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MOLECULAR PHYSICS, 2000, VOL. 98, NO. 20, 1597± 1601

Editors’ Note

This paper draws attention to the advantages that would be obtained by adopting a new convention for the sign of g factors that would make the g factor for a negative quantity (g 2), rather than a positive quantity as generally adopted at present. The editorsº ¡ are aware that the proposal made in this paper concerning the conventional sign of the g factor for electron spin will be seen by some readers as controversial. We have nonetheless agreed to publish this paper in the hope that it will stimulate discussion. The editors would welcome comments on this proposal in the form of short papers, which they will then be happy to consider for publication together at a later date.

Remarks on the signs of g factors in atomic and molecular Zeeman spectroscopy

J. M. BROWN1, R. J. BUENKER2, A. CARRINGTON3, C. DI LAURO4, R. N. DIXON5, R. W. FIELD6, J. T. HOUGEN7, W. HUÈ TTNER8*, K. KUCHITSU9, M. MEHRING10, A. J. MERER11, T. A. MILLER12, M. QUACK13, D. A. RAMSAY14, L. VESETH15 and R. N. ZARE16 1 Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, UK 2 Bergische UniversitaÈ t-Gesamthochschule Wuppertal, Theoretische Chemie, Gauû straû e 20, D-42097 Wuppertal, Germany 3 Department of Chemistry, University of Southampton, Southampton SO9 5NH, UK 4 Universita degli Studi Federico II, Dipartimento di Chimica Farmaceutica e Tossicologica, Via Domenico Montesano 49, I-80131 Napoli, Italy 5 School of Chemistry, University of Bristol, Bristol BS8 1TS, UK 6 Department of Chemistry, MIT, 77 Massachusetts Ave., Cambridge, MA 02139, USA 7 Optical Technology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 8 Abteilung Chemische Physik, UniversitaÈ t Ulm, D-89069 Ulm, Germany 9 Department of Chemistry, Josai University, Keyakidai, Sakado 350-0295, Japan 10 II. Physikalisches Institut, UniversitaÈ t Stuttgart, Pfa€ enwaldring 57, D-70550 Stuttgart, Germany 11 Department of Chemistry, The University of British Columbia, 2036 Main Mall, Vancouver BC, Canada V6T 1ZI 12 Department of Chemistry, The Ohio State University, Newman and Wolfrom Laboratory, 100 West 18th Avenue, Columbus, OH 43210, USA 13 Laboratorium fuÈ r Physikalische Chemie, ETH ZuÈ rich (Zentrum), CH-8092 ZuÈ rich, Switzerland 14 Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario, Canada K1A 0R6 15 Department of Physics, University of Oslo, 0316 Oslo 3, Norway 16 Department of Chemistry, Stanford University, Stanford, CA 94305-5080, USA

* Author for correspondence. e-mail: [email protected]

Molecular Physics ISSN 0026± 8976 print/ISSN 1362± 3028 online # 2000 Taylor & Francis Ltd http://www.tandf.co.uk/journals 1598 J. M. Brown et al.

(Received 1 March 1999 ; revised version accepted 6 June 2000 )

Various magnetic moments, associated with rotational, vibrational, nuclear spin, electron orbital and electron spin angular momenta, can contribute to the Zeeman e€ ect in atoms and molecules. They are considered in this paper in the context of the e€ ective Hamiltonian where relativistic and other corrections as well as the e€ ects of mixing with other electronic states are absorbed in appropriate g factors. In spherically symmetric systems, the magnetic moment arising from a speci® c can be written as the product of three factors: the nuclear or (which is positive), the g factor (which may be positive or negative), and the corresponding angular momentum (which is a vector). A con- vention is discussed, in which the sign of the g factor is positive when the dipole moment is parallel to its angular momentum and negative when it is antiparallel. This would have the advantage that it could be applied consistently in any situation. Such a choice would require the g factors for the electron orbital and electron spin angular momenta to be negative. This concept can easily be extended to the case of a general molecule where the relation between the dipole and angular momentum vectors has tensorial character.

1. Introduction implies that the direction of the moment is chosen The g factor, ® rst introduced by Lande [1] to explain such that the torque imposed by the ® eld acts to align anomalous Zeeman e€ ects in atomic spectra, now has a m in the direction of B. The same sign is used in classical more wide-ranging signi® cance. It is generally utilized to electrodynamics, and therefore is in agreement with the represent the linear relationship between magnetic direction of macroscopic moments induced by any elec- moments and angular momenta. For nuclei, g factors tric current distribution. A common example is the are usually positive but are occasionally negative, e.g. permanent magnetic dipole moment of a small planar 3He, 17O. The underlying convention is that for positive conducting loop, which can be expressed as g factors the of the nucleus is oriented m iAn 2 parallel to its angular momentum but for negative g ˆ ? … † factors the reverse is true. Such is the usage in particle where i is the current in the wire, A is the area enclosed physics and in nuclear magnetic resonance (NMR). In by the loop, and n its normal vector de® ned by the electron spin resonance spectroscopy (ESR) the opposite right hand rule [4]. The? direction of i is de® ned in accor- convention is used by most authors, that is the g factor, dance with the transport of positive charges. e gS, is chosen to be positive even though the electron spin Zeeman e€ ects in diamagnetic species observed at and magnetic moment vectors are antiparallel, as ® rst high resolution are caused by rotational, vibrational, shown by Dirac [2]. The results of many magnetic ® eld and nuclear magnetic moments (neglecting magnetiz- experiments do not depend on the relative orientations ability contributions). They all have in common that a of the two vectors. For example, the theory of magnetic linear relation holds between the magnetic moment and oscillations in metals (de Haas± van Alphen e€ ect, used the angular momentum. Thus, a rotational moment mrot e for probing Fermi surfaces) shows that the `sign of gS is can be formulated [5]: indeterminate and it is natural to assume that it is posi- rot rot [ ] mi ·Ngij Jj ; 3 tive’ 3 . In this paper, we wish to discuss the situation ˆ … † where several di€ erent kinds of moment are present, and where J is the rotational angular momentum in units of where their relative orientations are of great importance. h-, grot is the rotational g tensor, i and j refer to the In particular, we explore the implications of the sign of molecule-® xed axes, and summation is implied over ge (and of ge , the orbital electronic g factor) on terms in S L j x;y;z. The , · eh-=2m [6], is the molecular Hamiltonian, for high resolution Zeeman ˆ N ˆ p a positive scaling factor (e and m are the charge and studies of molecular species. p mass of the , h- is the divided by 2º). The other angular momenta considered below are, like J, all understood to be given in units of h-. 2. The eŒective Zeeman Hamiltonian Linear and symmetric-top molecules in degenerate The basic Hamiltonian for the magnetic ® eld inter- vibrational states can exhibit a vibrational angular action is momentum pz about the ® gure axis which gives rise to H m B; 1 a magnetic moment [7] ˆ ¡ · … † where m is the magnetic dipole moment and B mvib · gvibp ; 4 the magnetic ¯ ux density. Various contributions to m z ˆ N z … † are considered below. The minus sign in equation (1) where the vibrational g factor, gvib, is a scalar. Signs of g factors 1599

rot vib Experiments have shown [8] that gii and g can be orbital , are both caused by circulating nega- vib positive or negative depending on the details of the tive charges (a similar remark applies to mz , equation nuclear± electron couplings in di€ erent molecules. Clas- (4)). sically speaking, the sum of all nuclear currents induced The situation with the electron in a rovibrating molecule may or may not be out- is at ® rst sight more di cult since the particle can be weighted by that of the electronic ones in forming the considered point-like, without spatial extension [12]. total m, according to equation (2). Thus, the classical picture of a spinning charged Atomic nuclei also show measurable magnetic dipole sphere does not apply. However, Dirac has shown that moments of either sign when their nuclear spin angular the relativistic version of the SchroÈ dinger equation gives momenta, I, do not vanish: a half-odd spin angular momentum component Sz, and nuc nuc a magnetic moment [2] m ·Ng I; 5 ˆ … † eh- where gnuc, a scalar, means the nuclear g factor. The me S ; 8 S ˆ ¡ m magnetogyric ratio, ® · gnuc=h-, is frequently used as e … † ˆ N an alternative description in NMR spectroscopy because which shows that the spin magnetic moment points anti- ! ®B gives the angular transition frequency between parallel to the spin. This again can be written ˆ neighbouring states directly [9]. me · ge S ; 9 S ˆ B S … † In equations (3), (4) and (5), we note the common where ge 2. Later it was found that elec- S ˆ ¡ principle that the g factors determine the orientations of trodynamic corrections have to be applied [13, 14], and the magnetic dipole moments relative to those of the that small amounts of directional dependence must be angular momenta. taken into account in a molecule [15], but this does not e Paramagnetic species contain unpaired so a€ ect our considerations (gS of a bound electron needs that electron spin magnetic moments contribute to the also, in principle, a relativistic correction, and the radia- tive corrections might di€ er from those of the free total magnetic moment. In cases of su cient symmetry e electron). The various small corrections to gL (equation (e.g. cylindrical symmetry in linear molecules), con- e served orbital angular momenta may also occur. An (7)) and gS (equation (9)) frequently have been discussed orbiting electron is equivalent to a current i e=T in the literature in terms of e€ ective Zeeman Hamilto- ˆ where T is the time needed for one orbit. The¡corre-… † nians. For example, extensive calculations have been sponding magnetic moment is carried out in atomic and molecular open-shell systems in [16] and [17]. me · L 6 L ˆ ¡ B … † Summarizing this part, we see that, if we follow the - line that the sign of a g factor should determine the where ·B eh=2me is the Bohr magneton, de® ned as a ˆ direction of its associated magnetic moment, we end positive quantity [6] (me is the mass of the electron and L is the angular momentum of the electron). Following the up with negative electronic g factors for both the orbital lead of equations (3± 5) it is tempting to write this as and spin magnetism of the electron. Lande [1] intro- duced a formula for the g factor of a many-electron me · ge L; 7 L ˆ B L … † atom in the adiabatic approximation which usually with ge 1 for the g factor of the unperturbed (clas- yields positive values. This formula was developed L ˆ ¡ sical) orbiting electron. In order to meet the precision before studies of molecular Zeeman e€ ects. If one achievable in high resolution experiments, relativistic wishes to keep consistency between atomic and molecu- corrections [10], nuclear motion corrections [11] and lar studies, the signs of Lande g factors would need to be non-adiabatic corrections must be applied, which reversed. cause ge to deviate from unity in the e€ ective Hamil- j L j tonian of an atomic or molecular paramagnetic species. 3. Discussion Many authors prefer, therefore, to use an orbital g It is perhaps helpful to put the proposals in equations factor instead of writing the operator in the form of (7) and (9) into historical perspective and to review equation (6). Note that equation (6) is the basic operator usage in the literature. The basic form of equation (6), which also gives rise to the electronic contribution of as obtained from electrodynamic theory, is rot m and which appears as an additive Van Vleck per- eh- turbation term in the tensor components grot which is me L: 10 ij L ˆ ¡ 2m always negative in the electronic [5]. It e … † e seems appropriate to ascribe the same sign to gL because The important characteristic… of† this relation in the the two phenomena, electronic molecular rotational and present discussion is that the magnetic dipole moment 1600 J. M. Brown et al. is oriented in the opposite direction to the angular It is now becoming commonplace to measure mag- momentum because the motion involved is that of a netic e€ ects in molecules where there are several dif- negatively charged particle. Three di€ erent approaches ferent contributions to the experimental magnetic to the implementation of this property have been used in dipole moment; these are incorporated in an e€ ective the literature. Hamiltonian. For example, laser resonance studies have recently been carried out (in zero ® eld) on a (i) Negative Bohr magneton helium atom in which one electron was replaced by an -4 Here the Bohr magneton is de® ned to be negative, antiproton, p He‡ [31]. It might soon be possible for re¯ ecting the negative charge e on the electron: similar exotic molecular systems to be formed con- ¡ taining a positron. In this situation, it is desirable to eh- · 0 establish a universal convention for the sign of a mag- B ˆ ¡ 2m e: netic dipole moment relative to its associated angular This approach is used by Messiah [18] and by Cohen- momentum. Tanoudji et al. [19] in their texts on . It has already been mentioned in the introduction that However, it does not conform to the IUPAC [6] and magnetic resonance results often do not depend on the IUPAP [20] de® nition of the Bohr magneton, and is orientations of the relevant magnetic moments. Workers not widely used. in such ® elds will not see the need for a general g factor convention. However, there are a number of situations (ii) Positive electronic g factor where the sign is indeed . One such example is In this approach, equation (7) is rewritten as the hyper® ne interaction in an atom which, in a spheri- e e cally averaged environment, is proportional to the term mL ·BgL L e nuc 2 e ˆ ¡ ·BgS·Ng ÁA 0 . The product of the electron gS e j j nuc with gL now equal to + 1. There is a similar equation and the nuclear… †g g factors determines the sign… of† e corresponding to equation (9) with gS written as + 2. the hyper® ne interaction.… † Moreover, in a multi-electron This has been used widely by atomic physicists [21± 23] system and in particular in molecules the expectation and by workers in the ® eld of open-shell molecules, value of the wavefunction at the nuclear site, Á 0 2, j A j where the electronic contribution to the magnetic is replaced by the spin density which may be either… † moment dominates the smaller nuclear contributions positive or negative depending on the electronic state. (see, e.g. [24± 26]). This convention is appealing in the It is of considerable importance for an understanding of following respect. It attempts to de® ne all g factors as the electronic state to determine the sign of the spin positive by the introduction of an explicit positive or density. A common consensus of the sign of the g factors negative sign in the relationship between the magnetic would immediately give a clue about the sign of the spin dipole moment and the associated angular momentum, density without any theory. The sign of the hyper® ne depending on whether the particle which generates the interaction can be determined experimentally from magnetic ® eld carries a positive or negative charge (see NMR Knight shifts in metals [32] or from triple equation (5) for a proton and equation (10) for an elec- ENDOR measurements [33] in organic radicals or tron). Its weakness is that it is not possible to apply it other paramagnetic materials. consistently in all situations. There are, for example, many nuclei which have negative g factors [6] despite carrying a positive charge. In addition, it gives no indi- 4. Conclusion cation of the sign of the g factor of a neutral particle In conclusion, any of the three possible conventions such as the . given above can be used to describe Zeeman e€ ects in atoms and molecules correctly. They need only to be (iii) Negative electronic g factors used consistently within a given piece of work in This is the convention given above in equations (7) which the convention followed is made explicit so that and (9). This approach has also been used before, for others can correctly use the results. In the present com- example by Townes and Schawlow [27], Frauenfelder munication, we wish to point out that approach (iii), in and Henley [28] and Sard [29], and has also recently which the sign of the g factor depends on the relative been endorsed by the Committee on Data for Science orientation of the magnetic dipole moment and the and Technology (CODATA) [30]. It has the merit of angular momentum which produces it, has the merits being consistent with the accepted description of nuclear of consistency and simplicity. and rovibrationally induced magnetic moments, and Finally, we wish to add that in any application in gives the orientation of the magnetic moment relative which the results are sensitive to the sign of g factors to its associated angular momentum directly. it is imperative that the convention followed should be Signs of g factors 1601 made clear and explicit, whether or not the one pro- [18] MESSIAH, A., 1989, Quantum Mechanics (Amsterdam: posed here is adopted. North-Holland). [19] COHEN-TANOUDJI, C., DIN, N., and LALOEÈ , F., 1977, Quantum Mechanics (Paris: Herman). [20] COHEN, E. R., and GIACOMO, P., 1987, Physica A, 146, 1. References [21] YANG, F., and HAMILTON, J. H., 1996, Modern Atomic [1] LANDEí, A., 1923, Z. Phys., 15, 189. and Nuclear Physics (New York: McGraw-Hill). [2] DIRAC, P. A. 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