In general, the two approaches are expected to yield similar results because the two frameworks are connected by an orthogonal unitary transformation. However, it is shown that the two frameworks cannot always be connected through an orthogonal transformation. Essentially there are two ways to derive this transformation.
It is important to emphasize that writing Eq. To continue along these lines, we refer the reader to Eq. One way to see it more explicitly is to convert Eq. This exponentiated line integral is used to solve Eq. In order to reveal the conditions for this situation to occur, we extend the integration in Eq. Next we review results derived in Chapter 1. This means that the electronic manifold is singlevalued and therefore implies that the exponentiated line integral as given in Eq. For the situation where the contour surrounds the abovementioned pathological points, the mathematical approach presented in Section 1.
However, it is conclusive according to the theory presented here. As can be seen from Eq. This possibility is guaranteed because the Curl equation [i. Before closing this section, I mention that the two-state case deserves special consideration because of its utmost importance. The two-state case is treated in detail in Section 3. We start by replacing, in Eq. The fact that these two transformations are carried out by the same matrix leads to several interesting consequences to be discussed next.
However, in Section 1. In what follows we derive the necessary condition for W to be singlevalued, and during this derivation that it becomes clear that, in fact, the singlevaluedness of the A matrix is not required. The proof19—20 is carried out for a region in configuration space for which the relevant electronic manifold forms a Hilbert subspace.
Short Summary We established that the necessary condition for the A matrix to yield a singlevalued diabatic potential as given in Eq. It is important to emphasize that for this to happen, the A matrix is not necessarily singlevalued because the B matrix, as was just proved, is not necessarily a unit matrix. We showed that the B matrix is identical to the D matrix, and therefore we refer, from now on, to both as the D matrix. We define this number as K and term it the topological number. In what follows we show that the two matrices are identical. Whereas the relation between the two electronic basis sets is presented explicitly in Eq.
In other words, the transformation from the adiabatic to the diabatic frameworks or vice versa discussed above does not affect the solution of this equation. Since A is the orthogonal transformation matrix that connects the two frameworks, it will be termed the adiabatic-to-diabatic transformation matrix and recognized by its acronym as the ADT matrix. In many studies the method employed to reach the diabatic framework follows the procedure as discussed in Section 2.
However, there are also numerous approaches that circumvent the use of the NACTs. Assuming smooth diabatic states, Macias and Riera44,45 apply such an approach to certain operators in the vicinity of the avoided crossing region assuming their corresponding diabatic presentation to be smooth as well. This approach was extended and generalized by Kryachko by introducing what he calls an equation of motion. This approach has so far been applied only for two-state systems.
Its extension to a multistate system does not seem to be obvious and so far has not been tried. A somewhat different approach was suggested by Rebentrost and others51,52 according to which the so-called ADT angle or also the mixing angle to be properly introduced in the next chapter is replaced by an angle that the open-shell orbital forms with the intermolecular axis.
In all cases reported so far the calculated ADT angles exhibit a reasonable functional form. Since this approach assumes a diabatic behavior at some parts in the region of interest, its application may encounter difficulties if applied within a region of numerous conical intersections see next chapter. A different approach is suggested by Sevryuk et al. Continuing as in Section 2. It is well known that such transformations do not affect the magnetic field, or, in other words that the magnetic field is invariant under this gauge transformation.
However, we do not consider magnetic fields but diabatic potentials, and therefore the above mentioned gauge invariance is observed with respect to the diabatic potential. Attaching a phase factor to the real eigenfunctions of an electronic Hamiltonian [see Eq. This transformation is shown not to affect the final diabatic potential [see Eq. To introduce the time dependence, the eigenfunctions are multiplied by time-dependent phase factors [see Eqs. The extension as in Eq. Continuing the derivation as in Section 2. The only change caused by the fact that the phases are now time-dependent in contrast to those described in Section 2.
The rest of the derivation is similar to the one given in the previous section. Moreover, it can be shown that here, too, the diabatic potential matrix is invariant with respect to the gauge transformation presented in Eqs. We would like to refer to one special type of phase, namely, any phase that is proportional to time t: As is noted from Eq.
It is noted that, because of the timedependent phases, the similarity between the two types of gauge transformations is enhanced. The diabatization process of Eq. However if N is the size of the relevant Hilbert subspace viz. As is well known, these functions fulfill the time-dependent eigenvalue equation: A similar expression is encountered in the simplified version of the present treatment when we attached time-dependent phase factors to the electronic eigenfunctions [see discussion in Section 2.
In addition, analyticity requires fulfillment of the following two conditions: The results of two consecutive differentiations of the A spatial coordinates, p and q, should not depend on the order of differentiation. This requirement can be shown to lead to the extended Curl equation similar to the one discussed in Chapter 1 [see Eqs. The results of two consecutive differentiations of the A to time and the other with respect to any spatial coordinate p, should not depend on the order of differentiation.
The corresponding four-component non-Abelian Curl equation, which guarantees the existence of an analytic solution for Eq. Consequently We differs from We.
We termed the first, the simplified version, the perurbative approach and the second, more general one, the nonperurbative approach. In both treatments we end up with a unitary ADT matrix reminiscent of the one encountered in the time-independent framework calculated by solving a vectorial first-order differential equation along contours. The main difference between the two approaches is as follows. Within the first approach we employ a time-independent electronic basis set so that the ADT matrix is time-independent and the resulting diabatic PES matrix is similar to the one encountered for a time-independent interaction.
Within the second approach we apply a time-dependent electronic basis set so that we end up with a time-dependent ADT matrix that leads to a somewhat more complicated diabatic potential energy matrix. However, this version may have its disadvantages, due to two contradictory requirements: It is true that within this process one encounters rectangular as opposed to square matrices, but the theory, as presented here, overcomes this obstacle in a reliable, coherent, and consistent way.
Similar expressions were derived in Section 2. Another interesting result is the way the Curl condition extends in case of a timedependent Hamiltonian [see Eq. We note that the extension is of the kind expected from a relativistic theory, although relativistic arguments are not explicitly mentioned in this context. As a final issue in this section, we mention that the ability to treat time-dependent Born—Oppenheimer systems is crucial because of the intensive studies of shaped and specially designed laser pulses to control molecular processes.
Next we check to what extent Eq. To A, conclude the proof, we consider Eq. For this sake we evaluate the first expression of Eq. This expression is used to replace, in Eq. A , A 62, A 23, B 11, L Werner and W, Meyer, J. Rebentrost, in Theoretical Chemistry: Advances and Perspectives, D. Jackson, Classical Electrodynamics, 2nd ed. Popescu, and D, Rohrlich, Nucl. B , Debreceniensis Series Physica et Chimica 34—35, A 99, A 61, Our task is to derive the corresponding ADT matrix A, and this is done in two ways: We employ the general solution of the differential equation as given in Eq.
For this purpose we have to derive the matrix G that diagonalizes the g matrix given in 3. Recalling that in the present case the matrix G is a constant matrix, the expression in Eq. In case n is an even number, Eq. In this case the quantization condition is more reminiscent of the spin of the electron this subject is discussed further in Section 5. In case the value of n [in Eq. The two possible outcomes for the D matrix imply that in general we encounter two kinds of contours.
In case the D matrix is the unit matrix, the situation is not clear because the eigenfunctions may or may not be singlevalued in such a case. In case there are no problematic points in the region surrounded by the contour, the eigenfunctions are singlevalued in that particular region. However, in case there is an even number of such points, the eigenfunctions are multivalued although n is even. The discussion related to these issues is extended in Sections 5. Employing the A matrix in Eq.
It is interesting to mention that the resulting D matrix is always a unit matrix that yields information regarding the way the three states are coupled. This type of coupling is discussed in Section 5. We do not present the corresponding topological D matrix as it is identical to the A matrix given in Eq. Next are discussed the conditions for which the relevant D matrix becomes diagonal. The difference between the case where n and are integers and the case where both are half-integers is as follows. These two situations are addressed later in Section 5. It is interesting to note that this is the first time in the present framework that quantization is formed by two quantum numbers.
This case is reminiscent of the two quantum numbers that characterize, for instance, the hydrogen atom. These two possibilities yield information regarding the way the four states are coupled with each other as is discussed in Section 5. The general theory demands that the matrix D as presented in Eq. Indeed, for the two-state case k was found to be either odd or even, for the three-state case it was found to be only even, and for the four-state case it was found, again, to be either odd or even.
Assuming that this pattern continues viz. The reason is that an odd antisymmetric matrix has, at least, one eigenvalue that is zero. Since all the diagonal terms are expected to have identical signs, this implies that the D matrix has to be the unit matrix and therefore k, in Eq. These two facts, based on our findings in Sections 3. Since A s is an orthogonal matrix, it can be written as in Eq. Since the two angles differ at most by an additive constant which is of a minor physical significance , we do not distinguish anymore between the two angles.
Both are termed ADT angles. A numerical example for a calculation of the n value in Eq. Since this equality is fulfilled for all potentials v x,y and w x,y that are analytic functions of x and y, the Curl condition in Eq. It is simple enough to be treated analytically and sufficiently general to have physical relevance. Since u 1 and u 2 are the adiabatic potential energy surfaces, it can be easily seen that each potential has a cone shape—one cone inverted with respect to the other—and have one common point at the origin see Fig.
The ci points were also mentioned in Section 1. It is noted that the curves have a double-hump structure. The humps are of equal size and increase as b increases. The b values are listed on the r. For this purpose we employ Eq. No other gridpoints are assumed. To continue we change the variable of integration in each integral in Eq. It is important to comment that for a given series of p—z points two or more successive points may be of the same kind, namely, either p— p or z—z points. Two such successive points produce an integral [of the kind given in Eq.
The final result for the integral in Eq. However, the quantization condition [see Eq. It is guaranteed to be zero if between any two consecutive z points we have an even number of p points. In case of a single ci m has to be at least 2, which implies that in order for the diabatic potentials in Eq. Choose a point in the region of interest that is the center for the circular contours. Count the remaining zp pairs to determine the number m.
Assuming that m is the number of zp pairs in this sequence, then the value of the line integral in Eq. It is important to note two facts: It may imply that it contains an even number of cis where half of them are positive cis and the other half are negative cis so that that their net contribution is zero. The number m may vary from one closed contour to another but not necessarily uniformly e. As an example, we consider again the elliptic Jahn—Teller model discussed in Section 3. Consequently the relevant contour surrounds one ci a similar situation is encountered for the regular Jahn—Teller model.
An exercise for studying a model with nine cis is presented at the end of this chapter see Section 3. Before closing our discussion of this subject, we mention two earlier studies on this issue: Next the Euler angles are employed for deriving the outcome due to a general rotation of a system of coordinates. In the following list we describe explicitly our two-state case see Section 3.
For the two-state case i. For the three-state case i.
We emphasize that n in Eq. Substituting the values of these parameters in Eq. For the four-state case i. It is important to emphasize that Eqs. The main issue to realize is that the original y component of the total angular momentum operator J is a special case of the quantized g matrix. Solution as Having the potentials in Eq. As for the zp points, the intersection between a circle and any horizontal x line forms a z point, and the intersection between a circle and any vertical y line forms a p point. Next, we consider, the ci distribution in regions surrounded by three circles: This fact implies that in the region surrounded by the inner circle is located one ci or, eventually, an odd number of cis points.
In fact, we see that we encounter one ci only. These intersection points form the following sequence of zp points: As a result line integral in Eq. Along the inner circle are located two z points and two p points and along the intermediate and the outer circles are located six z points and six p points.
Note that the internal distribution of the z and p points in both circles is different. In fact, we encounter five cis; four of them have identical signs, and one of them the central one possesses an opposite sign. Since each pair of the same kind i. The interpretation of this outcome is that in this enlarged region we find five cis with one sign and four with an opposite sign. Wigner, Gruppentheorie, Friedrich Vieweg, Braunschweig, Here we concentrate on their spatial distribution as obtained from ab initio treatments. Another concept that is frequently mentioned in this chapter is the conical intersection ci.
Although cis or points of cis were introduced in Section 3. Still, the fact that cis and NACTs are closely connected makes it impossible to refer to the one without mentioning the other. At this stage we emphasize that ci points are points at which two adiabatic electronic states become degenerate, and they are considered as the sources for forming NACTs. Therefore while studying the NACTs we concentrate on regions surrounding the various ci points. This concept is introduced in Section 2. Another aspect of the theory, of similar importance, is the approximate fulfillment of the Curl equation see Section 1.
However, this issue is discussed, later, in Chapter 6. It is important to realize that there exists only one plane that contains these three atoms Beyond Born—Oppenheimer: Still, for our purposes it suffices to consider only the body-fixed system of coordinates, where a point in configuration space is described in terms of three coordinates.
However, in order to simplify the search for the positions of the cis, we break up the three-dimensional configuration space and present it as a series of two-dimensional spaces that are chosen to be a series of parallel planes. In what follows we distinguish between the various planes as follows. It is important to mention that in order to obtain the NACTs for the three-dimensional configuration space the values of RBC are varied in some order and in this way to reveal all the cis of the given molecular system.
Doing it this way, it can be shown that the cis on the various planes are connected by continuous finite or infinite lines known by the term seams. So far we mentioned triatom systems only. In fact, we also present results for a tetraatomic system. There the search is done by fixing the positions of three atoms and leaving, again, one atom to probe the NACTs.
Since we limit our discussion to configurations where all four atoms form a plane, the free, test atom is assumed to move on that same plane as well. This system is discussed further in Section 4. It is important to mention that these studies are limited to an infinitely small i. The contribution of the present author and his collaborators to this issue is in extending the quantization concept to substantial regions of configuration space and this first by discussing it theoretically see Section 2.
In this way the quantization is an inherent feature of Born—Oppenheimer systems and applies to a group of N states that, in a given region of configuration space, forms a Hilbert subspace as introduced in Section 1. Here we briefly mention that the Renner—Teller model, as it is now termed,2,10 will not be discussed any further as this subject is beyond the scope of the present book.
Some confusion is created when quantization is connected with diabatization—a process required if quantum-mechanical nonadiabatic calculations are to be carried out see Chapter 2. It was established in Section 2. Thus quantization also, has a very important practical aspect, and it is this aspect that is usually ignored. Therefore part of the numerical study is devoted also to this subject.
The numerical study to be presented next emphasizes the two aspects of this issue; On one hand we show, numerically, that quantization of the two-state systems can take place at relatively large regions of configuration space and therefore for beyond the region covered, e.
The theory is based on calculus performed along contours in configuration space as discussed in Chapter 1. However, in so doing we reveal that this spatial distribution is affected by two other cis that couple the second and the third states and are located not too far from the just-mentioned D3h ci1 see Fig.
We note that these cis are located on the two sides of the symmetry line that connects the center of mass of the two fixed hydrogens and the 1,2 ci point. The distance between these two cis is 0. Thus, in order to properly account, for the Rydberg states, two diffuse functions—one s function and one p function—were added to the basis set, in an even-tempered manner,20 with the exponents of 0. We used the active space including all three electrons distributed on nine orbitals.
Additional information is supplied in Table 4. The main features to be noticed in Figure 4. However, once q exceeds this value, 0. In other words, we encounter strong disturbances, most probably, due to the next third state that acts via one or several 2,3 cis. However, it seems to be damaged by the third state once the region becomes large enough to include also the twin 2,3 cis. This situation is discussed further in Section 4.
We used the active space including all eight valence electrons distributed on 10 orbitals. Four different electronic states, including the two studied states, were computed by the abovementioned state-average CASSCF method with equal weights. To be more specific, we fixed the distance ROH between the oxygen and the central hydrogen i. A schematic representation of the configurations is given above each panel. The results of this study are presented in Fig. We note that the figure is arranged in columns—each column for one situation, specified by the abovementioned distance, ROH and a radius q of the a specific circle [RHX , the distance between H1 and the position of the 1,2 ci, is fixed to be at 1.
As for the shape of the curves, we note that they can be considered to be of the elliptic Jahn—Teller type see Section 3. Our interpretation for this phenomenon is that inside the region surrounded by this large circle the system does not form any 2,3 cis so that the two lower states can form a quasiisolated two-state Hilbert subspace see Section 2. In this respect we mention that our preliminary results for acetylene are presented in Refs. Following convergence tests we included in the calculations, in addition to the four studied states i.
In order to reveal the cis of this system, we restrict our study to a plane that contains all four atoms in this way we have to consider five internal coordinates. Next, fixing the CC distance and the position of the l. We present three fixed atoms, namely, one hydrogen on the l. As is usually the case, this free-moving hydrogen is used as a test particle to locate cis and to examine their spatial intensities. We encounter various shapes for the different NACTs, but the dominant shape is the double-hump shape.
A double hump is, as mentioned earlier, typical for the elliptic Jahn—Teller model see Section 3. In particular, we emphasize the encouraging results for the 1,2 NACT for which the double-hump structure survives even for relatively large q values. The results for the 1,2 NACT, as calculated along circles centered at the 1,2 ci, are presented in panels a — c ; the results for the 2,3 NACT as calculated along circles centered at the 2,3 ci, are presented in panels d — f ; the results for the 3,4 a NACT as calculated along circles centered at the r.
Also noted are the positions of the four ci points. The figure emphasizes the existence of the four cis and shows how close they are located to each other. The structure of the potential energy surfaces PESs and the corresponding ci points as presented in Figure 4. For this purpose, in Figure 4. Also, note the different scales used in the two types of figure panels viz.
Two additional observations are to be made: These small values indicate that, most likely, no 3,4 cis exist in the regions given in Figure 4. D3h ci was revealed by employing Eq. The results of this calculation are presented in Table 4. Returning now to the more general cases in Table 4. The situation improves significantly once a third state is added. Adding the two upper states i. The main reason for this breakup is, as discussed earlier, the missing 3,4 cis in the region of interest. In fact, the quality of the four diagonal D-matrix elements deteriorated because the fourth state is strongly coupled with the fifth state [recall the existence of the 4,5 ci], and this coupling is ignored, thus affecting the conditions to achieve the quantization.
This is clearly evident while inspecting the five diagonal elements of the D matrix for each region as presented in the last column. This addition improves not only the four-state quantization but even the three-state quantization—in particular the quantization for the larger regions cf. Doing this shift enabled the increase of the relevant circular region without getting too close to the fixed hydrogens axis. This implies that the slow A , deterioration of the nice quantization along the first three regions as q increases is not necessarily connected to the size of the region but can sometimes be attributed to the effect damage caused by the presence of the fixed atoms close to this region, in this case the two hydrogens.
The 1,2 a b X23 0. We used the active space, including all nine valence electrons distributed on nine orbitals full-valence active space. CC axis and at a distance of 0. It is important to mention that in order to solve for A and D, we employ the procedure described in Section 1. The integration along the contour is started with an A matrix that is the unit matrix. More details are given in Ref. The following is to be noted: The first three columns in the Table 4. We pay attention to two features: At this stage we are not yet fully able to discuss the difference between the two situations.
This is done in Section 5. In the fifth and the sixth columns of Table 4. Acta 52, Knowles, with contributions from J. Ingold, Nature London , Acta 34, 1 Acta 86, 97 It turns out that the Hellmann—Feynman theorem can be extended to a situation that yields a closed formula for the NACTs. This extension is attributed to Epstein,3 as was recommended by Singh and Singh in Proof To prove the theorem, we consider, following Eq.
To complete the derivation, we do the following: We recall that He is an operator that also acts on the l. In what follows we briefly elaborate on the meaning of Eq. It can be seen that in order to guarantee the quantization as presented in Eq. So far we have discussed a general case where j and k refer to any two states. In fact, not any, arbitrary, pair of states forms a degeneracy point.
For this purpose we have the following lemma. Proof To prove this lemma, we consider the following two nonadjacent states: This situation leads to a formation of a single three-state degeneracy. In Chapter 1 we discussed in some detail the extended Curl equation and referred to the possibility that the component H of F [see Eq. Since the NACTs are singular at the degeneracy points, their derivatives cannot be formed and therefore H and also F becomes undefined at these degeneracy points. In what follows we assume that the opposite is also valid; specifically, at the pathological points where H cannot be formed, we attribute it to a singularity of one or several NACTs.
Again, we emphasize that these singularities are all assumed to be simple poles. In what follows we refer to these points as ci points, or simply cis although the potentials may not be conical as, e. However, our studies, discussed in this book, are carried out on planes. As is shown later, this single atom is used as a test particle to expose the cis on the particular plane. Having defined the plane, which contains the cis, it is important to emphasize that the theory developed here does not apply for the whole plane but for a given region in this plane, usually defined by a closed contour.
Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections. Michael Baer. ISBN: Apr pages. Beyond Born–Oppenheimer: Conical Intersections and Electronic Nonadiabatic Coupling Terms. Author(s). Michael Baer. First published
We also discussed to some extent the conditions for that to happen; however, this discussion was limited, and it is our intention to extend it in the present section. As is presented next, the magnitudes to play a role in determining the extent a Hilbert space breaks up into Hilbert subspaces are the NACTs. The basic assumption is as follows. We consider a region in configuration space and a given distribution of cis.
Assuming that the Hilbert space contains N states and breaks up into L subspaces where the Pth subspace contains N P states so that N is given in the form: Each two adjacent states form at least one ci. The dotted lines were drawn to differentiate the subspaces.
The question to be asked is whether these two requirements are compatible with the conditions given in Eq. In other words, is it enough to require that the upper and lower states of a group of states do not form any cis with their external neighbors to guarantee the breakup as presented in Eqs.
The answer to this question can be found only in Section 5. They found that these functions, when surrounding a point of degeneracy, may acquire a phase that leads to a flip of sign of these functions. In particular, Herzberg and Longuet-Higgins2 demonstrated this feature with respect to eigenfunctions of the Jahn—Teller model. This interesting observation implies that if a molecular system possesses a ci at a point in configuration space, the two relevant electronic eigenfunctions may become multivalued.
In the discussion that follows Eq. In this sense the findings due to Longuet-Higgins et al. In case of several cis, this conclusion is generalized as will be done next. The line integral in Eq. The value of the line integral is not known at this stage because of the two possible signs , but it is obvious that the value of n, in Eq. In other words, the eigenfunctions flip their sign when this particular contour surrounds an odd number of cis but no sign-flips occur in case of an even number of cis.
Returning to Section 3. Short Summary We showed that sign flips of the eigenfunctions and the signs along the diagonal of the D matrix are closely connected in view of Eq. It is known from numerous calculations that for a sufficiently small region surrounding a ci the two states that produce this ci form a two-state Hilbert subspace5,6 see Chapter 4. This also applies in the present case for the two lower states, namely, states 1 and 2 and their 1,2 ci and for the two upper states, namely, states 2 and 3 and their 2,3 ci.
For this reason we employ Eq. Consequently the appropriate D matrix takes the following form: To prove our statement, we consider the expression as derived in Section 1. Consequently, the integral in Eq. Therefore we intend to obtain the D matrix in a different way. For this purpose we introduce two constant matrices G12 and G23 that enable the presentation of Eq.
Summary We know from numerical calculations that tracing any contour that surrounds a single ci causes the sign flip of the two states that form the ci. In the present section we proved, analytically, that surrounding two cis as given in Lemma 5. Thus the sign related to the intermediate state is left unaffected. The conclusion of this study is that tracing any closed contour within the three-state Hilbert subspace yields at most two sign flips however if the contour does not surround any ci, no sign flip takes place.
This finding can be extended, in a straightforward way, to Hilbert subspaces of arbitrary dimensions, as will be done in the next section. The geometric approach is based on the analytical derivation in Section 5. Two adjacent states may form more than one ci, but here, to simplify the discussion, we assume one ci for any two adjacent states. The extension is done at later stages. A contour that surrounds n consecutive cis i. For instance, 1,4 surrounds C1 and C3 but not C2 see Fig. These notations can be extended, in case of larger subspaces, to include other possibilities.
Having these notations, we return to describe the various possible sign flips. C1 and the C2 see Fig. The curves are presented as a function of q.
The conclusion for three-state systems is that we can have either no sign flip when the does not surround any ci or three cases where two different functions flip signs. We have two such situations: Then we have the contour 14 that surrounds all three cis see Fig. Finally, we have the case where the contour 1,4 surrounds C1 and C3 but does not surround C2 see Fig.
In this case all four functions flip signs none of the four functions get their sign flipped twice. As an example, we present this particular situation in terms of a product of two matrices. Extending the situation in Eq. Based on Figure 5. We revealed six different types of contours that lead to the sign flip of two functions and one kind that leads to the sign flip of all four functions.
We present two examples: For the contour that surrounds one 2,3 ci and one 3,4 ci see Fig. For the contour that surrounds one 2,3 ci and two 3,4 cis see Fig. There is some confusion regarding the indices because the analysis is done with respect to three excited state, ignoring the ground state usually assigned as state 1. The change in K is either 2 [when the two functions flip their original signs, e.
Thus any change in K caused by including or excluding cis is accompanied by a flip of an even number of signs only. Next, since the smallest value of K is zero and never 1 , K may attain even integers only. As is noted K determines the number of possible sign flips along a given contour. Information regarding the number of different groups of functions with identical number of flipped signs is still missing. Short Summary Extending Lemma 5. Here we would like to refer to cases where at a given point three or more states form one degeneracy point.
In fact we may even expect higher multistate degeneracy when several more than two seams cross each other all at the same point. The issue is how to incorporate this situation into the already existing theoretical framework. In what follows we assume that each two adjacent states form only one ci the extension to several cis is straightforward. We start by analyzing a three-state degeneracy point and consider the following situation: Extending this situation to a degeneracy point formed by N states does not change the final result, namely, only two functions flip signs: This general result is at odds with some more recent models.
First, it contradicts the conclusions given in Sections 3. Then it also contradicts a model discussed by Child and Manolopoulos. We explain these outcomes as follows. The Three-State System To explain this case, we assume the existence of four cis: We showed that in case the contour surrounds an even number of cis, no sign flip takes place. Here we consider the extension of this situation for a three-state system where each two adjacent states form a pair of cis and the contour surrounds all four cis. To determine what the signs of the D matrix diagonal elements are in such a situation, we apply a product similar to Eq.
In case of a single three-state degeneracy point, since any contour surrounds all four cis, we always encounter the same series. Therefore, enforcing the four-state degeneracy causes the shift of all six cis to the same point so that any contour surrounds the six cis.
The second case where all four functions flip their signs is somewhat more complicated. Here states 1,2 and 3,4 each form one ci, but the two intermediate states, 2,3 , form two cis. Therefore the four corresponding matrices to be employed are D12 and D34 given in Eqs. A different model was studied by Child and Manolopoulos. These eigenfunctions were then followed along closed contours formed by fixed q values to determine their signs at the end of the contour.
In other words, the model produces more than the minimal number of two cis one ci for each adjacent pair of states. This conclusion is also supported by an analysis carried out by Pistolesi and Manini. Thus in both cases more than just the minimal number of three cis are involved again, one ci for each adjacent pair of states.
In this respect we also mention a model applied by Varandas et al. In other words, regardless of the value of N , only two functions flip their signs, namely, those relate to the lowest and the highest states. This magnitude, like any other classical angular momentum, is a vector. Next, pricisely as in the case of the ordinary angular momentum, these variables—for a given spin—are allowed to have a limited number of discrete values. Usually these values are added as a subscript to the wavefunction that describes the state of the particle.
This implies that in fact a wavefunction of a particle with a nonzero spin is not a single function but a set of functions that are characterized by their spin subscripts. As was discussed, the number n in this equation has to be an integer but not necessarily an even integer as required by the spatial Bohr— Sommerfeld quantization law2 and therefore can also be an odd integer.
In case the two states are coupled by several cis, the value for the total spin for a region surrounded by a given closed contour is simply the algebraic sum of the individual spin values in the same way as is done in case of the ordinary electronic spins. Thus in this case the meaning of the ordinary spin is conserved.
See also a more detailed discussion in Section 5. As we have shown, the magnitude that most characterizes a Hilbert subspace is its cis. In the present section we are interested mainly in the topological features produced by the cis. For instance, a two-state system may have several cis but it is immaterial how many cis it possesses, the topological number K is either 2 or 0 see discussion in Section 5. This difficulty, as we discussed in previous sections of this Chapter , is resolved by the topological D matrix.
In order to introduce the topological spin for an N -state system, we need to introduce some order regarding the various numbers that define the N -state system in a given region.
We start by defining Nc , as the smallest possible number of cis, which an N -state Hilbert subspace is capable of forming. It is important to emphasize that in general, one may encounter more than one ci between a pair of states3—9 so that the number of cis is usually larger than Nc. On the other hand, K does not depend on how many cis exist in the N -state system, but it is constrained in two ways: Thus, each Hilbert subspace is now characterized by a spin quantum number S, which is related to the number of states that form the Hilbert subspace and the topological effects that take place within the assigned region in configuration space, namely, how many possible or different sign flips may occur while following all possible contours in this assigned region.
The reason for not having a unique relation with Nc is the fact that in every sign flip two functions are involved regardless of whether the number of cis, Nc , is even or odd. In this situation a magnetic dipole interaction should theoretically be formed because of the existence of the topological spin as, for instance, in case of an ordinary electronic spin. It is obvious that the molecular system, in a field-free situation, is not affected by the sign flip of one function or another as the diabatic potential is a well-defined magnitude of the subspace as a whole.
However, exposing the molecular system to a cyclic, electromagnetic field causes parts of the molecule to move, one with respect to the other, along various closed contours. This motion may cause some of the eigenfunctions to flip their sign. For a given molecule, the number of flipped signs is determined by the contours enforced by the electromagnetic field. This situation causes the creation of the topological spin, which in turn interacts with the magnetic field.
In other word the issue is not how many different sign flips may be formed along a particular contour but rather the probability of producing a certain number of sign flips in a given molecule or subspace , and this probability is related to the number of possible bundles of contours to form this number of sign flips. So let us summarize as follows. The multiplicity of the D matrix due to an electromagnetic field depends on what happens along a given contour viz.
As examples, we consider the cases of four and five states where each two adjacent states are coupled by a single ci. The matrix has to be orthogonal at any point in configuration space see Section 1. It is important to realize that the first condition is due to the fact that A is a solution of Eq. In what follows this matrix is labeled A N.
Before describing the general situation, we discuss two cases: Once we find that A 2 and A 3 fulfill the abovementioned, conditions we continue to the general case. Next we consider the D matrix, which is presented in Eq. It is important to note that deriving the matrix A 3 in this way guarantees that the matrix is orthogonal i.
Since the diagonality of the D 3 matrix is guaranteed by the quantization condition see Sections 2. To obtain the three angles, Eq. Because of the product in Eq. Case 1 can occur if the contour surrounds an even number of conical intersections for each pair of adjacent states or does not surround any conical intersection , whereas case 2 applies for all other situations. It is straightforward to see that according to this model, K is either 2 or 0 as, indeed, is required.
Moreover, we note that for the results in Figure 5. The diagonal elements of the D matrix calculated solving the Euler angles are identical to the one listed in Table 4. The matrix A N as presented in Eq. In order to obtain the matrix D N , one replaces, in Eq. The only nonzero elements, if these requirements are fulfilled, are the diagonal ones, which are made up of products of cosine functions [see also Eq. Our next assignment is to prove the following lemma. This implies that, in general, sign flips can occur only to an even number of terms of the D N matrix.
Short Summary At the beginning of this section we listed three requirements for the extended Euler matrix to be a model matrix for the ADT matrix. In this section we proved that these Euler matrices fulfill these requirements and therefore can be considered as suitable for presentation of a general ADT matrix.
However, a consistent convergence study of this kind could not be done for ab initio systems because numerical instabilities are enhanced as N increases. In this section we present such a study, but employing eigenfunctions derived from the appropriate Mathieu equation. Because of its simplicity, we face no instabilities, and the wall time it takes to produce a sizable electronic manifold is relatively short.
This choice of the interaction term has several numerical advantages, discussed below. To simplify the forthcoming treatment somewhat, we introduce a new parameter, x, defined as follows: With all these changes, Eq. For our purposes and in the notation of Ref. This feature may affect the rate of convergence for points on the real axis.
For this reason, the convergence in each case has to be treated with care. In this respect it is important to mention that we had to include terms in the series in Eq. In order to obtain the D matrix for arbitrary x values, we solve Eq. Short Summary We showed here that the more extended is the region in configuration space expressed in terms of x , the larger is the required size N of a group of states in order to become a Hilbert subspace. Our study is not typical for Born—Oppenheimer states because the various ci points for this system are located at the same point, but nevertheless this study illuminates some aspects of the relation between x and N.
Hellmann, Einfuhrung in die Quantenchemie, Deuticke, Leipzig, III, Krieger, Malabar, B 37, A , 1 Ebert, and K Ruedenberg, J. London A, , 45 A 62 , Acta, 75, 33 On the basis of this feature, we present a new approach to treat the NACTs, namely, to consider their spatial distribution as fields where at least some of them are produced by source points reminiscent of electromagnetic fields formed by charged particles electrons, protons, etc.
It will be shown that the source points in our case are the ci points—the points at which certain NACTs become singular see Section 5. Since cis arrange themselves along semi infinite long contour lines—seams2—4 —and since electric charges do not exist within the Born—Oppenheimer framework, formation of an analogy with the magnetic component of the electromagnetic field is suggested.
However, from experiments it is known that solenoids produce magnetic fields only inside the solenoid whereas outside the solenoid the field is zero. Having formed this similarity, we are Beyond Born—Oppenheimer: Two seams are shown: These assumptions were justified by detailed numerical treatments as discussed throughout the present chapter. For this purpose we first summarize our knowledge regarding the solenoid and its field. Thus, if for a given value of z the x—y plane is chosen in such a way that the axis of the solenoid is perpendicular to it and therefore is along the z axis , then H possesses only a z component and A, only the nonzero components A x and A y.
Because of this difference, the simulation may fail. However, in Corollary 1. The A vector produced by a solenoid possesses a cylindrical symmetry whereas the spatial distribution of a NACT formed even by a single ci is rarely circular. For instance, in case of a three-atom system it is frequently elliptic15 see Section 3. Both types will be studied employing the ordinary mathematical tools as applied within field theory. It seems as if Eq. To continue, we consider the case of one single ci. In order to guarantee the quantization in Eq.
To continue, we perform the integrations on both sides of Eq. We continue to elaborate on Eq. The solution of Eq. Since q is a radius, it is always positive and therefore Eq. The virgin functions are used as boundary conditions to solve the C-D equations. Moreover, except for its singlevaluedness in configuration space see Section 6. We are aware that this is a far-reaching assumption and therefore, this assumption has been under detailed scrutiny in more recent numerical treatments. Consequently the line integral in Eq.
At first it was not clear why the two unit vectors n vectors posses opposite signs. About the Author Michael Baer is one of the foremost authorities on molecular scattering theory. He wrote the seminal paper in the field of electronic nonadiabatic molecular collisions in and has continued to make fundamental contributions to electronic nonadiabatic processes in molecular systems. He also contributed significantly to developing numerical methods to treat, quantum mechanically, reactive-exchange processes and is a co-author of the negative imaginary potential approach to decoupling the dynamics in different arrangement channels, which is now used worldwide.
Baer, who received his M. Before that he was a theoretical physicist and an applied mathematician for almost 40 years at the Soreq Nuclear Research Center, Israel. The author was a visiting scientist in many foreign universities and scientific institutes, among them Harvard University and the University of Oxford. He has published more than scientific articles and edited several books. Table of contents Reviews Preface. The Abelian and the non-Abelian Curl Equation. The Abelian and the non-Abelian Div-Equation.
The Study of the Abelian Case. The Study of the non-Abelian Case. The Integral Equation along an Open Contour. The Integral Equation along an Closed Contour. Solution of the Differential Vector Equation. Diabatization and Topological Matrix. The Transformation for the Electronic Basis Set. The Transformation for the Nuclear Wave-Functions. Implications due to the Adiabatic-to-Diabatic Transformation. Application of Complex Eigenfunctions. Introducing Time-Independent Phase Factors. Introducing Time-Dependent Phase Factors. The Time Dependent Treatment. The Time-Dependent Perturbative Approach.
The Time-Dependent non-Perturbative Approach. Treatment of Analytical Models. The Adiabatic-to-Diabatic Transformation Matrix. The Diabatic Potential Matrix. Treatment of the General Case. The Elliptic Jahn-Teller Model. The Wigner Rotation Matrices. Studies of Molecular Systems. The Construction of Hilbert Subspaces. The Sign Flips of the Electronic Eigenfunctions.