Subbarrier fusion reactions and many-particle quantum tunneling
Quantum mechanics is indispensable in understanding microscopic systems such as atoms, molecules, and atomic nuclei. One of its fundamental aspects is a quantum tunneling, where a particle penetrates into a classically forbidden region. This is a wave phenomenon and is frequently encountered in diverse processes in physics and chemistry.
The importance of quantum tunneling has been recognized from the birth of quantum mechanics. For instance, it was as early as 1928 when Gamow, and independently Gurney and Condon, applied quantum tunneling to decays of atomic nuclei and successfully explained the systematics of the experimental half-lives of radioactive nuclei.
In many applications of quantum tunneling, one only considers penetration of a one-dimensional potential barrier, or a barrier with a single variable. In general, however, a particle which penetrates a potential barrier is never isolated but interacts with its surroundings or environments, resulting in modification in its behavior. Moreover, when the particle is a composite particle, the quantum tunneling has to be discussed from a many-particle point of view. Quantum tunneling therefore inevitably takes place in reality in a multi-dimensional space. Such problem was first addressed by Kapur and Peierls in 1937. Their theory has further been developed by e.g., Banks, Bender, and Wu, Gervais and Sakita, Brink, Nemes, and Vautherin, Schmid, and Takada and Nakamura.
When the quantum tunneling occurs in a complex system, such as the trapped flux in a superconducting quantum interference devices (SQUID) ring, the tunneling variable couples to a large number of other degrees of freedom. In such systems, the environmental degrees of freedom more or less reveal a dissipative character. Quantum tunneling under the influence of dissipative environments plays an important role and is a fundamental problem in many fields of physics and chemistry. This problem has been studied in detail by Caldeira and Leggett. This seminal work has stimulated lots of experimental and theoretical works, and has made quantum tunneling in systems with many degrees of freedom a topic of immense interest during the past decades.
In nuclear physics, one of the typical examples of tunneling phenomena is heavy-ion fusion reaction at energies near and below the Coulomb barrier. Fusion is defined as a reaction in which two separate nuclei combine together to form a compound nucleus. In order for fusion reaction to take place, the relative motion between the colliding nuclei has to overcome the Coulomb barrier formed by a strong cancellation between the long-ranged repulsive Coulomb and the short-ranged attractive nuclear forces (as a typical example, Fig.1 shows the internucleus potential between O and Sm nuclei as a function of the relative distance). Unless under extreme conditions, it is reasonable to assume that atomic nuclei are isolated systems and the couplings to external environments can be neglected. Nevertheless, one can still consider intrinsic environments. The whole spectra of excited states of the target and projectile nuclei (as well as several sorts of nucleon transfer processes) are populated in a complex way during fusion reactions. They act as environments to which the relative motion between the colliding nuclei couples. In fact, it has by now been well established that cross sections of heavy-ion fusion reactions are substantially enhanced due to couplings to nuclear intrinsic degrees of freedom at energies below the Coulomb barrier as compared to the predictions of a simple potential model. Heavy-ion subbarrier fusion reactions thus make good examples of environment-assisted tunneling phenomena.
Theoretically the standard way to address the effects of the couplings between the relative motion and nuclear intrinsic degrees of freedom on fusion reactions is to numerically solve the coupled-channels equations which include all the relevant channels. In the eigen-channel representation of coupled-channels equations, the channel coupling effects can be interpreted in terms of a distribution of fusion barriers. In this representation, the fusion cross section is given by a weighted sum of the fusion cross sections for each eigen-barrier. Those eigen-barriers lower than the original barrier are responsible for the enhancement of the fusion cross section at energies below the Coulomb barrier. Based on this idea, Rowley, Satchler, and Stelson have proposed a method to extract barrier distributions directly from experimental fusion excitation functions by taking the second derivative of the product of the fusion cross section and the center of mass energy with respect to , i.e., . This method was tested against high precision experimental data of fusion cross sections soon after the method was proposed. The extracted fusion barrier distributions were sensitive to the effects of channel-couplings and provided a much more apparent way of understanding their effects on the fusion process than the fusion excitation functions themselves. It is now well recognised that the barrier distribution approach is a standard tool for heavy-ion subbarrier fusion reactions.
The aim of this paper is to review theoretical aspects of heavy-ion subbarrier fusion reactions from the view point of quantum tunneling of composite particles. To this end, we mainly base our discussions on the coupled-channels approach. Earlier reviews on the subbarrier fusion reactions can be found in Refs. ?), ?), ?), ?), ?). See also Refs. ? and ? for reviews on subbarrier fusion reactions of radioactive nuclei, and e.g., Refs. ? and ? for reviews on fusion reactions relevant to synthesis of superheavy elements, both of which we do not cover in this article.
The paper is organized as follows. We will first discuss in the next section a potential model approach to heavy-ion fusion reactions. This is the simplest approach to fusion reaction, in which only elastic scattering and fusion are assumed to occur. This approach is adequate for light systems, but for fusion with a medium-heavy or heavy target nucleus the effects of nuclear excitations during fusion start playing an important role. In Sec. 3, we will discuss such nuclear structure effect on heavy-ion fusion reactions. To this end, we will introduce and detail the coupled-channels formalism which takes into account the inelastic scattering and transfer processes during fusion reactions. In Sec. 4, light will be shed on the fusion barrier distribution representation of fusion cross section defined as . It has been known that this approach is exact when the excitation energy of the intrinsic motion is zero, but we will demonstrate that one can generalize it unambiguously using the eigen-channel approach also to the case when the excitation energy is finite. In Sec. 5, we will turn to a discussion on the present status of our understanding of deep subbarrier fusion reactions. At these energies, fusion cross sections have been shown to be suppressed compared to the values of the standard coupled-channels calculations. This phenomenon may be related to dissipative quantum tunneling, that is, an irreversible coupling to intrinsic degrees of freedom. In Sec. 6, we discuss an application of the barrier distribution method to surface physics, more specifically, the effect of rotational excitations on a dissociative adsorption process of H molecules. We then summarize the paper in Sec. 7.
2 One dimensional potential model
2.1 Ion-ion potential
Theoretically, the simplest approach to heavy-ion fusion reactions is to use the one dimensional potential model where both the projectile and the target are assumed to be structureless. A potential between the projectile and the target is given by a function of the relative distance between them. It consists of two parts, that is,
where is the nuclear potential, and is the Coulomb potential given by
in the outside region where the projectile and the target nuclei do not significantly overlap with each other. Figure 1 shows a typical potential for the -wave scattering of the O + Sm reaction. The dotted and the dashed lines are the nuclear and the Coulomb potentials, respectively, while the total potential is denoted by the solid line. One can see that a potential barrier appears due to a strong cancellation between the short-ranged attractive nuclear interaction and the long-ranged repulsive Coulomb force. This potential barrier is referred to as the Coulomb barrier and has to be overcome in order for the fusion reaction to take place. in the figure is the touching radius, at which the projectile and the target nuclei begin overlapping considerably. One can see that the Coulomb barrier is located outside the touching radius.
There are several ways to estimate the nuclear potential . One standard method is to fold a nucleon-nucleon interaction with the projectile and the target densities. The direct part of the nuclear potential in this double folding procedure is given by
where is an effective nucleon-nucleon interaction, and and are the densities of the projectile and the target, respectively. The double-folding potential is in general a non-local potential due to the anti-symmetrization effect of nucleons. Usually, either a zero-range approximation or a local momentum approximation is employed in order to treat the non-locality of the potential.
A phenomenological nuclear potential has also been employed. For instance, a Woods-Saxon form
has been widely used, where the parameters were determined from a least-squares fit to the experimental data of heavy-ion elastic scattering.
A nuclear potential so constructed has been successful in reproducing experimental angular distributions of elastic and inelastic scattering for many systems. Moreover, the empirical value of surface diffuseness parameter, 0.63 fm, is consistent with a double folding potential. Recently, a value of the surface diffuseness parameter has been determined unambiguously using heavy-ion quasi-elastic scattering at deep subbarrier energies. It has been confirmed that the experimental data are consistent with a value around 0.63 fm.
In marked contrast, recent experimental data for heavy-ion subbarrier fusion reactions suggest that a much larger value of diffuseness, ranging from 0.75 to 1.5 fm, is required to fit the data. The Woods-Saxon potential which fits elastic scattering overestimates fusion cross sections at energies both above and below the Coulomb barrier, having an inconsistent energy dependence with the experimental fusion excitation function. A reason for the large discrepancies in diffuseness parameters extracted from scattering and fusion analyses has not yet been fully understood. However, it is probably the case that the double folding procedure is valid only in the surface region, while several dynamical effects come into play in the inner part where fusion is sensitive to.
We summarize the relation between the surface diffuseness parameter of a nuclear potential and the parameters of the Coulomb barrier, that is, the curvature, the barrier height, and the barrier position in Appendix A for an exponential and a Woods-Saxon potentials.
2.2 Fusion cross sections
In the potential model, the internucleus potential, , is supplemented by an imaginary part, , which mocks up the formation of a compound nucleus. One then solves the Schrödinger equation
for each partial wave , where is the reduced mass of the system, with the boundary conditions of
Here, and are the outgoing and the incoming Coulomb wave functions, respectively. is the nuclear -matrix, and is the wave number associated with the energy .
If the imaginary part of the potential, , is confined well inside the Coulomb barrier, one can regard the total absorption cross section as fusion cross section, i.e.,
In heavy-ion fusion reactions, instead of imposing the regular boundary condition at the origin, Eq. (2.2), the so called incoming wave boundary condition (IWBC) is often applied without introducing the imaginary part of the potential, . Under the incoming wave boundary condition, the wave function has a form
at the distance smaller than the absorption radius , which is taken to be inside the Coulomb barrier. Here, is the local wave number for the -th partial wave defined by
The incoming wave boundary condition corresponds to the case where there is a strong absorption in the inner region so that the incoming flux never returns back. For heavy-ion fusion reactions, the final result is not sensitive to the choice of the absorption radius , and it is often taken to be at the pocket of the potential. With the incoming wave boundary condition, in Eq. (2.2) is interpreted as the transmission coefficient. Equation (2.2) is then transformed to
where is the penetrability for the -wave scattering defined as
For a parabolic potential, Wong has derived an analytic expression for fusion cross sections, Eq. (2.2). We will discuss it in Appendix B.
The incoming wave boundary condition, Eq. (2.2), has two advantages over the regular boundary condition, Eq. (2.2). The first advantage is that the imaginary part of nuclear potential is not needed, and the number of adjustable parameters can be reduced. The second point is that the incoming wave boundary condition directly provides the penetrability and thus the round off error can be avoided in evaluating . This is a crucial point at energies well below the Coulomb barrier, where is close to unity. Notice that the incoming wave boundary condition does not necessarily correspond to the limit of , as the quantum reflection due to has to be neglected in order to realize it. The incoming wave boundary condition should thus be regarded as a different model from the regular boundary condition.
2.3 Comparison with experimental data: success and failure of the potential model
Let us now compare the one dimensional potential model for heavy-ion fusion reaction with experimental data. Figure 2 shows the experimental excitation functions of fusion cross section for N+C (the left panel) and O+Sm (the right panel) systems, as well as results of the potential model calculation (the solid lines). One can see that the potential model well reproduces the experimental data for the lighter system, N + C. On the other hand, the potential model apparently underestimates fusion cross sections for the heavier system, O + Sm, although it reproduces the experimental data at energies above the Coulomb barrier, which is about 59 MeV for this system.
In order to understand the origin for the failure of the potential model, Fig. 3 shows the experimental fusion excitation functions for O + Sm reactions and a comparison with the potential model (the solid line). To remove trivial target dependence, these are plotted as a function of center of mass energy relative to the barrier height for each system, and the fusion cross sections are divided by the geometrical factor, . With these prescriptions, the fusion cross sections for the different systems are matched with each other at energies above the Coulomb barrier, although one can also consider a more refined prescription. The barrier height and the result of the potential model are obtained with the Akyüz-Winther potential. One again observes that the experimental fusion cross sections are drastically enhanced at energies below the Coulomb barrier compared with the prediction of the potential model. Moreover, one also observes that the degree of enhancement of fusion cross section depends strongly on the target nucleus. That is, the enhancement for the O + Sm system is order of magnitude, while that for the O + Sm system is about a factor of four at energies below the Coulomb barrier. This strong target dependence of fusion cross sections suggests that low-lying collective excitations play a role, as we will discuss in the next section.
The inadequacy of the potential model has been demonstrated in a more transparent way by Balantekin et al.. Within the semi-classical approximation, the penetrability for a one-dimensional barrier can be inverted to yield the barrier thickness. Balantekin et al. applied such inversion formula directly to the experimental fusion cross sections in order to construct an effective internucleus potential. Assuming a one-dimensional energy-independent local potential, the resultant potentials were unphysically thin for heavy systems, often multi-valued potential. This result was confirmed also by the systematic study in Ref. ?. These analyses have provided a clear evidence for the inadequacy of the one-dimensional barrier passing model for heavy-ion fusion reactions, and has triggered to develop the coupled-channels approach, which we will discuss in the next section.
In passing, we have recently applied the inversion procedure in a modified way to determine the lowest potential barrier among the distributed ones due to the effects of channel coupling. The extracted potential for the O + Pb scattering is well behaved, indicating that the channel coupling indeed plays an essential role in subbarrier fusion reactions.
3 Coupled-channels formalism for heavy-ion fusion reactions
3.1 Nuclear structure effects on subbarrier fusion reactions
The strong target dependence of subbarrier fusion cross sections shown in Fig. 3 suggests that the enhancement of fusion cross sections is due to low-lying collective excitations of the colliding nuclei during fusion. The low-lying excited states in even-even nuclei are collective states, and strongly reflect the pairing correlation and shell structure. They have thus strongly coupled to the ground state, and also have a strong mass number and atomic number dependences. As an example, the low-lying spectra are shown in Fig. 4 for Sm. The Sm nucleus is close to the (sub-)shell closures (=64 and ) and is characterized by a strong octupole vibration. Sm, on the other hand, is a well deformed nucleus, and has a well developed ground state rotational band. Sm is a transitional nucleus, and there exists a soft quadrupole vibration in the low-lying spectrum. One can clearly see that there is a strong correlation between the degree of enhancement of fusion cross sections shown in Fig. 3 and e.g., the energy of the first 2 state.
Besides the low-lying collective excitations, there are many other modes of excitations in atomic nuclei. Among them, non-collective excitations couple only weakly to the ground state and usually they do not affect in a significant way heavy-ion fusion reactions, even though the number of non-collective states is large. Couplings to giant resonances are relatively strong due to their collective character. However, since their excitation energies are relatively large and also are smooth functions of mass number, their effects can be effectively incorporated in a choice of internuclear potential through the adiabatic potential normalization (see the next section).
The effect of rotational excitations of a heavy deformed nucleus can be easily taken into account using the orientation average formula. For an axially symmetric target nucleus, fusion cross sections are computed with this formula as,
where is the angle between the symmetry axis and the beam direction. is a fusion cross section for a fixed orientation angle, . This is obtained with e.g., a deformed Woods-Saxon potential,
which can be constructed by changing the target radius in the Woods-Saxon potential, Eq. (2.1), to . See Ref. ? for a recent application of this formula to fusion of massive systems, in which the formula is combined with classical Langevin calculations.
The left panel of Fig. 5 shows the potential for the O+Sm reaction obtained with the deformation parameters of =0.306 and . The deformation of the Coulomb potential is also taken into account (see Sec. 3.4 for details). The solid line shows the potential for . For this orientation angle, the potential is lowered by the deformation effect as compared to the spherical potential shown by the dotted line, because the attractive nuclear interaction is active from relatively large values of . The opposite happens when as shown by the dashed line. The potential is distributed between the solid and the dashed lines according to the value of orientation angle, . The solid line in the right panel of Fig. 5 shows the fusion cross sections obtained by averaging the contribution of all the orientation angles through Eq. (3.1). Since the tunneling probability has an exponentially strong dependence on the barrier height, the fusion cross sections are significantly enhanced for those orientations which yield a lower barrier than the spherical case. It is remarkable that this simple calculation accounts well for the experimental enhancement of fusion cross sections at subbarrier energies. Evidently, nuclear structure effects significantly enhance fusion cross sections at energies below the Coulomb barrier, which make fusion reactions an interesting probe for nuclear structure.
3.2 Coupled-channels equations with full angular momentum coupling
The nuclear structure effects can be taken into account in a more quantal way using the coupled-channels method. In order to formulate the coupled-channels method, consider a collision between two nuclei in the presence of the coupling of the relative motion, , to a nuclear intrinsic motion . We assume the following Hamiltonian for this system,
where and are the intrinsic and the coupling Hamiltonians, respectively. In general the intrinsic degree of freedom has a finite spin. We therefore expand the coupling Hamiltonian in multipoles as
Here are the spherical harmonics and are spherical tensors constructed from the intrinsic coordinate. The dot indicates a scalar product. The sum is taken over all values of except for , which is already included in the bare potential, .
For a given total angular momentum and its component , one can define the channel wave functions as
where and are the orbital and the intrinsic angular momenta, respectively. are the wave functions of the intrinsic motion which obey
Here, denotes any quantum number besides the angular momentum. Expanding the total wave function with the channel wave functions as
the coupled-channels equations for read
where the coupling matrix elements are given as
Notice that these matrix elements are independent of .
For the sake of simplicity of the notation, in the following let us introduce a simplified notation, , and suppress the index . The coupled-channels equation (3.2) then reads,
These coupled-channels equations are solved with the incoming wave boundary conditions of
where denotes the entrance channel. The local wave number is defined by
whereas . Once the transmission coefficients are obtained, the inclusive penetrability of the Coulomb potential barrier is given by
The fusion cross section is then given by
where we have assumed that the initial intrinsic state has spin zero, . This equation for fusion cross section is similar to Eq. (2.2) except that the penetrability is now influenced by the channel coupling effects.
3.3 Iso-centrifugal approximation
The full coupled-channels calculations (3.11) quickly become intricate if many physical channels are included. The dimension of the resulting coupled-channels problem is in general too large for practical purposes. For this reason, the iso-centrifugal approximation, which is sometimes referred to as the no-Coriolis approximation or the rotating frame approximation, has often been introduced. In the iso-centrifugal approximation to the coupled-channels equations, Eq. (3.11), one first replaces the angular momentum of the relative motion in each channel by the total angular momentum , that is,
This corresponds to assuming that the change in the orbital angular momentum due to the excitation of the intrinsic degree of freedom is negligible. Introducing the weighted average wave function
where we have suppressed the index for simplicity, and using the relation
one finds that the wave function obeys the reduced coupled-channels equations,
These are nothing but the coupled-channels equations for a spin-zero system with the interaction Hamiltonian given by
In solving the reduced coupled-channels equations, similar boundary conditions are imposed for as those for ,
Since the reduced coupled-channels equations in the iso-centrifugal approximation are equivalent to the coupled-channels equations with a spin-zero intrinsic motion, the complicated angular momentum couplings disappear. A remarkable fact is that the dimension of the coupled-channels equations is drastically reduced in this approximation. For example, if one includes four intrinsic states with 2, 4, 6, and 8 together with the ground state in the coupled-channels equations, the original coupled-channels have 25 dimensions for , while the dimension is reduced to 5 in the iso-centrifugal approximation. The validity of the iso-centrifugal approximation has been well tested for heavy-ion fusion reactions, and it has been concluded that the iso-centrifugal approximation leads to negligible errors in calculating fusion cross sections.
3.4 Coupling to low-lying collective states
3.4.1 Vibrational coupling
Let us now discuss the explicit form of the coupling Hamiltonian for heavy-ion fusion reactions. We first consider couplings of the relative motion to the 2-pole surface vibration of a target nucleus. In the geometrical model of Bohr and Mottelson, the radius of the vibrating target is parameterized as
where is the equivalent sharp surface radius and is the surface coordinate of the target nucleus. To the lowest order, the surface oscillation is approximated by a harmonic oscillator and the Hamiltonian for the intrinsic motion is given by
Here is the oscillator quanta and and are the phonon creation and annihilation operators, respectively. The surface coordinate is related to the phonon creation and annihilation operators by
where is the amplitude of the zero point motion. The deformation parameter can be estimated from the experimental transition probability using (see Eq. (3.34) below)
The surface vibration of the target nucleus modifies both the nuclear and the Coulomb interactions between the colliding nuclei. In the collective model, the nuclear interaction is assumed to be a function of the separation distance between the vibrating surfaces of the colliding nuclei, and thus is given as
If the amplitude of the zero point motion of the vibration is small, one can expand this equation in terms of and keep only the linear term,
This approximation is called the linear coupling approximation. The first term of the right hand side (r.h.s.) of Eq. (3.30) is the bare nuclear potential in the absence of the coupling, while the second term is the nuclear component of the coupling Hamiltonian. Even though the linear coupling approximation does not work well for heavy-ion fusion reactions, we employ it in this subsection in order to illustrate the coupling scheme. In Sec. 3.5, we will discuss how the higher order terms can be taken into account in the coupling matrix.
The Coulomb component of the coupling Hamiltonian is evaluated as follows. The Coulomb potential between the spherical projectile and the vibrating target is given by
where is the charge density of the target nucleus and the electric multipole operator defined by
and have assumed that the relative coordinate is larger than the charge radius of the target nucleus. If we assume a sharp matter distribution for the target nucleus, the electric multipole operator is given by
up to the first order in the surface coordinate .
up to the first order of . Here, is the coupling form factor, given by
where the first and the second terms are the nuclear and the Coulomb coupling form factors, respectively. Transforming to the rotating frame, the coupling Hamiltonian used in the iso-centrifugal approximation is then given by (see Eq. (3.21)),
Notice that the coupling form factor has the value
at the position of the bare Coulomb barrier, , and the coupling strength is approximately proportional to the charge product of the colliding nuclei.
In the previous subsection, we showed that the iso-centrifugal approximation drastically reduces the dimension of the coupled-channels equations. A further reduction can be achieved by introducing effective multi-phonon channels. In general, the multi-phonon states of the vibrator have several levels, which are distinguished from each other by the angular momentum and the seniority. For example, for the quadrupole surface vibrations, the two-phonon state has three levels (), which are degenerate in energy in the harmonic limit. The one-phonon state, , couples only to a particular combination of these triplet states,
It is thus sufficient to include this single state in the calculations, instead of three triplet states. In the same way, one can introduce the -phonon channel for a multipolarity as
See Appendix C for the case of two different vibrational modes of excitation (e.g., a quadrupole and an octupole vibrations).
If one truncates the phonon space up to the two-phonon state, the corresponding coupling matrix is then given by
where is defined as .
The effects of deviations from the harmonic oscillator limit presented in this subsection on subbarrier fusion reactions have been discussed in Refs. ? and ?.
3.4.2 Rotational coupling
We next consider couplings to the ground rotational band of a deformed target. To this end, it is convenient to transform to the body fixed frame where the axis is along the orientation of the deformed target. The surface coordinate is then transformed to
where , and are the Euler angles which specify the body-fixed frame, thus the orientation of the target. If we are particularly interested in the quadrupole deformation (=2), the surface coordinates in the body fixed frame are expressed as
If we further assume that the deformation is axial symmetric (i.e., ), the coupling Hamiltonian for the rotational coupling reads (see Eq. (3.35))
In order to obtain this equation, we have used the relation
The coupling Hamiltonian in the rotating frame is thus given by
where is the angle between and , that is, the direction of the orientation of the target measured from the direction of the relative motion between the colliding nuclei. Since the wave function for the state in the ground rotational band is given by , the corresponding coupling matrix is given by
when the rotational band is truncated at the first 4 state. Here, is the excitation energy of the first 2 state, and is defined as as in Eq. (3.41).
One of the main differences between the vibrational (3.41) and the rotational (3.49) couplings is that the latter has a diagonal component which is proportional to the deformation parameter . The diagonal component in the rotational coupling is referred to as the reorientation effect and has been used in the Coulomb excitation technique to determine the sign of the deformation parameter. Notice that the results of the coupled-channels calculations are independent of the sign of for the vibrational coupling.
The effects of the deformation on subbarrier fusion were studied in Ref. ?. If there is a finite deformation, the coupling Hamiltonian in the rotating frame becomes
Higher order deformations can also be taken into account in a similar way as the quadrupole deformation. For example, if there is an axial symmetric hexadecapole deformation in addition to quadrupole deformation, the coupling Hamiltonian reads
where is the hexadecapole deformation parameter.
3.5 All order couplings
In the previous subsection, for simplicity, we have used the linear coupling approximation and expanded the coupling Hamiltonian in terms of the deformation parameter. However, it has been shown that the higher order terms play an important role in heavy-ion subbarrier fusion reactions. These higher order terms can be evaluated as follows. If we employ the Woods-Saxon potential, Eq. (2.1), the nuclear coupling Hamiltonian can be generated by changing the target radius in the potential to a dynamical operator
For the vibrational coupling, the operator is given by (see Eq. (3.37)),
The matrix elements of the coupling Hamiltonian can be easily obtained using a matrix algebra. In this algebra, one first looks for the eigenvalues and eigenvectors of the operator which satisfies
This is done by numerically diagonalising the matrix , whose elements are given by
for the vibrational case, and