSICS OF MASS IN THE LINEAR ABSORPTION SPECTRUM THE RESIDUE OF THE LINEAR POLARIZABILITY IS SEEN FROM EQ (43) TO EQUAL (44) LIM ?F0F0 ? ? = 0 ˆFF ˆ WHERE F DENOTES THE FINAL STATE IN THE ONE-PHOTON ABSORPTION PROCESS WE HAVE TREATED THE PERTURBING ELECTRIC FIELD AS CLASSICAL, WHICH MEANS THAT THE LIFETIMES OF THE EIGENSTATES N OF THE UNPERTURBED HAMILTONIAN ARE INFINITE THE LINEAR ABSORPTION FROM THE GROUND STATE TO THE EXCITED STATES IS IN THIS CASE DESCRIBED BY DIRAC DELTA FUNCTIONS PEAKED AT THE EXCITATION ENERGIES, AND MONOCHROMATIC LIGHT SOURCES NEED TO BE EXACTLY TUNED FOR TRANSITIONS TO OCCUR IN REALITY, AND WHEN TREATING ALSO THE ELECTRIC FIELD QUANTUM MECHANICALLY, THERE IS NO SUCH THING AS AN EXACT ZERO EXTERNAL FIELD, AS WELL AS THERE ARE NO INFINITE LIFETIMES OF EXCITED STATES EVEN OF PERFECTLY ISOLATED SYSTEMS A PHENOMENOLOGICAL WAY TO INCORPORATE THE SITUATION OF FINITE EXCITED STATE LIFETIMES INTO OUR DESCRIPTION IS TO INTRODUCE POPULATION DECAY RATES N0 THAT EQUAL THE INVERSE LIFETIME, OR THE LIFETIME BROADENING, OF THE EXCITED STATE N THE EXPRESSION FOR THE LINEAR POLARIZABILITY IS THEN WRITTEN AS (45) ? = ? N ? 0NN ?1+ NA FIRST-ORDER HYPERPOLARIZABILITY JUST AS AN EXPLICIT FORMULA FOR THE LINEAR POLARIZABILITY IS IDENTIFIED FROM THE LINEAR POLARIZATION, WE ARE ABLE TO RETRIEVE THE CORRESPONDING FORMULA FOR THE FIRST-ORDER HYPERPOLARIZABILITY FROM THE SECOND-ORDER POLARIZATION WE OBTAIN THE SECOND- ORDER POLARIZATION FROM EQ (35) BY INSERTION OF THE FIRST-ORDER AND SECOND-ORDER PHYSICS OF MASS CORRECTIONS TO THE WAVE FUNCTION, WHERE 1 AND 2 ARE GIVEN BY EQS (38) AND (39), RESPECTIVELY, IN COMBINATION WITH EQ (31) LET US FIRST CONSIDER THE THREE TERMS IN EQ (35) ONE AT A TIME: 0 ˆ2 = 0EIE0T/?ˆ (47) × N N0 ?1 ?2 ?I2P0 ?2 ?I ×EIN0?1?2TE2TE?IENT/?N = N0 ?1 ?2 ?I2P0 ?2 ?IE?I1+2TE2T 1 ˆ1 = (48) × N ? N0 ?1 +I E?IN0?1TETEIENT/?N ˆ × P P0 ?2 ?IEIP0?2TETE?IEPT/?P (49) = N0 +1 +IP0 ?2 ?I E?I1+2TE2T 2 ˆ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–(50) TO BE INTRINSICALLY SYMMETRIC WITHOUT ALTERING THE PHYSICAL POLARIZATION; WE DO SO BY OPERATING WITH THE OPERATOR 1/212 WHICH PERFORMS THE SUMMATION OF TERMS OBTAINED BY PERMUTING THE PAIRS 1 AND 2 THE FACTOR OF ONE HALF OBVIOUSLY CAUSES THE POLARIZATION TO PHYSICS OF MASS (51) = NP 0 ˆNN ˆ PP ˆ N0 ?1 ?2 ?I2P0 ?2 ?I + 0 ˆ NN ˆPP ˆ E?I1+2TE2T AND, BY A DIRECT COMPARISON TO EQ (5), WE IDENTIFY THE EXPRESSION FOR THE FIRST-ORDER HYPERPOLARIZABILITY TO BE ? 12 = N0 ? ?I2P0 ?2 ?I (52) + 0 ˆ NN ˆPP ˆ N0 + +I2P0 +2 +I IN CONNECTION WITH THE LINEAR RESPONSE FUNCTION, WE DISCUSSED BRIEFLY THE POSSI- BILITY OF INCORPORATING ABSORPTION OF LIGHT THROUGH IMAGINARY DAMPING TERMS THAT PARALLELS THE POSITIVE INFINITESIMALS IN THE EXPRESSIONS FOR POLARIZABILITIES AND HYPER- POLARIZABILITIES WITH A REASONABLE DETUNING, WE ARGUED THAT THE LINEAR ABSORPTION IS NEGLIGIBLE AND THE LINEAR RESPONSE FUNCTION IS REAL ANALOGOUSLY, IF ALL ONE- AND TWO-PHOTON FREQUENCIES ARE NONRESONANT THEN THE FIRST-ORDER HYPERPOLARIZABILITY WILL BE REAL, WHICH IS EQUIVALENT TO LETTING EQUAL ZERO IN EQ (52) IN THE NONRESO- NANT REGION, THEREFORE, THE EXPRESSION FOR THE FIRST-ORDER HYPERPOLARIZABILITY CAN BE WRITTEN IN A MORE COMPACT FORM: ? (53) = WHERE ?12 DENOTE THE SUM OF THE SIX TERMS ONE GETS BY PERMUTING PAIRS OF ?, 1 AND 2 THE VERIFICATION OF EQ (53) IS DONE IN A STRAIGHT- FORWARD MANNER BY A DIRECT COMPARISON OF THE SIX TERMS IN EQ (52) WITH = 0 THE FIRST-ORDER HYPERPOLARIZABILITY TENSOR IS SAID TO POSSESS FULL PERMUTATION SYMMETRY IN THE NONRESONANT REGION BEFORE CLOSING THE DERIVATION OF THE FIRST-ORDER HYPERPOLARIZABILITY, WE WISH TO REMOVE THE APPARENT DIVERGENCES OF EQ (53) IN THE LIMIT OF NON-OSCILLATING PERTURBING PHYSICS OF MASS FIELDS JUST AS WE DID FOR ? , IT IS OUR INTENTION TO DERIVE AN EQUIVALENT EXPRESSION FOR ? 12 THAT EXCLUDES TERMS INVOLVING THE GROUND STATE IN THE SUMMATION LET US BEGIN BY SPLITTING EQ (53) ACCORDING TO ? 12 = NP ? 0 ˆNN ˆ PP ˆ (54) + P ? 0 ˆ00 ˆ PP ˆ + N ? 0 ˆNN ˆ 00 ˆ + 0 ˆ00 ˆ 00 ˆ SINCE (55) THE SIX PERMUTATIONS GENERATED FROM THE LAST TERM IN EQ (54) WILL CANCEL EACH OTHER FURTHERMORE, WITH THE FULL PERMUTATION OPERATOR ON THE OUTSIDE, WE ARE FREE TO INTERCHANGE ANY TWO PAIRS OF INDICES IN THE RESPECTIVE TERMS IN EQ (54): IN THE SECOND TERM WE INTERCHANGE ? AND 1, AND IN THE THIRD TERM WE INTERCHANGE 2 AND 1 THE EXPRESSION FOR THE FIRST-ORDER HYPERPOLARIZABILITY THEREBY APPEARS AS ? 12 = (56) + N ?0 ˆNN ˆ 00 ˆ 1N0 ?2 ? ? N ? 0 ˆN0 ˆ 0N ˆ WHERE ˆ IS THE FLUCTUATION DIPOLE MOMENT OPERATOR ˆ?0 ˆ0 THIS EQUATION REPRE- SENTS AN APPROPRIATE FORM OF THE QUANTUM MECHANICAL FORMULA FOR THE FIRST-ORDER PHYSICS OF MASS HYPERPOLARIZABILITY IN THE NONRESONANT REGION AND INCLUDING NON-OSCILLATING EXTERNAL FIELDS THE MOLECULAR PARAMETERS THAT ENTER THIS EXPRESSION ARE THE EXCITATION ENER- GIES, THE GROUND TO EXCITED STATE TRANSITION MOMENTS, THE EXCITED TO EXCITED STATE TRANSITION MOMENTS, AND THE PERMANENT DIPOLE MOMENT OF THE GROUND AS WELL AS THE EXCITED STATES ALL THIS INFORMATION IS IN PRINCIPLE CONTAINED IN ? 12, AND IT CAN BE EXTRACTED FROM THE SECOND-ORDER RESPONSE FUNCTION BY A RESIDUE ANALYSIS THE FIRST-ORDER HYPERPOLARIZABILITY HAS BOTH SINGLE AND DOUBLE RESIDUES FROM EQ (53), WE SEE THAT ONE OF THE FIRST-ORDER RESIDUES BECOMES LIM 2?F0F0 ?2 ? 12 = (57) = N 0 ˆNN ˆ FF ˆ + 0 ˆ NN ˆFF ˆ F ˆ LIM 2?G0G0 ?2 ? = (58) = 0 ˆ FF ˆGG ˆ SECOND-ORDER HYPERPOLARIZABILITY WE MUST NOW ONCE MORE RETURN TO THE PERTURBATION EXPANSION OF THE MOLECULAR POLARIZATION AND CONSIDER THE THIRD-ORDER POLARIZATION IN EQ (36) FROM WHICH WE WILL IDENTIFY A FORMULA FOR THE SECOND-ORDER HYPERPOLARIZABILITY IN ANALOGY WITH PHYSICS OF MASS WHAT WE HAVE ACHIEVED FOR THE POLARIZABILITY AND THE FIRST-ORDER HYPERPOLARIZABILITY THE FIRST-, SECOND-, AND THIRD-ORDER CORRECTIONS TO THE WAVE FUNCTION ARE GIVEN BY EQS (38), (39), AND (40), RESPECTIVELY, IN COMBINATION WITH EQ (31) THE FOUR TERMS THAT CONTRIBUTE TO THE THIRD-ORDER POLARIZATION ARE: 0 ˆ3 = 0EIE0T/?ˆ N N0 ?1 ?2 ?3 ?I3M0 ?2 ?3 ?I2P0 ?3 ?I ×EIN0?1?2?3TE3TE?IENT/?N = N0 ?1 ?2 ?3 ?I3M0 ?2 ?3 ?I2P0 ?3 ?I ×E?I1+2+3TE3T 1 ˆ2 = N ? N0 ?1 +I E?IN0?1TETEIENT/?N ˆ (60) × M M0 ?2 ?3 ?I2P0 ?3 ?I ×EIM0?2?3TE2TE?IEMT/?M = N0 +1 +IM0 ?2 ?3 ?I2P0 ?3 ?I ×E?I1+2+3TE3T 2 ˆ1 = N ? N0 ?1 ?2 +I2M0 ?2 +I (61) ×E?IN0?1?2TE2TEIENT/?N ˆ × P P0 ?3 ?IEIP0?3TETE?IEPT/?P = N0 +1 +2 +I2M0 +2 +IP0 ?3 ?I ×E?I1+2+3TE3T PHYSICS OF MASS 3 ˆ0 = N (63) = E?I1+2+3TE3T PHYSICS OF MASS AND, BY A DIRECT COMPARISON TO EQ (5), WE IDENTIFY THE EXPRESSION FOR THE SECOND- ORDER HYPERPOLARIZABILITY AS ? 123 = N0 + +I3M0 +2 +3 +I2P0 +3 +I JUST AS FOR THE LOWER-ORDER RESPONSES, WE HAVE KEPT THE IMAGINARY TERMS IN THE DENOMINATORS IN ORDER TO MAINTAIN CONVERGENCE OF THE SECOND-ORDER HYPERPOLARIZ- ABILITY AT ALL FREQUENCIES, AS WELL AS TO INDICATE THE POSSIBILITY OF INCLUDING DAMPING IN THE NEAR-RESONANT AND RESONANT REGIONS OF THE FREQUENCY SPECTRUM IN THE NONRES- ONANT REGION, HOWEVER, THE IMAGINARY TERMS MAY BE LEFT OUT, AND THE EXPRESSION FOR ? 123 CAN BE WRITTEN MORE COMPACTLY AS ? 123 = WHERE ?123 DENOTES THE SUM OF THE 24 TERMS ONE GETS BY PERMUTING PAIRS OF ?, 1, 2, AND 3 A DIRECT COMPARISON OF EQ (64) WITH = 0 AND EQ (65) VERIFIES THIS RESULT THE SECOND-ORDER HYPERPOLARIZABILITY TENSOR IS SAID TO POSSESS FULL PERMUTATION SYMMETRY IN THE NONRESONANT REGION THE SECULAR DIVERGENCES IN EQ (65), I E THE SINGULARITIES DUE TO TERMS IN WHICH ONE OR MORE OF THE STATES EQUAL 0, CAN BE REMOVED IN A SIMILAR MANNER AS WE DID FOR THE LOWER-ORDER PROPERTIES IN FACT, EXPLICIT NONDIVERGENT FORMULAS HAVE BEEN DERIVED FOR MOLECULAR HYPERPOLARIZABILITIES UP TO FOURTH ORDER [5], AND FOR THE SECOND-ORDER HYPERPOLARIZABILITY, THE RESULTING FORMULA CAN BE WRITTEN AS ? 123 = PHYSICS OF MASS (67) = (68) = OUT OF THE 24 PERMUTATIONS IN EQ (65) FOR ? 123 ONLY THOSE OBTAINED BY PERMUTING THE PAIRS ? AND 2 CONTRIBUTE TO THIS DOUBLE RESIDUE THE RESIDUE IS TO BE EVALUATED AT 1 = ?FG/2 IN ORDER TO RELATE TO MONOCHROMATIC TWO-PHOTON ABSORPTION IN THE EXCITED STATE G PHYSICS OF MASS (69) = (70) = (71) = 0 ˆNN ˆ F T = WHERE FTF0 ?1 IS A DIMENSIONLESS FUNCTION THAT DEPENDS ON THE TIME T AND THE SEPARATION OF THE FREQUENCY OF THE PERTURBING FIELD FROM THE TRANSITION FREQUENCY PHYSICS OF MASS N F ˆ NN ˆ 0F = WHERE WE HAVE SYMMETRIZED THE SUMMATIONS OVER DUMMY INDICES THIS SYMMETRIZA- TION OBVIOUSLY DOES NOT EFFECT THE VALUE OF THE AMPLITUDE D2 AS LONG AS WE COMPENSATE WITH A FACTOR OF ONE-HALF IN FRONT THE COMPLEX CONJUGATION ASSOCIATED WITH REVERSING THE ORDER OF THE STATES IN THE NUMERATOR IS IRRELEVANT SINCE TRANSITION PROBABILITIES DEPEND ON THE ABSOLUTE SQUARE OF THE MATRIX ELEMENTS THE READER SHOULD COMPARE THE EXPRESSION FOR THE TWO-PHOTON MATRIX ELEMENT TO THE FIRST-ORDER RESIDUE OF THE FIRST-ORDER HYPERPOLARIZABILITY IN EQ (57) ONE OF THE INTERESTING CHARACTERISTICS OF TWO-PHOTON ABSORPTION IS THAT SINCE THE SQUARE OF THE SECOND-ORDER AMPLITUDE, I E THE PROBABILITY OF TWO-PHOTON ABSORPTION, DEPENDS ON THE PERTURBING ELECTRIC FIELD TO THE FOURTH POWER, IT WILL EXHIBIT A SQUARE DEPENDENCE ON THE LIGHT INTENSITY TWO-PHOTON ABSORPTION IS THEREFORE NONLINEAR TO FIRST ORDER WITH RESPECT TO THE INTENSITY OF THE INCIDENT LASER BEAM CARRIED OUT TO YET A HIGHER ORDER IN THE PERTURBATION, WE WILL BE ABLE TO OBTAIN AN EXPRESSION FOR THE THREE-PHOTON ABSORPTION MATRIX ELEMENT THE THIRD-ORDER AMPLI- TUDE, TAKEN FROM EQ (40), IS D ONE-PHOTON TRANSITION MATRIX ELEMENTS BETWEEN THE GROUND STATE 0 AND THE EXCITED STATE F ? WHERE THE OPERATOR 123 DENOTE THAT THE SUMMATION OF TERMS OBTAINED BY PERMUTING THE PAIRS 1, 2, AND 3 IS PERFORMED THE CONNECTION TO THE FIRST-ORDER RESIDUE OF THE SECOND-ORDER HYPERPOLARIZABILITY IN EQ (67) IS VERIFIED BY EXPLICIT COMPARISON OF THE SIX TERMS IN EACH CASE WE SEE THAT THE PROBABILITY OF THREE-PHOTON ABSORPTION DEPEND ON THE LIGHT INTENSITY TO THE THIRD POWER IN TABLE WE HAVE SUMMARIZED SOME OF THE MOLECULAR PROPERTIES THAT CAN BE RETRIEVED FROM THE LINEAR AND NONLINEAR RESPONSE FUNCTIONS 4 THE SEPARATION OF ELECTRONIC AND NUCLEAR DEGREES OF FREEDOM THE EXPRESSIONS PRESENTED IN THE PREVIOUS SECTION APPLY TO EXACT STATES IN PRACTICE, IT IS NOT POSSIBLE TO OBTAIN THE COMPLETE SET OF EXACT STATES AND WE THEREFORE NEED WAYS TO OBTAIN APPROXIMATE SOLUTIONS THE MOST IMPORTANT STEP WHEN DEVELOPING STRATEGIES FOR EVALUATING LINEAR AND NONLINEAR OPTICAL PROPERTIES IS THE INTRODUCTION OF THE BORN–OPPENHEIMER APPROXIMATION [15, 16] AS WE WILL BRIEFLY OUTLINE, AND AS DESCRIBED IN MUCH MORE DETAIL IN CHAPTER 3, THE BORN–OPPENHEIMER APPROXIMATION PHYSICS OF MASS 2ME I I>J > ZZ R +V EL K R PHYSICS OF MASS WHERE THE SUMMATIONS RUN OVER ALL NUCLEI IN THE MOLECULE, AND M IS THE MASS OF NUCLEUS THERE ARE 3N DEGREES OF FREEDOM IN THIS EQUATION, OF WHICH THREE DEGREES OF FREEDOM CORRESPOND TO THE TRANSLATION OF THE CENTER OF MASS OF THE MOLECULE AND THREE (OR TWO IN THE CASE OF LINEAR MOLECULES) DEGREES OF FREEDOM CORRESPOND TO THE OVERALL ROTATION OF THE MOLECULE ABOUT THE CENTER OF MASS THE REMAINING 3N ?65 COORDINATES DESCRIBE THE RELATIVE MOTIONS OF THE NUCLEI OF THE MOLECULE, AND CONSTITUTE THE VIBRATIONS OF THE MOLECULE THE TRANSLATION OF THE MOLECULAR CENTER OF MASS IS NOT QUANTIZED AND THEREFORE NOT OF ANY INTEREST TO OUR DISCUSSION TO A GOOD FIRST APPROXIMATION, THE ROTATIONAL MOTION OF THE MOLECULE IS DECOUPLED FROM THE VIBRATIONAL MOTION, AND WE MAY THUS TREAT THIS SEPARATELY CONSIDERING THE PRODUCT FORM OF THE TOTAL MOLECULAR WAVE FUNCTION, LET US BRIEFLY RETURN TO THE LINEAR POLARIZABILITY GIVEN IN EQ (43), AND LET US ASSUME THAT THE COMPLETE SET OF VIBRONIC PRODUCT STATES CAN BE CONSIDERED AS THE EXACT STATES OF THE MOLECULAR SYSTEM THE LINEAR POLARIZABILITY CAN THEN BE WRITTEN AS [9, 10, 28] ? = ? KK ? 00 ˆKK KK ˆ KK ? (84) + KK + WHERE THE PRIME INDICATES THAT THE SUMMATION RUNS OVER ALL ELECTRONIC K AND VIBRA- TIONAL STATES K APART FROM THE VIBRONIC GROUND STATE AND ?KK DENOTES THE ENERGY DIFFERENCE BETWEEN THE INTERMEDIATE AND GROUND VIBRONIC STATES TO REDUCE THE COMPLEXITY OF EQ (84), WE MAKE THE OBSERVATION THAT ENERGIES INVOLVED IN ELECTRONIC TRANSITIONS ARE IN GENERAL MUCH LARGER THAN THE VIBRATIONAL EXCITATIONS THAT CONTRIBUTE SIGNIFICANTLY TO THE SUMMATION WE MAY THEREFORE ASSUME THAT KK ?K0, WHICH ALLOWS US TO PARTITION THE COUPLED SUMMATIONS IN EQ (84) INTO TWO CONTRIBUTIONS, ONE OVER ELECTRONIC EXCITED STATES INVOLVING ONLY THE GROUND VIBRATIONAL STATE, AND ONE OVER THE VIBRATIONALLY EXCITED STATES INVOLVING ONLY THE ELECTRONIC GROUND STATE ? = (85) = 0 ˆKK ˆ K0 ? + K0 + K0 ? + K0 + WHERE, IN THE FORMER TERM, THE CLOSURE OVER (K) HAS BEEN CARRIED OUT AND, IN THE LATTER TERM, WE HAVE INTRODUCED THE NOTATION 00 = 0 ˆ0 THE LATTER OF THE TWO TERMS ONLY INVOLVES EXCITATIONS WITHIN THE VIBRATIONAL MANIFOLD AND IS FOR THIS REASON REFERRED TO AS THE PURE VIBRATIONAL CONTRIBUTION [3, 6] IN RECENT YEARS, VARIOUS EFFICIENT APPROACHES HAVE BEEN DEVELOPED FOR THE CALCULATION OF PURE VIBRATIONAL CONTRIBUTIONS PHYSICS OF MASS