# Barrier penetration in a discrete-basis formalism

G.F. Bertsch  Department of Physics and Institute for Nuclear Theory, Box
351560, University of Washington, Seattle, Washington 98915, USA  K. Hagino
Department of Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

The dynamics of a many-particle system are often modeled by mapping the
Hamiltonian onto a Schrödinger equation. An alternative approach is to solve
the Hamiltonian equations directly in a model space of many-body
configurations. In a previous paper the numerical convergence of the two
approaches was compared with a simplified treatment of the Hamiltonian
representation. Here we extend the comparison to the nonorthogonal model
spaces that would be obtained by the generator-coordinate method. With a
suitable choice of the collective-variable grid, a configuration-interaction
Hamiltonian can reproduce the Schrödinger dynamics very well. However, the
method as implemented here requires that the barrier height is not much larger
than the zero-point energy in the collective coordinates of the
configurations.

##  I Introduction

Ever since the pioneering work of Hill and Wheeler HW , low-energy fission has
been parameterized by their barrier penetration formula, based on a one-
dimensional Schrödinger equation. See Refs. si16 ; ko23 ; be23 ; kawano2023
for recent extensions of that model. A kinetic energy operator and potential
energy function for the Schrödinger equation can be derived in the Generator
Coordinate Method of many-body theory, but beyond the extension to two
dimensions2D  the generalization to other degrees of freedom presents
formidable obstacles GY . In contrast, Hamiltonians constructed from the
configuration-interaction (CI) approach al2020 ; BY ; BH2022 ; BH2023  can in
principle deal with any mechanisms present in nuclear dynamics.

A CI basis is usually constructed from nucleonic Hamiltonians by solving the
Hartree-Fock or Hartree-Fock-Bogoliubov equations in the presence of shape
constraints. Those constraints map out a path for the barrier crossing. To
proceed further without leaving the CI basis one needs to understand how to
deal with the non-orthogonality of the constrained configurations. Also, as a
practical question, how closely do the configurations need to be spaced along
a fission path to reproduce the Schrödinger dynamics? In this work we apply
reaction theory as formulated in a discrete basis of states to investigate how
well that framework can reproduce the Schrödinger.

The focus of this study is the transmission probability $T$ for traversing an
isolated one-dimensional barrier, following up on the work of Ref. ha23 . We
assume that the barrier potential vanishes at large distances, so the wave
function satisfies ordinary plane-wave boundary conditions.

##  II Model space and Hamiltonian

###  II.1 Basics

The construction of the model space and the Hamiltonian within it closely
follows the treatment in Ref. BY . The states in the space are obtained by
self-consistent mean-field theory augmented by a $q$-dependent constraining
field. A finite basis is generated on a mesh of points $\\{q_{i}\\}$, making a
path along the collective coordinate. With those wave functions one computes
the overlaps

$\mbox{\boldmath$N$}_{ij}=\langle\psi_{i}|\psi_{j}\rangle.$ (1)

Here and hereafter, we use boldface symbols for matrices. The Hamiltonian
matrix elements are similarly computed with a Hamiltonian $H$ that contains a
nucleon-nucleon interaction,

$\mbox{\boldmath$H$}_{ij}=\langle\psi_{i}|H|\psi_{j}\rangle.$ (2)

Insight into the workings of this approach can be obtained by taking the
center-of-mass coordinate as a paradigm of a collective variableBY . One finds
that $\mbox{\boldmath$N$}_{ij}$ can be parameterized quite well as a Gaussian,

$\mbox{\boldmath$N$}_{ij}\approx
n_{|i-j|}=\exp\left(-(q_{i}-q_{j})^{2}/4s^{2}\right)$ (3)

where $s$ is a physical parameter associated with the size of the collective
wave packets. The states giving rise to the above overlaps have a separable
form

$\psi_{i}(q,\vec{\xi})=\psi_{\rm
int}(\vec{\xi})\exp\left(-(q-q_{i})^{2}/2s^{2}\right)$ (4)

where $\psi_{\rm int}$ depends only on intrinsic coordinates $\vec{\xi}$. One
assumes that the Hamiltonian can be separated into an intrinsic part and a
collective kinetic part given by

$\hat{H}^{0}=-\frac{\hbar^{2}}{2M_{q}}\frac{\partial^{2}}{\partial q^{2}}.$
(5)

Here $M_{q}$ is an inertial parameter associated with the collective
coordinate. The matrix elements of $\hat{H}^{0}$ are parameterized as

$\mbox{\boldmath$H$}^{0}_{ij}\approx
h_{|i-j|}=n_{|i-j|}E_{q}\left(1-(q_{i}-q_{j})^{2}/2s^{2}\right)$ (6)

where $E_{q}=\hbar^{2}/(4M_{q}s^{2})$ is the zero-point energy of the
configuration.

In using a discrete-basis representation in reaction theory, it is helpful to
understand how to represent noninteracting plane waves. For this purpose, we
choose a grid of uniformly spaced points separated by $\Delta
q=q_{i+1}-q_{i}$. The eigenstates of $H^{0}$ are given by

$\Psi_{k}(q,\vec{\xi})=\sum_{n}\psi_{n}(q,\vec{\xi})\,r^{n},$ (7)

where $r=e^{ik\Delta q}$ and $k$ is a momentum index. Note that the space
states only support momenta in the range $-\pi<k\Delta q<\pi$.

The kinetic energy $E_{DB}$ of the plane-wave state in the discrete basis is
given by BY

$E_{DB}(k)=\frac{h_{0}+2\sum_{j>0}h_{j}\cos{jk\Delta q}}{1+2\sum_{j>0}n_{j}}.$
(8)

In practice $H$ will be treated as a band-diagonal matrix with matrix elements
$\mbox{\boldmath$H$}_{ij}$ set to zero for $|i-j|>N_{od}$. In the simplest
version of the theory, $N_{od}=1$ and the matrix is tridiagonal with
interactions only between nearest neighbors. The quality of the energy fit to
the Schödinger energy $E_{s}=k^{2}\hbar^{2}/2M_{q}$ depends on $N_{od}$ and on
the dimensionless ratio $\Delta q/s$. The computed $E_{DB}(k)$ does not go
exactly to zero at $k=0$, since the sum of the Gaussians in Eq. (7) still has
some variation as a function of $q$. To keep the energy comparisons
consistent, we compare the excitation energy

$E=E_{\rm DB}(k)-E_{\rm DB}(0)$ (9)

with the Schródinger energy $E_{s}$.

Fig. 1 shows the comparison with some examples. In Ref. BY  the choice $\Delta
q=5^{1/2}s$ was advocated for discrete-basis Hamiltonians of tridiagonal form.
The derived eigenenergies for the tridiagonal and next-to-nearest-neighbor
approximations in the range $0<k<\pi/s$ are shown in Fig. 1(a). One can see
small differences between the two, but overall it appears that the tridiagonal
approximation is acceptable up to energies $\sim 2E_{q}$.

Figure 1: Energies of plane-wave states in the discrete-basis formalism for
mesh parameters $\Delta q=5^{1/2}$ (upper panel) and $2/3\times 5^{1/2}$
(lower panel). Black : Schrödinger Hamiltonian
$\hat{H}^{0}=k^{2}\hbar^{2}/2M_{q}$; red dashed: discrete-basis Hamiltonian
with $N_{od}=1$; blue dotted: $N_{od}=2$. Energies are in units of $E_{q}$,
and momenta in units of $s$.

Fig. 1(b) shows the spectra for a somewhat smaller mesh spacing, $\Delta
q=\frac{2}{3}5^{1/2}\,s$. Here the tridiagonal treatment fails. On the other
hand, inclusion of next-to-nearest neighbors ($N_{od}=2$) restores a good
approximation to the energy curve and increases the range of $k$.

###  II.2 The barrier

We consider the transmission coefficient for a plane wave incident on a
barrier of the form

$V(q)=V_{0}\exp\left(-q^{2}/2\sigma^{2}\right).$ (10)

This simulates a quadratic barrier around $q=0$ but vanishes at large
distances. The matrix elements for wave functions in Eq. (4) are

$\displaystyle\langle\psi_{i}|\hat{V}(q)|\psi_{j}\rangle=\mbox{\boldmath$V$}_{ij}$
$\displaystyle=$ $\displaystyle
V_{0}\sqrt{\frac{2\sigma^{2}}{s^{2}+2\sigma^{2}}}\exp\left(-(q_{1}+q_{2})^{2}/4(s^{2}+2\sigma^{2})\right)\mbox{\boldmath$N$}_{ij}.$

Besides $\Delta q$ and $N_{od}$, a third numerical parameter in the discrete-
basis formulation is the dimension $N_{DB}$ of the $H$ and $N$ matrices. We
will see that the space needs to extend beyond the range of the barrier by
only a few states to produce fairly accurate transmission probabilities.

###  II.3 The transmission probability

The energies $E$ and eigenstates $\vec{\psi}_{E}=(f_{1},f_{2},...,f_{N_{od}})$
of the matrix Hamiltonian
$\mbox{\boldmath$H$}_{ij}=\mbox{\boldmath$H$}^{0}_{ij}+\mbox{\boldmath$V$}_{ij}$
satisfy the equation

$\mbox{\boldmath$H$}^{\prime}\vec{\psi}_{E}\equiv(\mbox{\boldmath$H$}-\mbox{\boldmath$N$}\,E)\vec{\psi}_{E}=0$
(12)

for rows $m$ that contain all of the possible elements of $\vec{H}^{\prime}$
in its band-diagonal construction. The rows beyond $m=N_{\rm grid}-N_{od}$
lack one or more matrix elements and must be modified to insure that the
transmitted wave satisfies an outgoing-wave boundary condition. The same
applies to the topmost rows with $m<N_{od}$. The boundary condition here
requires the wave function to be composed of a linear combination of incoming
and reflected plane waves. This is achieved by modifying the diagonal
$\mbox{\boldmath$H$}^{\prime}_{mm}$ and replacing Eq. (12) by the
inhomogeneous equation

$(\mbox{\boldmath$H$}^{\prime}+\mbox{\boldmath$\Delta
H$}^{\prime})\vec{\psi}_{E}=\vec{v}.$ (13)

The general expressions for $\mbox{\boldmath$\Delta H$}^{\prime}$ and the
vector $\vec{v}$ are derived in the Appendix. Following the numerical solution
of Eq. (13) the transmission probability111This method is an alternative to
standard $S$-matrix theorymi93 ; da95 ; mi10 . The present approach avoids the
necessity of calculating the real and imaginary parts of the coupling between
states in $H$ and the scattering channels. is evaluated as

$T_{DB}=1-\frac{|f_{1}e^{ik\Delta q}-f_{2}|^{2}}{|f_{1}e^{-ik\Delta
q}-f_{2}|^{2}}.$ (14)

In previous publications (e.g., Ref. BH2023 ) we examined the theory at the
level of the $N_{od}=1$ approximation. In this work we also consider the
$N_{od}=2$ approximation. The Appendix also includes the detailed formulas for
this case.

The discrete-basis formalism defined in this way satisfies an important check
on the theory. Eq. (13) can be solved analytically if $V$ vanishes, yielding
the plane-wave solution $\vec{\psi}_{k}$ of Eq. (7) with $k$ satisfying
$E=E_{DB}(k)$. Thus $T_{DB}=1$ trivially when there is no barrier.

One should be cautious in using the discrete-basis formalism at higher
energies even though they may still be in the allowed range of $E_{DB}(k)$. As
an extreme example, the energy is a maximum at $k\,\Delta q=\pi$ but the
corresponding wave function is a pure standing wave with amplitudes
$\vec{\psi}_{i}$ alternating in sign. Its transmission probability is zero.
This is easily seen from Eq. (14). The transmission probability reaches a
maximum somewhere inside the allowed range of $E_{DB}$ and decreases at higher
energies (see Fig. 4 below). Obviously the treatment is then unphysical.

##  III Numerical examples

We now compare calculated transmission probabilities $T_{DB}$ with those
obtained by integrating the Schrödinger equation222The resulting $T_{s}$ is
quite close to the Hill-Wheeler transmission probability
$T_{HW}=[1+\exp(-2\pi(E-V_{0})/\hbar\omega)]^{-1}$ if the curvature parameter
$\omega=(V_{0}/\sigma^{2}M_{q})^{1/2}$ is the same. A comparison is provided
in the Supplementary Material.

$\left(\hat{H}^{0}_{q}+V(q)\right)\phi=E\phi.$ (15)

The first example is a Hamiltonian with a moderately sized barrier; its
parameters are $V_{0}=1$ in energy units of $E_{q}$ and $\sigma=2$ in length
units of $s$. This barrier height is well within the domain of acceptable
energies. Also the barrier curvature parameter expressed as a harmonic
oscillator energy $\hbar\omega$ is within the domain. Fig. 2 displays the
calculated wave function for an incident energy just at the barrier top,
$E=V_{0}$. The numerical parameters are $\Delta q=(2/3)5^{1/2}\,s$,
$N_{od}=2$, and $N_{DB}=42$. The dimension of the matrices $N_{DB}$ is much
larger than necessary; the purpose is to exhibit the plane-wave character of
the solution outside the barrier region. The points show the real and
imaginary parts of the scattering wave function in the $q$-representation
calculated as

$\Psi(q)=\sum_{i}f_{i}\psi_{i}(q)=\sum_{i}f_{i}\exp(-(q-q_{i})^{2}/2s^{2}).$
(16)

Figure 2:  Scattering wave function partially transmitted across a barrier.
Physical parameters are $(V_{0},\sigma,E)=(1.0,2.0,1.0)$ in length and energy
units $s$ and $E_{q}$, respectively. The numerical parameters are $(N_{\rm
grid},N_{od},\Delta q)=(42,2,2\times 5^{1/2}/3)$. The real and imaginary parts
of the wave function are shown as solid blue and dashed red lines,
respectively.

The wave function on the right-hand side is clearly a traveling wave of the
form $e^{ikx}$ with $k>0$ as required for an outgoing flux. The wave on the
other side has both incoming and outgoing components that almost add together
for a standing wave pattern. This is somewhat deceptive. A pure standing wave
would have equal amplitudes of incoming and outgoing components implying a
reflection probability of one. The actual reflection probability in this
example is close to $1/2$, the expected value in the Hill-Wheeler formula.

We next compare the energy dependence of $T_{DB}$ with Schrödinger solutions,
taking the same barrier parameters as before. The numerical parameters are set
to $N_{DB}=10$ and $\Delta q=5^{1/2}s$ to show what can be achieved in a small
space. The Hamiltonian is defined on a range of $q$ that is long enough to
cover the barrier region and meet the criteria for plane-wave behavior near
the end points The calculated $T_{DB}(E)$ is shown in Fig. 3, both in linear
and logarithmic scales, together with that obtained by solving the Schrödinger
equation333See the Supplementary Material for computer codes to perform the
calculations.. The figure shows that the discrete-basis approach with
$N_{od}=2$ is in excellent agreement with the Schrödinger equation, even at
deep subbarrier energies. Also, the more economical $N_{od}=1$ treatment with
a somewhat larger mesh spacing is useful, given that the microscopic nuclear
Hamiltonians in current use have limited predictive power.

Figure 3: Transmission probabilities $T$ for barrier crossing as a function of
excitation energy $E_{\rm ex}$, plotted in linear (upper panel) and
logarithmic (lower panel) scales. The barrier has the form Eq. (10) with
$V_{0}=E_{q}$ and $\sigma=2\,s$. Shown are the transmission probabilities from
the Schrödinger equation (solid black line), the discrete-basis equation in
the tridiagonal approximation (blue dotted line), and the discrete-basis
equation with next-to-nearest neighbor interactions (red dashed line).

We now examine limits of the discrete-basis approach for higher barriers. Fig.
4 shows $T_{DB}(E)$ for $V_{0}=2$ and $3E_{q}$. At both barrier heights the
discrete-basis Hamiltonian is not useful above $E\sim 3E_{q}$.

Figure 4: Transmission coefficients for $V_{0}=2E_{q}$ (upper panel) and
$V_{0}=3E_{q}$ (lower panel). Blue dashed: Schrödinger equation; red solid:
discrete-basis method with $N_{od}=1$ and $\Delta_{q}=5^{1/2}$.

For a more quantitative assessment of the performance we examine energies
$E_{1/2}$ at which the transmission probability reaches $1/2$, i.e.
$T_{DB}(E_{1/2})=1/2$, and its slope $dT/dE$ at that energy. These are
presented in Fig. 5. The parameters are the same as before except for
$N_{od}$. One sees that $E_{1/2}$ is quite accurate up to $V_{0}=3E_{q}$. But
this is somewhat misleading because the full $T_{DB}(E)$ curve does not reach
close to $T=1$ at higher energies. From the lower panel one sees that the
transmission probability rises somewhat more sharply for the tridiagonal
Hamiltonian than for the Schrödinger $H_{s}$ in most of safe region of
energies. However, the $N_{od}=2$ treatment is quite accurate at low energies.

Figure 5: Panel a): $E_{1/2}$ vs $V_{0}$ for the Schrödinger Hamiltonian
(solid black) compared to the discrete-basis Hamiltonian (dashed blue).
$E_{1/2}=V_{0}$ on the dotted black line. Panel b): $dT/dE$ at $E_{1/2}$ vs
$V_{0}$. Schrödinger results are shown by the solid black line. The discrete-
results are shown by dotted blue and the dashed red lines for $N_{od}=1$ and
2, respectively. Other parameters are the same as in Fig. 3.

##  IV Conclusion

At a purely phenomenological level, the one-dimensional Hamiltonian proposed
by Hill and Wheeler leads to a simple formula that remains an integral part of
fission phenomenologyko23 ; ca09 . But fission theory at a microscopic level
relies on a many-particle formalism to create a matrix Hamiltonian or to
determine the parameters of a Schrödinger Hamiltonian. This work has shown
that the usual procedure for building a CI basis can mimic the Schrödinger
approach quite well. However, there is a important restriction on its
applicability. Namely, the barrier height cannot be much higher than a few
times the zero-point energies of the configurations as given by $E_{q}$ in Eq.
(6). In Ref. BH2023  the functional form of the equation is verified for a few
barrier-top configurations finding $E_{q}$ in the range444$E_{q}=h_{2}/2$ in
the notation of Ref. BH2023 .$1.5-2$ MeV. Whether this is too small depends on
the details of how the paths to a transition-state start out. The barrier
heights of well-known fissile nuclei are of the order of $6$ MeV above the
ground-state energy, somewhat outside the reach of the present approach.
However, CI configurations at more favorable energies in the compound nucleus
might be diabatically connected to the transition states, giving more scope to
the method.

Two ways come to mind for increasing the space of higher energy excitations in
the collective variables. In reaction theory of small clusters, excitations of
their center-of-masses can be introduced by algebraic operators in the
harmonic oscillator representationkr19 . However, that representation may not
be practical for heavy nuclei. Another approach would be to include momentum
constraints at the mean field level to increase the energies with respect to
collective coordinates re87 ; hi22 . This may have been explored for small-
amplitude shape changes, but to our knowledge has not been implemented in
codes for generating a CI basis in heavy nuclei.

If the space were large enough to use the discrete-basis approach with
confidence, a fundamental question in fission theory could be addressed.
Namely, what is the relative importance of collective flow versus diffusive
flow in large-amplitude shape changes? In one extreme, the shape changes go
mainly through collective coordinates that lead to a Schrödinger equation in
one- or a few-dimensions. In another extreme, the shape changes come about as
a random walk through non-collective intermediate configurations. There are
compelling arguments that diffusive flow dominates at energies much higher
than the barrier bu92 . On the other hand theory based on an adiabatic
collective coordinate does quite well at the far subbarrier energies
associated with spontaneous fission schunck2016 ; whitepaper . It seems to us
that some form of a CI approach is needed for treating both mechanisms on the
same footing.

As a final remark, the present analysis is based on the factorization
hypothesis contained in Eq. (4) and leading to the dynamics derived from Eq.
(6). The conclusions regarding the adequacy of the generated space of
configurations would certainly be affected.

###### Acknowledgements.

We thank T. Kawano and L.M. Robledo for helpful comments on the manuscript.
This work was supported in part by JSPS KAKENHI Grant Numbers JP19K03861 and
JP23K03414.

##  V Appendix

We derive here the equation for the wave function satisfying plane-wave
boundary conditions of the discrete-basis Hamiltonian. The outgoing wave
function has the form $f_{m}=Cr^{m}$ where $r=\exp(ik\Delta x)$ and $C$ is an
arbitrary constant. The equation for row $m_{0}$ in the matrix $H^{\prime}$
reads

$r^{m_{0}}\sum_{m=-N_{od}}^{N_{od}}h^{\prime}_{|m|}Cr^{m}=0$ (17)

providing that the Hamiltonian matrix elements are beyond the range of the
potential $V$ and that the row is within $N_{od}<m_{0}\leq N_{\rm DB}-N_{od}$.
The missing terms of rows $N_{DB}-N_{od}<m_{0}\leq N_{DB}$ are added to the
diagonal element in that row,

$\Delta
H^{\prime}_{m_{0},m_{0}}=\sum_{m=m^{\prime}}^{N_{od}}h^{\prime}_{m}r^{m}$ (18)

where $m^{\prime}=N_{DB}-m_{0}+1$. To deal with the missing entries in the
beginning rows, we consider the incoming channel amplitude $f_{0}$ on the site
adjacent to the first site $q_{1}$ in the matrix Hamiltonian and possibly
others if $N_{od}>1$. The contribution of $f_{0}$ is missing from rows of
$H^{\prime}\psi$ in the range $m_{0}\leq N_{od}$. Only the term
$h^{\prime}_{1}f_{0}$ is missing in the last of these rows. It is separated
out as an inhomogenous term in the Hamiltonian equation.

For $N_{od}=1$ the matrix $H^{\prime}$ is tridiagonal and the equation to be
solved is $(\mbox{\boldmath$H$}^{\prime}+\mbox{\boldmath$\Delta
H$}^{\prime})\vec{f}=-h_{1}\vec{v}$ with vector $\vec{v}=(f_{0},0,\cdots,0)$.
For the numerical solution, one can set $f_{0}=1$ and determine the rest of
the wave function by matrix inversion as in Ref. BH2023 .

The wave function around the first site will have outgoing as well as incoming
components for the full Hamiltonian with a barrier $V$. The amplitudes of
incoming and reflected components $(a_{\rm in},a_{\rm out})$ can be extracted
from the wave function amplitudes at $f_{0}$ and $f_{1}$,

$\left(\begin{matrix}a_{\rm in}\cr a_{\rm
out}\cr\end{matrix}\right)=\frac{1}{r-r^{-1}}\left[\begin{matrix}r&-1\cr-r^{-1}&1\cr\end{matrix}\right]\left(\begin{matrix}f_{1}\cr
f_{0}\cr\end{matrix}\right).$ (19)

This is the end of the story for $N_{od}=1$.

For $N_{od}>1$, there are other incomplete rows in the Hamiltonian matrix
requiring amplitudes $f_{0},..,f_{-N_{od}+1}$. These can be determined from
$(a_{\rm in},a_{\rm out})$ as

$\displaystyle f_{m}$ $\displaystyle=$ $\displaystyle a_{\rm
in}r^{-(1-m)}+a_{\rm out}r^{(1-m)},$ $\displaystyle=$
$\displaystyle\frac{1}{r-r^{-1}}\left((r^{-m}-r^{m})f_{1}+(r^{-m+1}-r^{m-1})f_{0}\right).$

The coefficients of terms with $f_{i}$ on the second line are added to the
Hamiltonian matrix element $\mbox{\boldmath$H$}^{\prime}_{1,m_{0}}$ while the
terms with $f_{0}$ are added to vector $\vec{v}$ in Eq. (13). In detail, the
matrix elements added to $\mbox{\boldmath$H$}^{\prime}$ for $N_{od}=2$ are

$\displaystyle\mbox{\boldmath$\Delta
H$}_{11}=-h^{\prime}_{2}\left(1/(1-r^{2})+1/(1-r^{-2})\right),$ (22)
$\displaystyle\mbox{\boldmath$\Delta
H$}_{N_{DB},N_{DB}}=h^{\prime}_{1}r+h^{\prime}_{2}r^{2},$ (23)
$\displaystyle\mbox{\boldmath$\Delta
H$}_{N_{DB}-1,N_{DB}-1}=h^{\prime}_{2}r^{2},$ (24)

and the nonzero components of the vector $v$ are

$\displaystyle v_{1}$ $\displaystyle=$ $\displaystyle
h^{\prime}_{1}+h^{\prime}_{2}\left(r^{-1}/(1-r^{2})+r/(1-r^{-2})\right),$ (25)
$\displaystyle v_{2}$ $\displaystyle=$ $\displaystyle h^{\prime}_{2}.$ (26)

The final inhomogenious equation to be solved is

$(\mbox{\boldmath$H$}^{\prime}+\mbox{\boldmath$\Delta
H$}^{\prime})\vec{f}=-\vec{v}f_{0}$ (27)

with an arbitrary nonzero $f_{0}$.

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