Schematic model for induced fission in a configuration-interaction approach

states, we multiply the bare matrix elements by the suppression factor N (q, q ) to the matrix elements. In addition, one

has to take into account the diabatic interaction between those

configurations which are connected diabatically. A simple formula for the diabatic interaction has been derived in Ref. [22]

based on a self-consistent separable interaction. Based on the

GOA, the formula reads

E (q j) + E (q j)

q j|vdb |q j

− h2 (q − q )2 ,

q|q 

(9)

where E (q j) = k j d + V (q) is the energy of the configuration

(q j). In the previous work [11], the value of h2 was estimated

to be about 1.5 MeV with the Gogny HFB calculations for

236

U. A typical value of single-particle spacing d is around 0.5

MeV when it is corrected for the effective mass (see Table III

in Appendix A). Combining these together, we estimate h2 =

3d in this model.

The first term on the right hand side of this equation ensures that the Green’s function (2) transforms properly under

a shift in energy scale E  = E − , that is G (E  ) = G(E ).

D. Width matrices

The matrices a (a = n, cap, and fis) in Eq. (2) can be in

principle derived with the generalized Fermi golden rule [23]

(a )kk  = 2π

k|v|lk  |v|lδ(El − E ),

(10)

l∈a

C. Off-diagonal couplings

The interaction between different Q blocks is responsible

for a shape change and is thus crucial to the modeling. It is

clear that the interaction is somewhat suppressed due to the

imperfect overlap of orbitals built on different mean-field

reference states. The size of the suppression is determined

by the overlap kernel, N (q j, q j  ), which is given by a

determinant of orbital overlaps. For simplicity, we assume

that the change of the single-particle orbitals between nearby

reference configurations is small. With this assumption, the

configurations with the same index j in neighboring Q blocks

are diabatically connected with the overlap matrix elements

approximated by

q j|N|q j   = N (q, q )δ j, j  ,

(8)

where l labels states in the decay channel a. Due to the

nonorthogonality of the configurations, the matrix a is in

general nondiagonal. In this work, we take a separable approximation and parametrize it as1

(a )kk  = γa

(N 1/2 )k,l (N 1/2 )k  ,l ,

(11)

l∈a

where (N )k,l is the square root of the norm kernel and γa

is the mean decay width. Here, the indices k and l label both

the deformation q and the excitation j. See Appendix B for a

derivation of Eq. (11).

1/2

(7)

where N (q, q ) is the overlap between the reference

configurations.

In Ref. [10], we used N instead of N 1/2 in the decay matrices.

We consider that N 1/2 is a more physical choice because of the

connection to orthogonal bases as we discuss in Appendix B.

024319-3

K. UZAWA AND K. HAGINO

PHYSICAL REVIEW C 108, 024319 (2023)

III. RESULTS

Let us now numerically evaluate the transmission coefficients and discuss the dynamics of induced fission. To this

end, we consider a chain of three Q blocks, q = q−1 , q0 , and

q1 , with the same spacing q, that is, q±1 = q0 ± q. We set

them to q = −1, 0, and 1 for convenience. Thus the overlaps

between adjacent Q blocks is N (q, q ± 1) = e−1 by Eq. (8)

with the chosen value of λ. For the barrier, we set V (q =

±1) = 0 and V (q = 0) = 4d, giving a barrier height Bh =

4d. In each Q block, the energy cutoff for the many-body

configurations is set to be Ecut = V (q) + 5.5d. The neutron

absorption and the gamma decay occur prior to the fission

barrier, so the incident and the capture channels couple to the

internal states by Eq. (11) at q = −1. Likewise, the fission

channel is coupled at q = 1. All the states at these endpoints

are coupled to individual decay channels. Since the relation

n < cap < fis is known empirically in the actinide region

[24], we set γn = 0.001d, γcap = 0.01d, and γfis = 0.1d in the

following calculations. As we show in Sec. IIIC below, the

transmission dynamics is not sensitive to the value of γfis .

A. Orthogonal basis

We first consider the case where all configurations are

orthogonal so that the norm kernel reads

q j|N|q j   = δq,q δ j, j  .

(12)

In this case, the suppression factor in the off-diagonal couplings are disregarded; that is, the off-diagonal couplings are

fully taken into account without the suppression factor. This is

a useful limit to study the role of the pairing interaction, since

the diabatic interaction does not contribute.

It is a well-known fact that the pairing correlation modifies

drastically the dynamics of spontaneous fission, particularly

through a reduction of the collective mass [25–27]. Another

important aspect of the pairing correlation is that it is responsible for a hopping of Cooper pairs from one configuration to

the neighboring one [28]. On the other hand, the role of pairing correlation in induced fission has not yet been understood

well, partly because the pairing correlation is considered to

be effective only in the vicinity of the ground state. However,

odd-even staggerings have been observed in fission fragments

in low-energy induced fission, which suggests that the pairing

correlation cannot be completely ignored.

Figure 2 shows the transmission coefficients for the fission channel, calculated with two different values of Gpair .

The strength of the neutron-proton random interaction is set

to be vnp = 0.03d. One can see that the pairing correlation

enhances the transmission probabilities far below the barrier,

while its effect is not important at the barrier top and above.

This is to be expected, since the number of configurations

with high seniority numbers increases as the excitation energy

increases and the pairing correlation becomes weaker.

To study systematically the role of pairing in induced

fission, we introduce an energy-averaged transmission coefficient. It is defined as

 E +

E /2

Tn,fis (E ) =

dE  Tn,fis (E  ).

(13)

E E −

E /2

FIG. 2. The transmission coefficients from the incident channel

to the fission channel as a function of the excitation energy, E , in

the model with an orthogonal configuration space. The solid and the

dashed lines are obtained with Gpair = 0 and 0.1d, respectively, for

the strength of the pairing interaction. The strength of the neutronproton interaction and the barrier height are set to be vnp = 0.03d

and Bh = 4d, respectively.

Table I summarizes the energy averaged transmission coefficient at E = Bh = 4d for several sets of (vnp , Gpair ). The

energy window for the energy average is set to be E = d.

Without the neutron-proton interaction; that is, vnp = 0, the

fission probability increases as the pairing strength increases.

Note especially that the transmission coefficient Tn,fis (E )

is zero when there is no interaction at all. As the value of

vnp increases, the dependence of Tn,fis (E ) on Gpair becomes

milder. For vnp = 0.06d, the transmission coefficient is almost

insensitive to the value of Gpair . This suggests that induced

fission is more sensitive to the neutron-proton random interaction as compared with the coherent pairing interaction.

B. Nonorthogonal basis

Let us now examine the dependence on the interactions for

a model having a nonorthogonal basis. In this case, diabatically connected configurations have a off-diagonal coupling

due to the one-body terms in the Hamiltonian, in addition to

the couplings due to the two-body residual interactions. This

corresponds to the diabatic interaction (9) parametrized with

a quantity h2 . To avoid an artifact due to the degeneracy of

the single-particle energies, we introduce an offset energy to

the q = 1 block, taking V (q)/d = (0, 4, 0.5) in Eq. (3). We

mention that this problem appears much more prominently

TABLE I. The averaged transmission coefficient for a fission

process, Tn,fis , for several sets of the interaction parameters and

assuming that the configurations are orthogonal. The barrier height

and the incident energy are both set to be 4d.

Gpair

Model

II

III

024319-4

vnp

0.03d

0.06d

0.107

0.318

0.1d

0.0441

0.161

0.331

0.2d

0.0589

0.173

0.331

SCHEMATIC MODEL FOR INDUCED FISSION IN A …

PHYSICAL REVIEW C 108, 024319 (2023)

FIG. 3. The transmission coefficient Tn,fis (E ) with two different

values of vnp . The pairing interaction is set to zero, i.e., Gpair = 0.

The other parameters are λ = 1.0, h2 = 3d, and Bh = 4d.

with the nonorthogonal basis as compared with calculations

with the orthogonal basis, which could not be cured merely

by introducing random numbers to the Hamiltonian kernel.

Figure 3 shows the transmission probability for fission with

two different values of vnp . In these calculations, the pairing

interaction is switched off by setting Gpair = 0, while the parameter h2 for the diabatic transitions is set to be 3d. From the

figure, one notices that the peaks are lowered and broadened

as the value of vnp increases. This can be understood easily

since the random interaction spreads the spectrum in each Q

block as is indicated in Fig. 1. The effect of vnp is not only to

broaden the peaks in the transmission coefficients but also to

increase the energy averaged transmission coefficients, as will

be discussed in Table II below.

Figure 4 shows an average fission-to-capture branching ratio α −1 as a function of the energy E . We define the average as

dE  Tn,fis (E  )

(14)

α −1 = 

dE  Tn,cap (E  )

where the range of the integration is the same as that in

Eq. (13). To simplify the discussion, we once again set the

pairing interaction to be zero. The solid and the dashed lines

TABLE II. The transmission coefficient for fission Tn,fis  and the

branching ratio α −1 for several sets of interactions. The parameters

shown for models I–V1 are the only ones that differ from the base

model. The overlap parameter is λ = 1.0 and the averaged observables are calculated at a central energy E = 4d. Interaction strength

parameters are in units of d. For model VI, the GCM Hamiltonian is

constructed such that the orthogonal physical Hamiltonian Eq. (B15)

is diagonal.

Model

vnp

Gpair

h2

Tn,fis 

α −1

Base

II

III

IV

VI

0.03

0.3

0.0

0.417

0.413

0.372

0.429

0.294

0.172

0.000

1.62

1.49

1.14

2.27

0.739

0.291

0.000

0.0

0.05

0.0

0.0

0.0

0.0

0.0

0.0

0.0

FIG. 4. The average fission-to-capture branching ratios as a function of the energy with two different values of h2 . The random

neutron-proton interaction is taken into account with the strength of

vnp = 0.03d, while the pairing interaction is set to be zero.

are obtained with h2 = 3d and h2 = 0, respectively. The

branching ratios increase with the excitation energy, and we

confirmed that the energy dependence becomes stronger as

the number of Q blocks increases. This would be an expected

behavior from a quantum barrier transmission. Furthermore,

one sees that the diabatic interaction increases the branching

ratios, that is consistent with the result in Ref. [11].

Table II summarizes the transmission coefficients and the

branching ratios for several parameter sets. The results of

models I, II, and III indicate that both the neutron-proton interaction and the pairing interaction enhance the transmission

coefficients as well as the branching ratios. They also indicate

that the transmission coefficients are more sensitive to the

neutron-proton interaction than to the pairing interaction. This

is consistent with the results of the orthogonal basis shown in

Table I, even though the degree of enhancement is smaller

than in Table I due to the overlap factor N (q j, q j  ) in the

off-diagonal matrix elements. In model IV, h2 is set to be zero.

The result indicates that the transmission coefficient and the

branching ratio significantly decreases without the diabatic

transitions, as has been already observed in Ref. [11]. See also

Fig. 4 for the sensitivity of the branching ratios to the value

of h2 . Finally, in model V, all the interaction strengths vnp ,

Gpair , and h2 are set to zero. Even in this case, the transmission

coefficient is not zero because the corresponding Hamiltonian

in the orthogonal physical basis is not diagonal in this case. As

we show in Appendix B, one can actually construct the GCM

Hamiltonian which is diagonal with the orthogonal basis.

With such a GCM Hamiltonian, we have confirmed that the

transmission coefficient becomes zero within the numerical

error (see model VI in the table).

C. Validity of the transition-state hypothesis

In the Bohr-Wheeler theory for induced fission [3], the

decay width is calculated as a sum of transmission coefficients

Ti across the barrier via transition states i,

1 

Ti ,

(15)

BW =

2π ρ i

024319-5

K. UZAWA AND K. HAGINO

PHYSICAL REVIEW C 108, 024319 (2023)

TABLE III. Estimated orbital level spacing in 236 U. The first two

are from potential models and the last extracted from the Fermi gas

formula and measured level densities.

d (MeV)

0.45

0.51

0.33

FIG. 5. The branching ratios at E = 4d as a function of γfis .

The interaction strengths are (v pn , Gpair , h2 ) = (0.03, 0.3, 3)d. A

parabolic fission barrier is employed with the barrier height of 4d.

The solid and the dashed lines show the results of the 3Q model,

while the dot-dashed line shows the results of the 7Q model. While

the nonorthogonality of the configurations is neglected in the solid

line, it is taken into account in the other lines with λ = 1.0. For the

sake of presentation, the branching ratios are multiplied by factors of

0.2 and 0.04 for the solid and the dashed lines, respectively.

where ρ is the level density of a compound nucleus. The

formula indicates that the transition states entirely determine

the decay rate, and that the details of the dynamics after

crossing the barrier are unimportant. The branching ratio in

the Bohr-Wheeler theory would be expressed as

Ti .

(16)

α −1 (E ) =

2π ρ(E )cap i

The solid and the dashed lines in Fig. 5 show the branching

ratios at E = 4d as a function of γfis for a model with the

pairing interaction switched off. For the calculations with

the orthogonal basis shown by the solid line, the branching

ratio is almost independent of the fission decay width γfis , in

agreement with the insensitivity property of the Bohr-Wheeler

theory. On the other hand, with the nonorthogonal basis, the

branching ratio increases gradually as a function of γfis , even

though the insensitivity property may be realized at large

values of γfis . To check the dependence on the number of Q

blocks, we repeat the calculations with 7Q blocks, parametrizing V as V (q)/d = 4 − 4q2 /9 ranging from q = −3 to q = 3

with q = 1. In this case, the branching ratio changes by

less than a factor of two while the fission decay varies by

an order of magnitude. All of these results indicate that the

hypothesis used in the Bohr-Wheeler theory is easily realized

in the present microscopic theory. See also Ref. [29] for a

similar study with random matrices.

Source

Woods-Saxon well

FRLDM [31]

FGM [32]

ties to the different types of interaction, as well as the validity

of transition state theory in a microscopic framework. We have

shown that the transmission coefficients are mainly sensitive

to the neutron-proton interaction, while the sensitivity to the

pairing interaction is much milder. The diabatic transitions

were also found to play a role. Depending on the interaction and the deformation-dependent configuration space, one

achieves conditions in which branching ratios depend largely

on barrier-top dynamics and are insensitive to properties

closer to the scission point. The insensitive property is one of

the main assumptions in the well-known Bohr-Wheeler formula for induced fission, but up to now it had no microscopic

justification.

The results in this paper indicate that the neutron-proton

interaction is an important part of a microscopic theory for

induced fission. To include it in realistic calculations based on

the density-functional theory will require a large model space,

however. See Table III in Ref. [11] for some estimates of the

dimensional requirements. Moreover, single-particle energies

are in general not degenerate in contrast to the schematic

model employed in this paper. This might require a different

energy cutoff, further enlarging the model space. To carry out

such large-scale calculations for induced fission, one will have

to either validate an efficient truncation scheme or develop

an efficient numerical method to invert matrices with large

dimensions. We leave this for a future work.

ACKNOWLEDGMENTS

We are grateful to G.F. Bertsch for collaboration on

the early stage of this work. This work was supported in

part by JSPS KAKENHI Grants No. JP19K03861 and No.

JP21H00120. This work was supported by JST, the establishment of university fellowships towards the creation of

science technology innovation, Grant No. JPMJFS2123. The

numerical calculations were performed with the computer

facility at the Yukawa Institute for Theoretical Physics, Kyoto

University.

APPENDIX A: ESTIMATION

OF PHYSICAL PARAMETERS

IV. SUMMARY

In this article, we have applied the CI methodology to

a schematic model for neutron-induced fission. The model

Hamiltonian contains the pairing interaction, the diabatic

interaction, and a schematic off-diagonal neutron-proton interaction. The model appears to be sufficiently detailed to

examine the sensitivity of the fission transmission probabili-

1. Orbital energy spacing

The single-particle level spacing d in the uniform model

sets the energy scale for the model and does not play any

explicit role in the model. However, it is required to determine

other energy parameters which are expressed in units of d.

Several estimates of d for 236 U are given in Table III. The

024319-6

SCHEMATIC MODEL FOR INDUCED FISSION IN A …

PHYSICAL REVIEW C 108, 024319 (2023)

TABLE IV. Characteristics of single-particle orbitals in a deformed Woods-Saxon potential corresponding to 236 U at deformation

(β2 , β4 ) = (0.274, 0.168). Dashed lines indicate the Fermi level.

Protons

2K

−1

−1

-----1

−1

Neutrons

εKπ (MeV)

2K

−3.39

−3.80

−4.93

−5.43

−5.53

−5.74

−1

−1

−1

-----1

−1

εKπ (MeV)

−4.15

−4.25

−4.40

TABLE V. Estimates of neutron-proton interaction strength.

Basis of estimate

G matrix

sd shell spectra

β decay

v0 (MeV fm3 )

Citation

530

490

395, 320

[21]

[35]

[36]

the same J = 0 state. This gives another factor of nearly two

reduction in the multiplicity. The remaining task is to estimate

the fraction of K = 0 configurations in the unprojected quasiparticle space. The distribution of K values is approximately

Gaussian with a variance given by

−5.07

−5.75

−5.82

K 2  = nqp K 2 sp ,

first is based on orbital energies in a deformed Woods-Saxon

potential with the parameters given in Ref. [30]; see Table IV

for the calculated orbital energies. In more realistic theory,

the momentum dependence of the potential tends to increase

the spacing, but the coupling to many-particle degrees of

freedom decreases the spacing of the quasiparticle poles. The

combined effect seems to somewhat decrease the spacing.2

2. Level density

It is important to know the composition of the levels in

the compound nucleus to construct microscopic models that

involve those levels. For a concrete example, consider the

levels at the neutron threshold energy Sn = 6.5 MeV in 236 U.

The predominant configurations at this energy should be k

subblocks at k ≈ Sn /d in the independent quasiparticle approximation. Another approach that is less sensitive to the

residual interaction is to estimate the total number of states

below Sn and compare it to the number obtained by summing

the Nk degeneracies in the Q-block spectrum. In the 236 U

example, the combined level spacing of J π = 3− and 4− is

about 0.45 eV at Sn [34]. At that excitation energy the level

density is the same for even and odd parities, and it varies

with angular momentum as 2J + 1. The inferred level spacing of J π = 0+ levels is thus about 7 eV. The accumulative

number of levels can be approximated by N = ρT where T

is the nuclear temperature, defined as T = d ln(ρ(E ))/dE .

A typical estimate for our example is T = 0.65 MeV, giving

N ≈ 1.0 × 108 . To estimate the level density in the present

model, we start with the set of quasiparticle configurations

including both parities and all K values. The resulting k blocks

have multiplicities that are well fit by the formula

Nk ≈ exp(−3.23 + 4.414k 1/2 ).

(A1)

Projection on good parity decreases this by a factor of two.

The projection on angular momentum J = 0 is more subtle.

The J = 0 states are constructed by projection from K = 0

configurations; other configurations do not contribute. However, there may be two distinct configurations that project to

We note that an energy density functional fit to fission data [33]

obtained an effective mass in the single-particle Hamiltonian very

close to 1.

(A2)

where nqp  ≈ 8 is the average number of quasiparticles in

the k block and K 2 sp ≈ 6 is an average over the orbital Ks

near the Fermi level. Including these projection factors, the

integrated number of levels up to Sn is achieved by including

all k subblocks up to k = 17 in the entry Q block.

3. Neutron-proton interaction

To set the scale for our neutron-proton interaction parameter vnp we compare it with phenomenological contact

interactions that have been used to model nuclear spectra. The

matrix element of the neutron-proton interaction is

n1 p1 |v|n2 p2  = −v0 I,

where

I=

d 3 rφn∗1 (r)φ ∗p1 (r)φn2 (r)φ p2 (r).

(A3)

(A4)

The parameter v0 is the strength of the interaction, typically

expressed in units of MeV fm3 . Some values of v0 from the

literature are tabulated in Table V. We shall adopt the value

v0 = 500 MeV fm3 to estimate the value of vnp .

If the wave functions of the eigenstates approach the compound nucleus limit, the only characteristic we need to know

is its mean-square average among the active orbitals. We

have used the Woods-Saxon model to calculate the integral

Eq. (A4) for all the fully off-diagonal matrices of the orbitals

within 2 MeV of the Fermi energy. Figure 6 shows a histogram

of their distribution.3 . The variance of the distribution is

I 2 1/2 = 5.22 × 10−5 fm−3 . Combining this with our estimate of v0 we find (n1 p1 |v|n2 p2 2 )1/2 = 0.025 MeV. This

implies vnp ∼ 0.05d with our estimated single-particle level

density.

APPENDIX B: REACTION THEORY IN A

NONORTHOGONAL BASIS

The space of configurations used in this work is not orthogonal. This causes some conceptual issues but does not cause a

significant computational burden in CI-based reaction theory.

If the orbitals are restricted only to those in Table IV, the histogram is more structured

024319-7

K. UZAWA AND K. HAGINO

PHYSICAL REVIEW C 108, 024319 (2023)

However, those basis states are not well localized with respect

to the GCM coordinate.

The relationship between the Hamiltonians in the physical

and GCM bases can be expressed

0.08

distribution

0.06

0.04

H˜ = N −1/2 HN −1/2

(B7)

H = N 1/2 H˜ N 1/2 .

(B8)

or

0.02

The physical resolvent is related to the GCM resolvent by

-0.0002 -0.0001

0.0001 0.0002

-3

I (fm )

G˜ = (N −1/2 HN −1/2 − E 1)−1

= N 1/2 (H − E N )−1 N 1/2 .

FIG. 6. Integrals I in Eq. (A4) of orbitals near the Fermi energy.

The theory is based on calculating the resolvent of H; in an

orthogonal basis it is given by

G = (H − E 1)−1 ,

(B1)

where 1 is the unit matrix, H is the Hamiltonian, and E is the

energy of the reaction. Non-orthogonal bases also arise in the

theory of spontaneous decays [37], and in electron transport

theory when wave functions are built from atomic orbitals.

See, for example, Refs. [38–42] for the formulation of the resolvent as commonly used in chemistry and condensed-matter

physics.

In a nonorthogonal basis the time-dependent Schrödinger

equation reads

,

(B2)

H = i h¯ N

dt

where N is the overlap matrix between basis states Ni j = i| j.

The corresponding resolvent is

G = (H − E N )−1 .

(B3)

There is hardly any difference from Eq. (B1) from a computational point of view. However, the couplings to reaction

channels should be treated with care.

To understand the couplings, we define a certain orthogonal basis which we call the physical basis. We call the vector

representing a wave function in that basis v phys and in the

nonorthogonal basis, v gcm . In the GCM the dot products of

basis elements satisfy

v gcm (i)∗ · v gcm ( j) = Ni j ,

(B4)

while those in the physical basis satisfy

v phys (i)∗ · v phys ( j) = δi j .

(B5)

A physical basis consistent with Eq. (B4) can then be defined

by setting

 1/2

(B6)

v phys (i) =

Ni j v gcm ( j).

This definition is not unique since the dot products are invariant under a unitary transformation of the physical basis.

Indeed, an orthogonal basis is usually constructed in the GCM

by diagonalizing N and using its eigenvectors as the basis.

(B9)

One see that the matrix inversion is the same as in Eq. (B1)

except for the replacement 1 → N. However, the matrix N 1/2

appears as pre- and postfactors.

In our applications of CI-based reaction theory we assume

that each channel is coupled to a single state (the “doorway”

state) in the internal space. Taking that state to be the basis

state d in the physical representation, the decay coupling

matrix  has elements4

1/2 ˜

(i, j) = Nid1/2 N jd

,

(B10)

where ˜ is the decay width of the physical state d into the

channel. Note that with this construction the transmission

coefficient in the physical basis

˜ )˜ b G˜ † (E )]

Ta,b = Tr[˜ a G(E

(B11)

is transformed to

Ta,b = Tr[a G(E )b G† (E )]

(B12)

in the GCM basis.

There is another reason for explicit construction of the

physical basis. The distinction between the GCM and physical

basis must be taken into account in Sec. III B where we

assessed the relevant importance of different interaction types

and we want to start with a Hamiltonian H˜ 0 for which the

transmission probability vanishes. One cannot simply set the

off-diagonal elements of H to zero if the overlap matrix N

connects the entrance and exit channels, even if the connection is indirect. It is the physical Hamiltonian H˜ that must

be diagonal. In two dimensions the construction is obvious.

Given the diagonal elements of H (i, i) = Ei , the Hamiltonian

that is diagonal in the physical basis is

(E1 + E2 )N12 /2

E1

H =

(B13)

(E1 + E2 )N12 /2

E2

Eq. (B13) can be viewed as a justification for the first term

in Eq. (9). The construction can be carried out in higher

dimensions using only linear algebra operations, but we have

no simple formula for the off-diagonal elements of H 0 . For

024319-8

A somewhat similar formula was used in Ref. [[10], Eq. (16)].

SCHEMATIC MODEL FOR INDUCED FISSION IN A …

PHYSICAL REVIEW C 108, 024319 (2023)

the base Hamiltonian treated in Sec. III B, N is given by

1.0

N = ⎝e−1

−4

e−1

1.0

−1

e−4

e−1 ⎠.

(B14)

Keeping only V (q) in H, H 0 is numerically found to be

0.0000 0.7571 0.1537

H 0 = ⎝0.7571 4.000 0.8547⎠.

(B15)

0.1537

1.0

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