THE CROSSIKG O F POTENTIAL SURFACES’ E. TELLER Department of Physics, George Washington Uniaersity, Washington, D. C. Received October 29, 1936
The crossing of potential curves or surfaces is important if reaction mechanisms are discussed, particularly if activated states are involved. Thus in photochemistry both the chemical yield and the fluorescence will depend on the question whether the molecules can get from a higher potential surface to a lower one, transforming in this way electronic excitation energy into kinetic energy of atoms and finally into heat. We know that in both diatomic and polyatomic molecules this can happen in several ways through the change of position of an electron, as discussed by Franck and Haber (l),or through dissociation and re-composition of the molecules. It is our purpose to discuss a difference between diatomic and polyatomic molecules with respect to the question whether electronic energy levels as functions of the atomic configuration may cross. I t is well known that potential curves of diatomic molecules will not cross unless the two electronic states in question differ in some essential m y . In fact two curves will cross only if the electronic states differ either in their symmetry properties or in their multiplicity or finally in the “position” of an electron which belongs to one atom or ion for the first curve but belongs to another atom or ion (sufficiently far removed from the first atom) for the second curve. These facts were first recognized by Hund ( 2 ) , who supported his statement b y the following argument: If two potential curves cross, the electronic state must be degenerate at the point of crossing. We should not expect, however, that degeneracy can be established by varying only one parameter, which would be of course the case if by varying the nuclear distance in a diatomic molecule we should as a rule encounter crossing points. Yeumann and Wigner (4) have given the mathematical proof that Hund’s qualitative argument is correct. The essence of this proof is as follows: We suppose that we already know the correct electronic proper-functions for all energy levels except for the two potential curves the crossing of which we want to investigate. Using for these last two the two arbitrary functions and $2, which are chosen to Presented a t the Symposium on Molecular Structure, held a t Princeton University, Princeton, New Jersey, December 31, 1936 to January 2, 1937, under the auspices of the Division of Physical and Inorganic Chemistry of the American Chemical Society. 109
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E. TELLER
be orthogonal to each other and t o the other proper-functions, the matrix of the electronic energy will be diagonal except for the elements a12 = a21*. (The asterisk signifies the conjugate complex value; u12 is the matrix element l$lH+z*, where H is the operator of the electronic energy.) Now the diagonal elements corresponding to and t+h,i.e., all and u22 are in general not equal, and also u12 will be different from zero. Degeneracy on the other hand would mean that all = a22 and aI2 = 0. By varying only one parameter we shall not expect that both these conditions can be satisfied. From this reasoning it is evident that if more than one parameter can be varied degeneracy may occur. This will be the case for polyatomic molecules where the whole atomic configuration (and not only a single interatomic distance) can be varied. In general three parameters are necessary, since a12 may be a complex quantity and two parameters are needed to make both its imaginary and its real part equal to zero. The diagonal elements of the energy matrix are aln-ays real, and only one more parameter is necessary to get all = a 2 ~ . If in our problem all magnetic forces may be neglected, which is justified for molecules containing no heavy atoms, then the electronic properfunctions can be chosen real. Therefore all matrix elements will become real, and only one parameter will be needed to obtain a12 = 0. I n this case it will be sufficient to change two parameters in order to get a common point on the two energy surfaces which represent the dependence of the electronic energy on the atomic configuration. If, on the other hand, the electronic spin becomes important, as will be the case whenever heavier elements are present, a12 will be complex and three parameters are needed to obtain crossing. The crossing of energy surfaces n hich is thus shown to be possible is, however, of a different type from the simple crossing of tm-o potential curves in diatomic molecules. Let us first inyestigate the case of light atoms where crossing is brought about b y varying two parameters. It will then be sufficient to discuss the dependence of the surface from two parameters z and y. If we are only interested in the behavior of the surface in the immediate neighborhood of the crossing, then n-e may assume that all matrix elements will be linear functions of z and y. Furthermore we choose the coordinates and the electronic energy E in such a way that the crossing should take place a t z = y = 0 and E = 0. Finally we may choose the proper-functions in such manner that for y = 0 the energy matrix should be diagonal. The part of the energy matrix -11 which determines the crossing of the two surfaces will then be
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If al = a2 and PI = pz the sum of the energies of the two surfaces will remain the same as in the crossing point, i.e., it will remain zero. Since we are not interested in the slope of the surfaces but only in the way they cross, \ve may simplify the problem b y putting al = a2 = a and PI = pz = P. The energy surfaces nil1 now be determined by the secular equation
and we obtain for the electronic energy
E
This means that the energy surface will be a double cone. As an example for such an energy surface we may consider a degenerate electronic state of a molecule with trigonal symmetry (e.g., CH-1). It can be shown (3) that the degeneracy will be split by the non-symmetrical atomic displacements (for instance, the displacement of the carbon atoni perpendicular to the molecular axis). For the energy matrix and the splitting of the energy we shall obtain expressions similar to 1 and 2 . But, owing t o the symmetry of the problem, n e have p = 0 and a = h and the cone will become circular. Of course for such symmetrical molecules the crossing itself is caused b y symmetry, whereas in the general case the crossing conies about at any point in which the parameters of the problem happen to have suitable yalues. If magnetic forces cannot be neglected and three parameters (x,y, and z ) must be used, then with ,einiplifications similar to those introduced for the two-parameter case we obtain for the secular equation and energy values the exprecLslolls '
The internal motion of the molecule caii be represented by a point moving on our potential surface. Each position of the point n-ill correspond to a configuration of the atoms in the molecule. If, t o begin n-ith, the molecule is in a certain configuration and on a certain electronic energy level, and in the elid is in the same configuration but on another energy level, then the point representing the internal motion mu5t ha\ e croswd from oiic energy surface to another. \Ye i u x
+ by I
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We may therefore write
d2rn2(E
+ uz + Py) - d 2 m 2 ( E - uz - P y )
-
2(az
+ P y ) (14)
The factor
is the reciprocal value of the velocity v a t the point z = y = 0. If the velocity is sufficiently high, the factors
@(E
-
-
fly)
and
p+
+ PY)
(which are not exactly equal to the classical velocities on the double cone) may also be substituted by v. Thus expression 3 can be written in the form
These equations contain y only as a parameter and they may be solved for each y = const. separately. If furthermore lye introduce instead of z the time t
x
= 2!
we obtain
These, hon-ever, are the equations n hich Zener ha. solved, so that our procedure of taking his rebults and integrating ox cr the different orbits y = const. is justified. *\part from the types of c r o 4 i g dibcuskecl here n e may also expect for polyatoniic molecules the simpler type of crossing in n hich only one parameter must be ~ a r i e din order to arrive at a conimoii point of the tn-o surfaces. This nil1 again be the cace if the tn-o energy states differ in symmetry, multiplicity, or electronic position. However, the difference in syninietry propertiec of the electronic proper-functions is less important for polyatomic inolecules than for diatomic molecules, because in a diatomic molecule the cliange of interatomic distance does not destroy the molecular
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symmetry; for a polyatomic molecule, however, the whole symmetry is frequently destroyed b y the internal atomic motion. On the other hand we have for polyatomic molecules the special type of “conical crossing” discussed above. The transition probability for this type of crossing may become quite considezable. If we take for instance v = lo6 cm. sec.-* and a = b = 1 e.v. per Angstrom unit, we obtain from expression 6, P = 0.6 A.U. Of course the velocities may be for chemical reactions anything between lo4 and lo7 cm. sec.-l l l u c h will depend on the magnitude of a and 6. A large value for a and a small one for b, i.e., a cone which rises steeply in the direction of the incident waves but rises slowly perpendicularly to this direction will be favorable for the transition. Even if the transition probability is relatively small, the transition from one surface to another may take place after a few periods of molecular vibration if the potential surface is .cuch that the molecular configuration will keep returning into the neighborhood of the apex of the cone. This may be the case for many photochemical processes where the question is whether the molecule will go over through vibration into a lower energy surface before it radiates. Since when we are concerned with a transition from a higher to a lower potential surface the apex of the cone will frequently be the lon-est point of the upper surface and since the vibrational energy might easily be dispersed between the many internal degrees of freedom of a polyatomic molecule, the probable values for the parameters 5 and y will be close to zero. Xlthough in such cases the velocity may become too small and our method of calculation may therefore be inapplicable, still the orders of magnitude will be given by expressions 6 and 7. We shall then get a transition in a short time because though expressions 6 and 7 show that the transition probability decreases for small velocities this will be compensated b y the fact that for small kinetic energies the molecular configuration will return more frequently to the crossing point. REFERENCES
(1) FRBNCK, J., A X D HABER,F . : Berliner Akademieberichte, Band 13 (1931). (2) HUND,F.: Z . Physik 40,742 (1927). (3) J A H N , H., AND TELLER, E . : Proc. Roy. SOC.London, in press. (4) vox NECMAXN, I., ASD WIGNER,E.: Physik. Z. 30, 467 (1929). (5) ZENER,C.: Proc. Roy. Soc. London 137A, 696 (1932).
