Chemical Physics 265 ,2001) 153±163 www.elsevier.nl/locate/chemphys
Chemistry of vibronic coupling. Part 1: How to maximize vibronic coupling constants in a diabatic harmonic potential model?
Wojciech Grochala 1, Robert Konecny, Roald Homann *
Department of Chemistry and Chemical Biology, Cornell University, Ithaca, NY 18450, USA Received 21 April 2000
Abstract Using a simple diabatic model of two equivalent energy minima with crossing harmonic potential energy surfaces, we i i analyze the vibronic coupling constant keg ,e, g electronic states coupled, i coupling vibration). keg ,expressed in i i energy units) is a product of heg ,expressed in force units) and hugjQijvei ,expressed in distance units) where heg is the dynamic o-diagonal vibronic coupling constant, ug and ve are vibrational wave functions for normal mode i, in the g i and e state respectively, and Qi is normal coordinate of ith mode. We study the way keg depends on three parameters: the force constant ,k), the reduced oscillator mass ,m) and the displacement between two minima along the normal i i coordinate of the coupling mode Qi DQ. For each k there is only one DQ which maximizes keg. The maximum in keg i originates from the opposing behavior of heg and hgjdH=dQijei factors in k; DQ space. A strategy for obtaining large i kegs should adjust properly small DQs in systems with strong multiple bonds ,large ks). This condition appears to be i p i hard to ful®ll in experiment. It also emerges that keg correlates linearly with k=m. Hence, to maximize keg one should build a system of light elements. The results presented here may be useful in the experimental search for high-tem- perature superconductivity, delimiting the space of element combinations which might be investigated. Ó 2001 Pub- lished by Elsevier Science B.V.
1. Introduction damental role of electron±phonon coupling in that theory. Although it is clear that the community The vibronic coupling constant ,VCC) is an lost some faith in the explanative role of classical important molecular parameter which describes in BCS theory after discovery of high-temperature a quantitative way the phenomenon of vibrational superconductivity ,HTSC) [6] in early Õ80s, both coupling of electronic states. 2 Interest in VCCs theoretical and experimental investigations of VCCs increased substantially when the BCS theory of su- are common in literature [7,8]. Despite many ,con- perconductivity was born [5], because of the fun- tradictory) claims by proponents of this or that theory, the HTSC phenomenon remains till today quite mysterious. Also few of the many compet-
* ing theoretical approaches can be easily translated Corresponding author. Fax: +1-607-255-5707. into chemical language. E-mail address: [email protected] ,R. Homann). 1 Also corresponding author. Our investigations also have their source in the 2 On the theory of vibronic coupling see Refs. [1±4]. phenomenon of superconductivity. Although BCS
0301-0104/01/$ - see front matter Ó 2001 Published by Elsevier Science B.V. PII: S0301-0104,01)00280-4 154 W. Grochala et al. / Chemical Physics 265 22001) 153±163
The general plan of our ®ve-part study, entitled ``Chemistry of Vibronic Coupling'' is as follows: in this paper ,part 1) we show, using a simple model of two harmonic potential wells, how to maximize the VCC k 4 in the ``space'' of force constants, normal coordinates and reduced oscillator masses. In forthcoming papers we will then look closer at diagonal and o-diagonal VCCs h in a space of ``chemical parameters'' such as the electroneg- ativity and covalent/ionic radii of the elements involved. In part 2 of the series [14] we perform quantum mechanical ,QM) calculations of the dynamic diagonal VCC for closed shell diatomics MeX ,Me alkali metal or H, X halogen or H). In part 3 [15] we perform QM computations of the dynamic o-diagonal VCC for mixed-valence and
intermediate valence Me2X systems ,Me alkali metal or H, X halogen or H). Finally, in part 4 [16] we will focus on both diagonal and o-diag- onal VCCs, looking for rules which might help to understand the in¯uence of Me and X position in the periodic table on the value of VCC. In part 5 of the series [17], we will try to understand VCCs as molecular systems are expanded into solids. Fig. 1. Correlation of Tc predicted by BCS theory with the experimental value of Tc. Numerical data for the points shown We will be looking to build a bridge between ,Al, Pb, K3C60,Li0:9Mo6O17,Ba0:6K0:4BiO3, LaSr0:15CuO4 and the VCCs and electron±phonon coupling con- YBCO superconductors) are taken from Ref. [8]. stants. In our simple considerations of the VCC in the context of superconductivity we thus hope to theory does not allow a quantitative prediction of come all the way from simple molecular systems to the onset and parameters of HTSC, there still ex- solids. ists a qualitative correlation between experimental values of the critical temperature ,Tc) and those calculated within the BCS theory [9] up to 100 K 2. Methods of calculations ,see Fig. 1). BCS theory deals with the full polaron coupling problem, including terms that are both 2.1. Types of vibronic coupling constants diagonal and o-diagonal in the local electronic populations. Since BCS predicts higher Tcs for There are many types of VCCs ,vibronic cou- larger VCCs, it is reasonable to look systematically pling constants) used. It is therefore necessary to for higher values of the VCC in order to obtain a de®ne accurately our notation and point of inter- 5 higher Tc value. est. More generally, we are looking for a place for chemical intuition in thinking about HTSC, 3 so as to provide a reasonable direction to the experi- 4 For de®nitions of k=eV and k=eV=A see Section 2 of this mental search for new materials. paper. 5 For vibronic coupling in molecules and in solids see Refs. [18±20]; on the role of vibronic coupling for electronic and vibrational dynamics see Refs. [21,22]; for charge transfer 3 For the work of four groups with a chemical and physical processes in condensed media see Refs. [23,24]; on Jahn±Teller orientation see Refs. [10±13]. eect in molecules and in solid state see Refs. [25±28]. W. Grochala et al. / Chemical Physics 265 22001) 153±163 155
Most often a VCC is expressed in force units, and denoted as h. There may be diagonal and o- diagonal VCCs, and linear and nonlinear ones. Among them we will look closer at the dynamic o- i diagonal linear VCC ,heg) de®ned as
i heg QihgjdH=dQijei typical units meV=A 1 where g and e are the diabatic electronic wave functions of two vibronically coupled electronic states, and dH=dQi is the derivative of Hamilto- nian along the normal coordinate Qi through which coupling occurs. Usually g is assumed to be i the ground state of a system. heg depends on ge- ometry of a molecule; in this paper we will be in- i terested only in a dependence of heg on one normal coordinate, Qi. i If g e, then heg becomes a diagonal linear CC i ,hee):
i hee hejdH=dQijei meV=A: 2
i i The hee may be thought as a negative force Fe acting on the potential energy surface ,PES) of the excited state e along normal coordinate Qi: i i hee ÀFe : 3
i If one determines hee at the equilibrium geometry of the ground state ,g ), then hi becomes a de- Fig. 2. Schematic PESs for the ground ,g) and excited ,e) states 0 ee i along normal coordinate Qi: ,a) case of hee 6 0; ,b) case of rivative of the g ! e excitation energy along Qi, i hee 0. evaluated at g0: i h dEexc=dQi at g0: 4 ee motion of atoms whose atomic orbitals do not i This de®nition of hee is close to chemical intuition; contribute to the molecular orbitals involved in the i Eq. ,4) says that hee is large when the energy of g ! e transition. An illustration might be an or- g ! e excitation changes substantially between ganic molecule consisting of a chromophoric part opposite phases of normal vibration i ,extension and a long side aliphatic chain. The vibrations of and contraction of the bond from its ground state the aliphatic chain, separated as they are from the equilibrium value). Fig. 2a illustrates de®nition of chromophore part of a molecule, obviously have i hee given in Eq. ,4). no impact on the g ! e excitation energy. i Eq. ,4) helps us to understand the cases when When is hee large? Consider the g ! e transition i i hee 0. A given vibration i has no eect on the arising from a HOMO/LUMO transition. hee will electronic excitation energy when the PES mini- be large for vibrations involving a given Ca±Cb mum of the excited state e is not displaced along bond stretch when HOMO and LUMO dier by a coordinate Qi in comparison with the PES mini- node across the Ca±Cb bond. It is very likely that a mum of the ground state g ,Fig. 2b). One way this large Ca±Cb bond length dierence between the g might be realized is when vibration i involves and e states ,and consequently a strong dependence 156 W. Grochala et al. / Chemical Physics 265 22001) 153±163
i i of the g ! e excitation energy on the Ca±Cb bond keg and kee ,which is de®ned in a manner similar i length) would be a consequence of such a nodal to keg) have energy units. These parameters are change ,node in HOMO across the Ca±Cb bond, central to classical superconductivity explanations, no node in LUMO or vice versa). In other words, because of the BCS theory [5] relationship: if during an electronic transition the bond order e À1 for a given bond changes much, then only those kTc 1:14hmDeb exp kNF 6 vibrations i which involve stretching of this bond i where k is the Boltzmann constant, Tc the critical will have large hee. Let us now return for a moment to o-diagonal superconducting temperature, mDeb the cuto fre- i quency of the phonon spectrum, k the electron± CC hge. Looking at its de®nition ,Eq. ,1)) one can 6 easily predict when hi vanishes. hi 0 when: phonon coupling constant in solid decreased ge ge e ,i) the molecular vibration i does not have the by coulombic repulsion of electrons, and NF the proper symmetry to couple states g and e or density of states at the Fermi level. The product k e ,ii) the g ! e transition is spin-forbidden. Of NF is often called the pairing potential, since it course, there is an additional simple intuitive rule controls the condensation of electrons into boson i i pairs. In other words, phonons open energy gaps for hge, similar to that given for hee: ,iii) among normal vibrational modes which at the Fermi level ,the magnitude of the gaps may i vary for dierent vibrations of the lattice). For us may couple states g and e, the largest values of hge should be found for modes i involving movement it is essential to note that a large electron±phonon of atoms whose atomic orbitals contribute sub- coupling constant k leads to a high supercon- stantially to MOs engaged in the g ! e transition. ducting transition temperature Tc. The present paper is devoted to ki , a counterpart of BCS k in For example, let the g ! e transition be a p±pà eg transition in a large organic molecule. Let us molecular systems. concentrate at a certain C ±C bond, such that The remaining kind of CC which is often used is a b a dimensionless one ,denoted most often as gi ) contributions of pz ,p) orbitals of Ca and Cb car- eg bon atoms to both g and e states are large. Let the and de®ned as contribution of pz orbital of Ca be of the same sign i i geg 1=hxikeg 1 7 in g and e, and let the contribution of pz orbital of Cb be of the opposite sign in g and e. Under such where hx is the energy of vibration i. De®nition of i i conditions a large coupling constant hge should be i gee is analogous to that given in Eq. ,7). While we expected for those normal modes with a signi®cant i i are not directly interested in the geg and gee, they contribution from a Ca±Cb stretch. are widely used in resonance Raman studies [29], As we will show in the following papers, inter- theory of CT complexes [30] and in the Marcus esting correlations may be found for hi and hi for ge ee theory of electron transfer [31] when analyzing a broad range of simple molecules. contributions from various vibrations to distortion The CC which is most important in consider- of a molecule upon electron transfer ,most often ations of superconductivity ,usually denoted k)is i gee is point of interest in studies based on so-called expressed in energy units and de®ned as crude Born±Oppenheimer approximation).
i i i So far we have introduced here the coupling keg kegv heghugjQijvei meV 5 i i i constants for a single normal mode i hee; heg; keg; i i i kee; geg; gee. These are linear CCs. There are cases, where ug and ve are vibrational wave functions for when CCs of higher order than linear should be normal mode i,ing and e states respectively. The considered. The simplest quadratic o-diagonal hugjQijvei integral is computed taking ug with vib- CC is de®ned as rational quantum number equal to zero and full set of ve ,vibrational quantum number varies from 0to1). Again, most often g is here the ground 6 Both diagonal and o-diagonal vibronic coupling constants and e is a certain excited electronic state. enter BCS considerations as mentioned above. W. Grochala et al. / Chemical Physics 265 22001) 153±163 157
ij 2 2 heg hgjd H=dQidQjjei meV=A ; 8 with its diagonal counterpart
ij 2 2 hee hejd H=dQidQjjei meV=A : 9 Now, i and j are two normal vibrational modes. In ij case i 6 j, hee describes vibronic coupling of e and g states through an i and j combination mode. ii When i j, hee describes vibronic coupling of e and g through iÕs ®rst overtone. Of course, qua- ij ij dratic keg and kee may be easily derived from both ij ij heg and hee, respectively. In the present and forth- coming papers [14±16] in our series ``Chemistry of vibronic coupling'' we will not consider quadratic and higher CCs. i As noted earlier, our main interest will be in heg, i i hee and keg ,especially when a g ! e transition has charge-transfer character). The central focus of i 7 this paper is keg. The literature in this ®eld is diverse, traversing chemistry and physics. There are many current nomenclatures in place for h, g and k.Wehave taken time here, with apologies to the experts in the ®eld, to delineate clearly the relationships be- tween the dierent measures of vibronic coupling. Let us present now the simple model used for Fig. 3. Schematic PESs for electronic states g and e for the i keg calculations. model used in this paper; ,a) c 6 0, ,b) c 0. See text for fur- ther details. i 2.2. Model for calculations of the keg
Imagine two electronic states g and e described reduced mass for vibration i. In our diabatic model by harmonic potential curves along coordinate Qi we assume, similarly to Marcus [33], that the force ,Fig. 3a). We introduce the variables k, DQ, c,and constants in states g and e are the same. 8 When m. k has physical meaning of a force constant for c 0 we have the case of a degenerate double- vibration i in both e and g states, DQ is the dis- minima potential well, realized for asymmetric placement between the minima of e and g states mixed-valence compounds ,see Fig. 3b). We want i along Qi, c is the vertical ,energy) displacement to ®nd keg according to Eq. ,5), varying k, DQ, c, between the minima of e and g states, and m is a and m as widely as possible. One immediately notices that parameter c does not change the value i i of keg ,neither the heg nor the hugjQijvei term) in our model ,although it contributes to the g ! e 7 We consider here a special case when the same vibration is Herzberg±Teller active and involved in a geometry change upon electronic excitation. In this context note an opposite case, when a Herzberg±Teller active mode does not form progression 8 This approximation is usually ful®lled with 10% accuracy, in the electronic spectrum, and the progression-forming fre- but deviations as big as Æ30% are known. We will discuss later quencies do not eect vibronic perturbations. See an interesting the qualitative impact of these deviations on the value of k, contribution on the phenanthrene 3400 A system in Ref. [32]. while now we will remain within the above assumption. 158 W. Grochala et al. / Chemical Physics 265 22001) 153±163 excitation energy). Thus, we will exclude parame- Table 1 ter c from the space of variables, reducing it to k, Values of h and k as a function of k and DQ for three dierent DQ and m. Phenomenologically, these three pa- masses: m 1, 10, 100 amu rameters fully describe ki in any molecular system k ,hartree/ DQ ,bohr) h ,hartree/ k ,hartree) eg bohr2) bohr) within a diabatic harmonic potential model. We have performed a scan of values of the VCC m 1 i 5 0.27 1.35 0.26 keg in the space of the above molecular parameters. 20 0.19 3.8 0.52 For this purpose a short program in the Python 50 0.15 7.5 0.82 programming language has been written. Our al- 100 0.13 13 1.16 i 300 0.1 30 2 gorithm computes independently the heg term and hu jQ jv i term. The hi term is simply [34] 500 0.09 45 2.59 g i e eg 700 0.08 56 3.06 i 1000 0.07 70 3.66 heg kDQ; 10 m 10 which requires no integration and is obviously m- 5 0.15 0.75 0.08 independent. 50 0.08 4 0.26 The hugjQijvei term is found by a numerical 300 0.055 16.5 0.63 integration procedure. 9 The algorithm used was 500 0.05 25 0.82 RombergÕs extension to the trapezoidal rule for 700 0.045 31.5 0.96 1000 0.04 40 1.16 better computational eciency. A vibrational wave function for a quantum oscillator is m 100 5 0.09 0.45 0.025 2 Wv QiNvHv yexp Ày =2 11 50 0.05 2.5 0.08 300 0.03 9 0.2 where 500 0.025 12.5 0.26 1000 0.02 20 0.36 v À1 1=2 1=2 Nv f 2 v! P g 12 is the normalization factor.
v 2 v 2 v Hv y À1 exp y d exp Ày =dy 13 smaller integration limits of Æ3 to 5 bohrs were used. Finally, the hugjQijvei term was multiplied by is HermiteÕs polynomial and i i heg to get keg. y mx 2P=h1=2Q; 14 Our calculations were performed in atomic i units of h, atomic mass units, hartrees and bohrs. In these units: k 1 hartree/bohr2 1556:7N/ x k=m1=2 15 m 15.567 mDyne/A. Numerical results are shown in Table 1. where xi=2P is the frequency of vibration i,andv is the vibrational quantum number.
Our algorithm computes the hugjQijvei term taking up to 15th order Hermite polynomials. We 3. Results and discussion have observed that due to the normalization factor Fig. 4 presents a plot of ki and its components: Nv ,strongly decreasing with v) the contribution eg i from higher orders than the 15th is negligible. The hugjQijvei and heg as a function of DQ for k 5 and integration limits were usually ,À7, 7) bohrs with m 1. This plot is central to our considerations. a 0.01 bohr grid, providing fast convergence. For Its main features can be easily deduced without large values of k and small values of DQ, huge calculations. Indeed, this is how we really came a i integration limits were not necessary. Hence, long way toward understanding of keg, before any computations were performed. It may be seen in Fig. 4 that there appears a 9 i Analytical formulas have been obtained in Ref. [35]. maximum in keg, at about 0.25 bohr. This maxi- W. Grochala et al. / Chemical Physics 265 22001) 153±163 159
i Fig. 4. Plots of and its components: hugjQijvei and heg as a i function of DQ for k 5 and m 1; heg has been divided by a factor of 10 to present these three plots on the same scale, al- i though each of them has obviously dierent units. Fig. 5. DQ; k points providing maximum kegs. Three dierent values of reduced oscillator mass, m,,m 1, 10, 100 amu) have been assumed. i mum originates from the opposing behavior of heg ,which, since hi kDQ, grows with increasing eg the position of DQ sisamonotonic and decreas- DQ) and hujQijvi ,which decreases with increasing opt DQ). The product of these two terms has a maxi- ing function of k for all three reduced masses. The mum along DQ. Very similar plots ,not shown necessity of DQ tuning for given value of ksin i order to maximize ki may be easily understood. here) may be obtained for keg and its components: eg i For very steep parabolas describing the PES of hugjQijvei and heg as a function of k for DQ, m constant. Again, for small values of k k 0, the electronic states g and e, one needs to provide overall ki is close to 0, due to the small hi term. small DQopt to have hugjQijvei large enough. eg eg On the other hand, it is necessary to separate For large values of k, the hugjQijvei term is now i the minima of shallow parabolas to obtain large very small, giving rise to a small keg. Again, there is i hi ,and hence attain large ki ). Interestingly, the some optimal value of k, yielding maximum keg. eg eg i position of ``optimal'' DQopt for a given k varies The existence of a maximum in keg in Fig. 4 is very interesting in the context of superconductivity. It with m. At a given k, the larger m, the smaller means that one needs to ®nd an optimal value of DQopt. Or, alternatively, at given DQ, the larger m, the smaller k . The hyperbolic-like behavior of DQ ,hereafter called DQopt) for a given k and m,or opt optimal ,i.e. those providing highest ki s) DQ; k an optimal value of k ,hereafter called kopt) for a eg given DQ and m, in order to attain a maximum points may be understood considering four simple i relationships: value of the coupling constant keg. Let us now plot the DQ; k points providing i i keg 0 for k 0 ordinate axis in Fig: 5; maximum keg for three dierent values of m ,m 1 amu, 10 amu, 100 amu). As it may be seen in Fig. 5, 16a 160 W. Grochala et al. / Chemical Physics 265 22001) 153±163
i value which cannot be reached by any molecular keg 0 for DQ 0 abscissa axis in Fig: 5; or extended system. It is two orders of magnitude 16b larger than the force constant for the nitrogen 2 molecule N2 ,3 hartree/bohr ). That extreme up- i keg 0 for k !1 and DQ !1; 16c per k limit was not realistic, but chosen only to i illustrate clearly the trend for keg which holds re- gardless of the k; DQ ± space probing. Let us ki should be a continuous function of k and DQ: eg proceed, limiting ourselves to a reasonable space 16d of ks in the range from 0 to 5 hartree/bohr2. For a typical chemical system with k 1 hartree/bohr2 Computational data suggest that with simulta- i and m 10 amu, keg will decrease to 0.037 hartree neous k and DQ increase, the hugjQijvei term goes ,about 1 eV) for ``optimal'' DQ. For ``nonoptimal'' i to zero faster than the h term goes to in®nity ,Eq. i eg DQ, values of k will be smaller than this. ,16c)). eg i There is one more question that begs to be an- Fig. 5 does not tell us about keg values at ``op- i i swered: can heg be a monotonic indicator of large timal'' k; DQ points. Consider such keg, plotted as i p kegs? Let us put the problem in other way: if k a function of k=m ,Fig. 6). increases and DQ simultaneously decreases ,k and As may be seen in Fig. 6, ki is a linear function p eg DQ are optimal for each other), what happens to of k=m for ``optimal'' k; DQ points ,of course, it i p the heg values? The answer to this question is given is not a linear function of k=m for any k; DQ in Fig. 7. points). The largest ki of 3.7 hartree ,within the eg As may be seen in Fig. 7, if optimal k; DQ are space of k, m and DQ monitored by us) is reached provided, the increase in hi s is connected with for k 1000 hartree/bohr2 15:6 Â 103 mDyne/ eg A, for m 1 amu and at DQopt ,about 0.04 bohr). Of course, 1000 hartree/bohr2 is a force constant
i i Fig. 7. Illustration of the keg vs heg monotonic relationship for Fig. 6. Plot of ki values at ``optimal'' k; DQ points as a three masses ,m 1, 10, 100 amu) at k; DQ points optimal for p eg i function of k=m. given heg. W. Grochala et al. / Chemical Physics 265 22001) 153±163 161 simultaneous monotonic ,although less than lin- in a major way, usually leading to symmetrization i ear) increase of keg. Let us summarize the results of a molecule along Qi. Given adiabaticity as one obtained. The perspectives for maximizing the more independent parameter, it is not easy to i coupling constant keg are as follows: guess and provide the ``optimum'' value of DQ for i ,i) there exists a maximal value of keg which may given molecule. be found for a given force constant k by providing These examples show how dicult it will actu- i the appropriate DQopt; DQopt decreases with k in- ally be to design large keg by playing with DQ. creasing. Our observations of the existence of optimal DQ i ,ii) keg may be maximized by increasing the k=m for given k have an interesting counterpart in ratio with simultaneous adjustment of DQ. studies of high-temperature superconductors. As i ,iii) keg may be maximized for given m by in- experimental Tc vs doping ,x) and Tc vs external i creasing heg, but k should be precisely adjusted to pressure ,p) curves show, there exists an optimal DQ. level of doping and an optimal external pressure applied which allow T maximization [37,38]. Ei- These rules may be translated into chemical c ther too small or too large values of x and p language as follows: strongly in¯uence bond distances and lead to dis- ,iv) large values of ki in mixed-valence molec- eg appearance of the HTSC phenomenon. On the ular systems should be sought for in systems with other hand, experimental data ,mainly for the strong multiple bonds ,large ks) built of light ele- well-studied cuprates [39,40]) indicate a strong ments ,small ms). The DQ should be precisely ad- correlation of HTSC with lattice instability and justed for a given k; m pair. The larger the phase transitions in these complex solid state sys- expected values of ki , the greater precision in DQ eg tems. tuning is necessary. Let us now brie¯y discuss the qualitative impact One can see two serious problems in applying of deviations from diabaticity and harmonicity on the qualitative understanding reached, even for ®nal value of ki . molecules. These are connected with the necessity eg of precise DQ adjustment for large k. Under- or 3.1. The eects of deviations from diabaticity and overestimation of DQ results in a strong decrease i i harmonicity on keg of keg. The problems are as follows: ,1) Many molecules with large k 10 prefer to be In the simple model used, we have assumed symmetric. Exceptions from a ``self-symmetriza- harmonicity of the vibrations in g and e states tion'' of the system for large k are only found along Q ,with force constants equal in both states, among molecules with large hi such as symmetric i eg k k ) and diabaticity ,PESs for states g and e linear F or F H radicals [16]. Such molecules are, e g 3 2 cross at Q 0, see Fig. 3b). We would like to however, extremely unstable toward decomposi- i discuss now brie¯y and in a qualitative way the tion [16]. impact of breakdown of these assumptions on the ,2) Another complication is hidden in the real value of ki . adiabatic ,and not diabatic, as previously as- eg ,i) What if k 6 k ? Since we decided to look for sumed) character of the g and e states, which mix, e g large ki s among systems with strong multiple creating two nondegenerate states: the real ground eg bonds ,providing large ks), we should not be and excited state of a system ,see next section). bothered by eventual breakdown of the k k According to the simplest theory using adiabatic e g assumption. To demonstrate this, consider a typ- potentials [36], strong adiabaticity in¯uences DQ ical mixed-valence ,MV) system of type fMn À AmÀ À M n1g. Given the same r skeleton in two n mÀ 10 ``units'' diering by one electron ,M À A and We mean here k, which has been used by us to describe mÀ n1 curvature of parabolas in a diabatic potential model, and not a A À M ), the dierences in stretching force constants for the above units originate exclusively real force constant for antisymmetric stretch of F3 or F2H. Obviously the latter is imaginary for both systems. from the partial p bonding. Since p bonds are 162 W. Grochala et al. / Chemical Physics 265 22001) 153±163
i keg may be also hidden in symmetric systems, contrary to our predictions based on a simplistic diabatic potential model.
4. Conclusions
Using a simple diabatic model of two equivalent electronic states with harmonic PESs, we analyze i values of the dynamic o-diagonal VCC ,keg)asa function of three parameters: force constant ,k), reduced oscillator mass ,m) and displacement be- tween two minima along the normal coordinate of Fig. 8. Impact of ,a) anharmonicity and ,b) increasing adi- a coupling mode Qi ,DQ). We show that for each k abaticity, on PES and vibrational wave functions in a system i with degenerate double-minima potential wells. there is only one DQ which maximizes keg. Exis- i tence of a maximum in the keg originates from the i opposing behavior of heg and hgjdH=dQijei factors much weaker than r bonds, stretching force con- in k; DQ space. i stants for the above units will not dier much. In searching for large kegs we are then led to Hence, our assumption of ke kg will be violated adjusting properly small DQs in systems with only slightly. strong multiple bonds ,large ks). It also emerges i p ,ii) What if vibrations are not harmonic? The that keg correlates linearly with k=m. Hence, eect of anharmonicity of vibrations on our model one should build system of light elements so to i is shown in Fig. 8a. maximize keg. However, these conditions are very It seems that anharmonicity has a double im- hard to realize experimentally, because real sys- i i pact on keg: through heg and also through hugjQijvei. tems with large ks either have a tendency ,a) to- i While anharmonicity increases heg ,by comparison ward symmetrization along Qi, or ,b) to chemical with harmonic case), it diminishes the hugjQijvei decomposition. DQ and k are not independent i term. The result of the interplay of heg and parameters, as well. hugjQijvei ,though the in¯uence of these two seems The results presented here may be useful in the to cancel to some extent) is dicult to deduce. experimental search for HTSC, in delimiting the And, in experiment, is dicult to control. More space of element combinations which should be exact studies ,using Morse potentials for example) investigated. Of course, superconductivity is a very might be performed in order to determine quan- complex solid state physical phenomenon. It can- i titatively the role of anharmonicity for keg. not be explained exclusively by a simple diabatic, ,iii) What if the system is adiabatic? The eect harmonic potential model such as that we used in of increasing adiabaticity is illustrated in Fig. 8b our simulations. This is why in parts 2, 3 and 4 of [41]. Among factors discussed so far, adiabaticity this series we will computationally examine dy- ,measured by electronic coupling parameter D namic linear VCCs in real QM molecular systems. [34,36]) undoubtedly has the strongest impact on i keg. Again, its role is not easily quanti®ed. Adia- i baticity in¯uences values of heg, DQ and hugjQijvei, Acknowledgements i i.e. keg and D are not independent parameters. In our opinion, the most signi®cant in¯uence of This research was conducted using the resources i increasing adiabaticity on keg is connected with of the Cornell Theory Center, which receives fund- the tendency for symmetrizing a molecule even for ing from Cornell University, New York State, the i large heg ,a mixed-valence system becomes an in- National Center for Research Resources at the termediate-valence system). This means that large National Institutes of Health, the National Science W. Grochala et al. / Chemical Physics 265 22001) 153±163 163
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