- Email: [email protected]
On the description of high-resolution experiments in molecular physics
Volume 44, number 1
15 November 1976
CHEMICAL PHYSICS LETTERS
ON THE DESCRIPTION OF HIGH-RESOLUTION
EXPERIMENTS
IN MOLECULAR PHYSKS
R.G. WOOLLEY University Chemical Laboratory.
Cambniige CB2 IEW. UK
Rcceivcd 3 August 1976
We argue that afi high-reTolution experiments on small molecules some of which arrcntly “belong” to chemistry and others to physics, should be brought together and discussed in a unified way using the best theory available. namely quantum mechanics, and that preconceptions about “molecular structure” should bc avoided in this arca of phyGx1 science.
I have recently argued [I] that the application of the quantum theory to the description of molecular phenomena carries with it an essential and inescapable limitation on the domain of validity of the classical concept of molecular structure. This is because the quantum theory asserts that whether the description of any given experiment is based on the time-dependent quantum states associated with molecular structures, or uses the molecular eigenstates for which no structural interpretation is possible, must be governed by the nature of the experiment and the resolution (e.g. in energy) actually achieved, and not by preconceptions about the “properties of molecules” that are
there is the additional possibility of chcrnid reaction, although the resolution of the individual quantum states is as yet poorer than in the scattering of light particles from molecules; however we might also include these experiments in the second category, noting that we may have to sum over final states and average over initial states in order to make comparisons with the observed cross sections_ The drive towards continual improvement in resolution and the concomitant penetration further into the quantum regime appears to be an essential aspect of modern physical science in general, and one is bound to reconsider the theoretical framework used for the discussion of experiments in-
dcrivcd
volving
from clarsical
macroscopic
chemistry.
The prac-
tical effect of this theoretical principle is that the molecular structure description is correct for almost all of molecular science, but that there remains the area of high precision experiments on small molecules in the gas phase where a fully quantum-mechanical theory involving the molecular eigenstatcs is now likely to be appropriate. The experiments I have in mind include elastic and inelastic scattering by molecules of photons i.e. birefringence and optical spectroscopy in its various forms, and similar scattering experiments involving light charged particles such as electrons and positrons which may of course also involve the annihilation/creation of photons as for example in photoemission. Significant quantum effects, for example oscillations in differential cross sections, are also seen in atomatom and atom-molecule scattering in beams where
moteculcs
as
the expcrimcntal
acLvanccs are
made As shown in my earlier paper many of the underlying ideas still used in molecular spectroscopy predate the discovery of the quantum theory [I] _ Whilst some of these experiments arc conventionally regarded as part of physical chemistry (o.g_ the optical spectroscopy of small moIecuIcs), others such as the scattering of slow electrons by molecules seem to lie in the province of physics. However ale these experiments come together in for exampie the study of atmospheric physics and astrophysics, and I believe that it is highly desirable to attempt to interpret the experimental information on small molecules in a unified way using the best theory available, namely quantum mechanics. Large parts of the more “chemica1” end of this subject matter have always been interpreted in terms of 2 form of quantum theory which is strongly influenced by the results of the OId Quantum Theory, and involve 73
Yoiumc
44, number 1
CHEMICAL PHYSICS LETTERS
an Implicit assumption ofmicrcxctlpic rigidity that is simply not consistent with a gcncral quantunl-mechanical theory of molecular elgcnstntcs [I]. We shall argue elsewhere that microscopic ngidity, which lies at the heart of the concept of molecular structure, arises as an asymptotic approximation in quantum mechanics; that is, crudely speaking it only cmergcs through the trarlsition from quantum to classical description [2]. A simple ii:usttation;of the tension caused by the attempt to preserve molcc&ar strucfure at all costs in the face of the requirements of quantum mcch-
anics cat be found in the different ways the notion of “dipolar” mofecutes affects our thinking a3out some of the typical gas phase experimer.ts mentioned above. The experimental and theoretical investigation of the scattering of slow electrons by diatomic moiecules is a rapidly dcvclopfng area of contemporary physics, and one can expect these studies to be cxtended systematically to iargcr molecules. A beautiful discussion of the scattering of slow electrons by heteronucleur diatomic (i.e. “dipolar”) molecules has been given by Chang and Fano [3]. When the electron is close to the molecule (the near zone) it is natural to use molecule-fixed coordinates because m this reference frame the interaction berween t!lc charge and the molecule is dominated by the molccule’s apparent dipole moment; however the interaction between a charged particle ilnd a real dipole is of such long-range that the scattering cross section is infinite, and in order to obtain a satisfactory, finite cross section we must transform to space-fixed axes to describe the wave zone, where the scattered cicctron is detected, so as to incorporate the condition that the molecular cigenstatcs have no actual dipole moment because of parity collservation. We could use space-fixed axes to describe the entire experiment but the frame transformation method leads to a particularly cfear picture of the physical processes involved; foi a diatomic molecule no assumption of microscopic rigidity is involved, and the transformation from the “molecule-abed” to the “space-fixed” (inertial) reference frame is unitary. This is an esscntiaiiy quantummechanical account of the experiment. By contrast, consider another type of experiment in which a “dipoiar’” molecule is placed iu an inhomogeneous electric field and one measures the deflexion caused by the field, or dternatively the degree of 74
15
November 1976
ellipticity (optical anisotropy) induced in a light beam passing through the gas + field system given that the light was initially plane or circularly polarized. In a theoretical description of this type of experiment one regards both the static electric field and the light beam as perturbations because one wishes to relate the observablc (the deflexion or the cliipticity of the light) to matrix elements of the isolated molecule; the optical anisotropy measurement is conventionally performed on a thermal ensemble at temperature T and so one has also to consider the statrstical mechanics of the probiem. The deflcxion experiment is simply intcrprctcd in terms of the *‘moIccuIar dipoIe moment”, whilst the best discussion of the optical anisotropy measurement is a dctalled semiclassical trcatmcnt, i~lvolving a cIassica1 ~oIt~rna~n statistical average, that leads to an intcrprctation in terms of the “molecular electric quadrupolc moment” and rcsolvcs the difficulties of origin~cpendence for the quadr~ipoIe nioment that one would expect for a classical charge distribution with a “pcrmancnt” clcctric dipole moment [4J . It is easy to see that thcsc accounts are strongly in~;icilced by the molecular structure hypothesis, for we know from the discussion above that the isolated molecule has no real dipole moment, and indeed quantuln-rnechanicaliy, is known to interact with the static field through the polariz~bility tensor for the internal states I [l] _ Although a detadcd quantum mechanical treatment will not be given here one or two salient points deserve to be mentioned; in an inhorno~erleous electric field the total momentum of a motecufe is not conserved, and this manifests itself as a coupling between the over aI motion cf the molectdar ten tre-of-mass and the pcrturbcd internal states of the moiccule. If the simplest adiabatic approximation is valid one can consider the overal motion of the molecular centre-of-mass R as arising from motion under the potential Vi = a (i): E(K)E(R), where E(R) is the electric field evaluated at the point R. This is the mccIlanism that is responsible for electric deflexion, and since it must contribute to the generatlon of the optical anisotropy by the inhomogeneous field one would want to treat this aspect of the problem explicitly through an approximate solution of the appropriate Schrijdinger equation. Moreover within a completely quantummechanical theory it is pcjssiblc to maintain explicitly the mechanica conservation laws and gauge invariance, and no difficutties over “origin dependent multipoles”
Volume 44, number 1
CHEhIlCAL PHYSICS LETTERS
can arise, for the theory will lead directly to the required connection between the macroscopic measurement and the appropriate matrix clcments. One should also note that almost all treatments of molecular light scattering compute a classical Boltzmann average of the transition probability which includes the effects of external static fields through the interaction energies of the “permanent” molccular electric and magnetic dipoies with the external Fields; whilst this is the proper thing to do for Ziqui& where an isolated molecule (cigenstate) picture can never be expected to be appropriate, the example of the calculation of the molecular electric susceptibility as a function of tempcraturc 5 la van Vleck shows that the quantum-mechanical prescription of taking the trace with the density operator cxp(- B(/kT) does not lead to the same formulae as the classical Boltzmann average, for exampie the “low-frequency” part of the averaged olarizability contributes the matrix element (ild P Ii> rather than i
15 November 1976
of gas-phase experiments which can be sensibly discussed in terms of “isolated molccuIcs”. It appears therefore that much of the reIevant quantum mechanical theory required to do justice to these precision experimcnta! measurements on small molcculcs in the gas phase has yet to be worked out, and this situation is unlikely to change until it is sufficiently widely accepted that the inteIIcctua1 baggage of classical macroscopic (bulk) chemistry should bc thrown overboard since it is simply not relevant in this particular area of physical science. The financial support of Trinity MI, Cambridge is gratefully acknowledged.
References 111 KG. Woollcy. Advan. Phys. 25 (1976) 27. [2] R.T. Sutcliffc md R.G. Woolicy, to bc pubtishcd (1976). [ 31 F..S.Clrang and U. Fmo. Phys. Rev. A6 (1972) 173. 14 J A.D. Buckingham and H-C. Longuct-Higgins. hIaL Phys. 14 (1968) 63.
75
15 November 1976
CHEMICAL PHYSICS LETTERS
ON THE DESCRIPTION OF HIGH-RESOLUTION
EXPERIMENTS
IN MOLECULAR PHYSKS
R.G. WOOLLEY University Chemical Laboratory.
Cambniige CB2 IEW. UK
Rcceivcd 3 August 1976
We argue that afi high-reTolution experiments on small molecules some of which arrcntly “belong” to chemistry and others to physics, should be brought together and discussed in a unified way using the best theory available. namely quantum mechanics, and that preconceptions about “molecular structure” should bc avoided in this arca of phyGx1 science.
I have recently argued [I] that the application of the quantum theory to the description of molecular phenomena carries with it an essential and inescapable limitation on the domain of validity of the classical concept of molecular structure. This is because the quantum theory asserts that whether the description of any given experiment is based on the time-dependent quantum states associated with molecular structures, or uses the molecular eigenstates for which no structural interpretation is possible, must be governed by the nature of the experiment and the resolution (e.g. in energy) actually achieved, and not by preconceptions about the “properties of molecules” that are
there is the additional possibility of chcrnid reaction, although the resolution of the individual quantum states is as yet poorer than in the scattering of light particles from molecules; however we might also include these experiments in the second category, noting that we may have to sum over final states and average over initial states in order to make comparisons with the observed cross sections_ The drive towards continual improvement in resolution and the concomitant penetration further into the quantum regime appears to be an essential aspect of modern physical science in general, and one is bound to reconsider the theoretical framework used for the discussion of experiments in-
dcrivcd
volving
from clarsical
macroscopic
chemistry.
The prac-
tical effect of this theoretical principle is that the molecular structure description is correct for almost all of molecular science, but that there remains the area of high precision experiments on small molecules in the gas phase where a fully quantum-mechanical theory involving the molecular eigenstatcs is now likely to be appropriate. The experiments I have in mind include elastic and inelastic scattering by molecules of photons i.e. birefringence and optical spectroscopy in its various forms, and similar scattering experiments involving light charged particles such as electrons and positrons which may of course also involve the annihilation/creation of photons as for example in photoemission. Significant quantum effects, for example oscillations in differential cross sections, are also seen in atomatom and atom-molecule scattering in beams where
moteculcs
as
the expcrimcntal
acLvanccs are
made As shown in my earlier paper many of the underlying ideas still used in molecular spectroscopy predate the discovery of the quantum theory [I] _ Whilst some of these experiments arc conventionally regarded as part of physical chemistry (o.g_ the optical spectroscopy of small moIecuIcs), others such as the scattering of slow electrons by molecules seem to lie in the province of physics. However ale these experiments come together in for exampie the study of atmospheric physics and astrophysics, and I believe that it is highly desirable to attempt to interpret the experimental information on small molecules in a unified way using the best theory available, namely quantum mechanics. Large parts of the more “chemica1” end of this subject matter have always been interpreted in terms of 2 form of quantum theory which is strongly influenced by the results of the OId Quantum Theory, and involve 73
Yoiumc
44, number 1
CHEMICAL PHYSICS LETTERS
an Implicit assumption ofmicrcxctlpic rigidity that is simply not consistent with a gcncral quantunl-mechanical theory of molecular elgcnstntcs [I]. We shall argue elsewhere that microscopic ngidity, which lies at the heart of the concept of molecular structure, arises as an asymptotic approximation in quantum mechanics; that is, crudely speaking it only cmergcs through the trarlsition from quantum to classical description [2]. A simple ii:usttation;of the tension caused by the attempt to preserve molcc&ar strucfure at all costs in the face of the requirements of quantum mcch-
anics cat be found in the different ways the notion of “dipolar” mofecutes affects our thinking a3out some of the typical gas phase experimer.ts mentioned above. The experimental and theoretical investigation of the scattering of slow electrons by diatomic moiecules is a rapidly dcvclopfng area of contemporary physics, and one can expect these studies to be cxtended systematically to iargcr molecules. A beautiful discussion of the scattering of slow electrons by heteronucleur diatomic (i.e. “dipolar”) molecules has been given by Chang and Fano [3]. When the electron is close to the molecule (the near zone) it is natural to use molecule-fixed coordinates because m this reference frame the interaction berween t!lc charge and the molecule is dominated by the molccule’s apparent dipole moment; however the interaction between a charged particle ilnd a real dipole is of such long-range that the scattering cross section is infinite, and in order to obtain a satisfactory, finite cross section we must transform to space-fixed axes to describe the wave zone, where the scattered cicctron is detected, so as to incorporate the condition that the molecular cigenstatcs have no actual dipole moment because of parity collservation. We could use space-fixed axes to describe the entire experiment but the frame transformation method leads to a particularly cfear picture of the physical processes involved; foi a diatomic molecule no assumption of microscopic rigidity is involved, and the transformation from the “molecule-abed” to the “space-fixed” (inertial) reference frame is unitary. This is an esscntiaiiy quantummechanical account of the experiment. By contrast, consider another type of experiment in which a “dipoiar’” molecule is placed iu an inhomogeneous electric field and one measures the deflexion caused by the field, or dternatively the degree of 74
15
November 1976
ellipticity (optical anisotropy) induced in a light beam passing through the gas + field system given that the light was initially plane or circularly polarized. In a theoretical description of this type of experiment one regards both the static electric field and the light beam as perturbations because one wishes to relate the observablc (the deflexion or the cliipticity of the light) to matrix elements of the isolated molecule; the optical anisotropy measurement is conventionally performed on a thermal ensemble at temperature T and so one has also to consider the statrstical mechanics of the probiem. The deflcxion experiment is simply intcrprctcd in terms of the *‘moIccuIar dipoIe moment”, whilst the best discussion of the optical anisotropy measurement is a dctalled semiclassical trcatmcnt, i~lvolving a cIassica1 ~oIt~rna~n statistical average, that leads to an intcrprctation in terms of the “molecular electric quadrupolc moment” and rcsolvcs the difficulties of origin~cpendence for the quadr~ipoIe nioment that one would expect for a classical charge distribution with a “pcrmancnt” clcctric dipole moment [4J . It is easy to see that thcsc accounts are strongly in~;icilced by the molecular structure hypothesis, for we know from the discussion above that the isolated molecule has no real dipole moment, and indeed quantuln-rnechanicaliy, is known to interact with the static field through the polariz~bility tensor for the internal states I [l] _ Although a detadcd quantum mechanical treatment will not be given here one or two salient points deserve to be mentioned; in an inhorno~erleous electric field the total momentum of a motecufe is not conserved, and this manifests itself as a coupling between the over aI motion cf the molectdar ten tre-of-mass and the pcrturbcd internal states of the moiccule. If the simplest adiabatic approximation is valid one can consider the overal motion of the molecular centre-of-mass R as arising from motion under the potential Vi = a (i): E(K)E(R), where E(R) is the electric field evaluated at the point R. This is the mccIlanism that is responsible for electric deflexion, and since it must contribute to the generatlon of the optical anisotropy by the inhomogeneous field one would want to treat this aspect of the problem explicitly through an approximate solution of the appropriate Schrijdinger equation. Moreover within a completely quantummechanical theory it is pcjssiblc to maintain explicitly the mechanica conservation laws and gauge invariance, and no difficutties over “origin dependent multipoles”
Volume 44, number 1
CHEhIlCAL PHYSICS LETTERS
can arise, for the theory will lead directly to the required connection between the macroscopic measurement and the appropriate matrix clcments. One should also note that almost all treatments of molecular light scattering compute a classical Boltzmann average of the transition probability which includes the effects of external static fields through the interaction energies of the “permanent” molccular electric and magnetic dipoies with the external Fields; whilst this is the proper thing to do for Ziqui& where an isolated molecule (cigenstate) picture can never be expected to be appropriate, the example of the calculation of the molecular electric susceptibility as a function of tempcraturc 5 la van Vleck shows that the quantum-mechanical prescription of taking the trace with the density operator cxp(- B(/kT) does not lead to the same formulae as the classical Boltzmann average, for exampie the “low-frequency” part of the averaged olarizability contributes the matrix element (ild P Ii> rather than i
15 November 1976
of gas-phase experiments which can be sensibly discussed in terms of “isolated molccuIcs”. It appears therefore that much of the reIevant quantum mechanical theory required to do justice to these precision experimcnta! measurements on small molcculcs in the gas phase has yet to be worked out, and this situation is unlikely to change until it is sufficiently widely accepted that the inteIIcctua1 baggage of classical macroscopic (bulk) chemistry should bc thrown overboard since it is simply not relevant in this particular area of physical science. The financial support of Trinity MI, Cambridge is gratefully acknowledged.
References 111 KG. Woollcy. Advan. Phys. 25 (1976) 27. [2] R.T. Sutcliffc md R.G. Woolicy, to bc pubtishcd (1976). [ 31 F..S.Clrang and U. Fmo. Phys. Rev. A6 (1972) 173. 14 J A.D. Buckingham and H-C. Longuct-Higgins. hIaL Phys. 14 (1968) 63.
75