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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 14, Issue 1

# A New Algorithm for the Approximation of the Schrödinger Equation

Rong-an LIN
• Key Laboratory of Road and Traffic Engineering of the Ministry of Education, Tongji University. China Communications Construction Company Limited, China
• Other articles by this author:
/ Theodore E. Simos
• Corresponding author
• Department of Mathematics, King Saud University, Riyadh 11451, Saudi Arabia. Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, Tripolis, Greece
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• Other articles by this author:
Published Online: 2016-12-30 | DOI: https://doi.org/10.1515/phys-2016-0066

## Abstract

In this paper a four stages twelfth algebraic order symmetric two-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives is developed for the first time in the literature. For the new proposed method: (1) we will study the phase-lag analysis, (2) we will present the development of the new method, (3) the local truncation error (LTE) analysis will be studied. The analysis is based on a test problem which is the radial time independent Schrödinger equation, (4) the stability and the interval of periodicity analysis will be presented, (5) stepsize control technique will also be presented, (6) the examination of the accuracy and computational cost of the proposed algorithm which is based on the approximation of the Schrödinger equation.

## 1 Introduction

A new algorithm for the solution of the special second order problems: $φ″(x)=ζ(x,φ),φ(x0)=φ0andφ′(x0)=φ0′$(1)

is produced in this paper. Emphasis will be given on the problems (1) for which the solution behaves with periodical or/and oscillating form

The structure of the paper is:

• The basic theory for the construction of the new algorithm is given in Section 2.

• In Section 3 the new algorithm is constructed.

• The behavior of the error of the algorithm is shown in Section 4 using the Schrödinger equation as model problem.

• The stability analysis of the algorithm is shown in Section 5.

• In Section 6 the solution of a system of Schrödinger type equations is shown.

• Finally, in Section 7 we present remarks and conclusions.

## 2 Phase-lag and Stability Analysis of General Symmetric 2 k-Step Methods

We will present in this section the basic points of the phase-lag and the stability analysis of the general symmetric multistep algorithms. We consider the 2 k-Step algorithms: $∑i=−kkγiφn+i=h2∑i=−kkδiζ(xn+i,φn+i)$(2)

for approximation of (1) on the domain [α, β]. The domain is determined by the user taking into account the properties of the application. The 2 k-step algorithm (2) is applied to the problem (1) by dividing the domain [α, β] into 2 k areas of the same size i.e. ${\left\{{x}_{i}\right\}}_{i=-k}^{k}\in \left[\alpha ,\beta \right]$ h denotes the step length of the approximation procedure and is given by: $h=|xi+1−xi|,i=1−k(1)k−1.$(3)

We have the following definitions:

If for the algorithm (2) applies: $γ−i=γi,i=0(1)kδ−i=δi,i=0(1)k$

then we have a symmetric 2 k-step algorithm

The operator: $L(x)=∑i=−kkγiφ(x+ih)−h2∑i=−kkδiφ″(x+ih)$(4)

is related with the algorithm (2), where yC2.

[1] We call the algorithm (2) is of algebraic order p, if the operator L determined by (4) eliminates for every possible combination of linear form of the functions 1, x, x2, ... , xp+1.

In order to determine the stability polynomials for the specific algorithm (2) it is necessary to apply it to the problem $φ″=−ϕ2φ.$(5)

Then we achieve the following difference equation: $Ak(v)φn+k+…+A1(v)φn+1+A0(v)φn+A1(v)φn−1+…+Ak(v)φn−k=0$(6)

For the above equation we have that:

• v = ϕh,

• h is the stepsize and

• Aj (v)j = 0(1)m are the stability polynomials for the specific algorithm and for the problem (5).

An equation which is related with (6) is: $Ak(v)σk+…+A1(v)σ+A0(v)+A1(v)σ−1+…+Ak(v)σ−k=0.$(7)

which is known as the characteristic equation of the algorithm for the problem (5).

[6] If for the algorithm (2) we have the following roots σi , i = 1(1)2 k of its related equation (7) for all $v\in \left(0,{v}_{0}^{2}\right),$: $σ1=eiθ(v),σ2=e−iθ(v),and|σi|≤1,i=3(1)2k$(8)

then we say that the algorithm (2) has an non empty interval of periodicity $\left(0,{v}_{0}^{2}\right)$. In the formula (8) θ(v) denotes a real function of v

(see [6]) If the interval of periodicity of an algorithm (2) is computed as (0, ∞), then we call this algorithm P-stable.

If the interval of periodicity of an algorithm (2) is computed as (0, ∞)\S¹, then we call the algorithm singularly P-stable.

[4], [5] We call phase-lag of an algorithm (2) with the related characteristic equation (7) the dominant term of the expression $τ=v−θ(v).$(9)

The phase-lag is equal to t, if the relation τ = O(vt+1) as v → ∞ is hold.

[2] If for an algorithm (2) the phase-lag is zero, then this algorithm is called phase-fitted.

[4] In order to compute the order of the phase-lag t and the constant of the phase-lag c for an algorithm (2) with the related equation (7), we use the direct formula: $−cvt+2+O(vt+4)=2Ak(v)cos⁡(kv)+…+2Aj(v)cos⁡(jv)+…+A0(v)2k2Ak(v)+…+2j2Aj(v)+…+2A1(v)$(10)

## 3 The New Algorithm

We will examine the algorithm: $φ^n=φn−a0h2(ζn+1−2ζn+ζn−1)−2a1h2ζnφ^n+12=12(φn+φn+1)−h2[a2ζ^n+(18−a2)ζn+1]φ^n−12=12(φn+φn−1)−h2[a2ζ^n+(18−a2)ζn−1]φn+1+a3φn+φn−1=h2[b1ζn+1+ζn−1+b0ζn+b2(ζ^n+12+ζ^n−12)]$(11)

where ζi = φ″( xi,φi), i = −1 (1) 1 and aι, ι = 0(1)3 bλ λ = 0(1)2 are real numbers or functions of v, ${\stackrel{^}{\zeta }}_{n±\frac{1}{2}}={\phi }^{″}\left({x}_{n±\frac{1}{2}},{\stackrel{^}{\phi }}_{n±\frac{1}{2}}\right),{\stackrel{^}{\zeta }}_{n}={\phi }^{″}\left({x}_{n},{\stackrel{^}{\phi }}_{n}\right)$ and $a0=−1272150064$(12)

Application of the method (11) to the scalar test equation (5) leads to the difference equation given by (6) with: $A1(v)=1+v2(b1+b2(12+v2(−127a2v22150064+18−a2)−127v4a22150064))A0(v)=a3+v2(b0+b2(1+2v2a21+127v21075032+2a1v2))$(13)

The request of elimination of the phase-lag and phase-lag’s derivatives leads us to the following system of equations: $Phase−Lag(PL)=T0Tdenom=0$(14) $FirstDerivativeofthePhase−Lag=∂PL∂v=0$(15) $SecondDerivativeofthePhase−Lag=∂2PL∂2v=0$(16) $ThirdDerivativeofthePhase−Lag=∂3PL∂3v=0$(17) $FourthDerivativeofthePhase−Lag=∂4PL∂4v=0$(18) $FifthDerivativeofthePhase−Lag=∂5PL∂5v=0$(19)

where $T0=(−1075032+127v6b2a2+1075032a2−18⋅b2v4+−1075032b1−537516b2v2)cosv−2150064a1+1272150064a2b2v6−1075032b2v4a2+−537516b0−537516b2v2−537516a3Tdenom=−1075032+127v6b2a2+1075032a2−18b2v4+−1075032b1−537516b2v2$

The solution of the above equations gives us the following parameters of the algorithm: b0, b1, b2, a1, a2 and a3: $b0=12T1Tdenom1,b1=−3T2Tdenom1,b2=24T3Tdenom1,a1=T4Tdenom2,a2=44793T5Tdenom3,a3=−8T6Tdenom4$(20)

where Tj, j = 1(1)6, Tdenom1, Tdenom2, Tdenom3 and Tdenom4 are determined in Supplement Material A.

The avoidance of impossibility of definition of the parameters determined in (20) achieved via the the Taylor series expression given below: $b0=45053894−4067611271v455133728272480+542817613251v6771026811981208640+400038789030314774501v82920023166182228554796691200−83563301859074359249218492629v1057189780313016922673114208811398784000+…b1=10697788−4067611271v4330802369634880+180939204417v61542053623962417280+400038789030314774501v817520138997093371328780147200−41224781806998621124747166231v1012708840069559316149580935291421952000+…b2=−140649+4067611271v482700592408720−180939204417v6385513405990604320−400038789030314774501v84380034749273342832195036800+113646584530515487342985747177v1028594890156508461336557104405699392000+…a1=−131896+14231797v43350426302080−5636191273v623427297512910720+2595679492029592027v8328427588575781431622645760−589474110051330099313v102921854575569847650263762545600+…a2=7115600+3592051v41554891520000+1466269151v617791068771840000+28400625415155169381v815241919277361794361344000000+5695234233338513739511v10134997001620958863245341081600000+…$ $a3=−2+45469v14282893554944000+…$(21)

The plot of the parameters determined in (20) are shown in Figure 1.

Figure 1

Plot of the parameters determined in (20).

In (22) we determine the LTE of the algorithm (symbolized as 4 - St - 2S - 12 - 5D): $LTE4−St−2S−12−5D=−454691697361329664000h14(φn(14)−21ϕ4φn(10)−70ϕ6φn(8)−105ϕ8φn(6)−84ϕ10φn(4)−35ϕ12φn(2)−6ϕ14φn)+Oh16$(22)

## 4 Comparative Error Analysis

For the error analysis the following problem is used: $φ″(x)=V(x)−Vc+Γφ(x)$(23)

(1) V(x) is a function which is called potential, (2) Vc is real number denoting an approximate value of the function V(x) at x, (3) Γ = Vc - E, (4) The energy is denoted as E.

The above mentioned test equation is the time independent radial Schrödinger equation.

Our study will be focused on the local truncation errors of the following methods of the same family:

## 4.1 Classical Algorithm (i.e. the algorithm (11) with constant coefficients)

$LTECL=−454691697361329664000h14φn(14)+Oh16$(24)

## 4.2 Algorithm Produced in [23]

$LTE4−St−2S−12−1D=−454691697361329664000h14(φn(14)+6ϕ10φn(4)+5ϕ12φn(2))+Oh16$(25)

## 4.3 Algorithm Obtained in [25]

$LTE4−St−2S−12−1D=−454691697361329664000h14(φn(14)−15ϕ8φn(6)−24ϕ10φn(4)−10ϕ12φn(2))+Oh16$(26)

## 4.4 Algorithm Produced in Section 3

$LTE4−St−2S−12−5D=−454691697361329664000h14(φn(14)−21ϕ4φn(10)−70ϕ6φn(8)−105ϕ8φn(6)−84ϕ10φn(4)−35ϕ12φn(2)−6ϕ14φn)+Oh16$(27)

Our analysis is based on the following algorithm:

• The local truncation error (LTE) formulae are expressed via the test equation (23).

• More specifically,we substitute the derivatives of the function φ which are included in the LTE formulae, with results based on the problem (23). Some of the results are shown in Appendix A.

• The previous step leads to new formulae for the LTE which have the form:

$LTE=∑i=0mPiΓi$(28)

where Pi , i = 0, 1, . . . are quantities which can have two forms:

• real numbers in cases that the algorithm use coefficients which are real numbers or

• functions of v in the other cases.

• Two cases for the parameter Γ are examined:

• First Case: Vc E = Γ ≈ 0: The Potential and the Energy are closed each other. This leads to a form of the LTE given by (28) which is approximately equal to LTE = P0 (since Γj = 0, j = 1, 2, . . .). Consequently, all the formulae of the LTE of the algorithms under comparison are approximately the same i.e. LTEP0 for all the algorithms under comparison. This achievement leads to the conclusion that for these values of Γ, the algorithms under comparison are of comparable accuracy.

• Second Case: Γ >> 0 or Γ << 0. Consequently, |Γ| is a very big number. The Potential is much greater or much smaller than the Energy. Consequently, for the algorithms under comparison, the most accurate one is the algorithm with asymptotic form of the expression of its LTE given by (28) with the minimum value of m.

The above achievements lead to the following asymptotic forms of the LTE formuale:

## 4.5 Classical Algorithm

$LTECL=−454691697361329664000h14(φxΓ7+⋯)+Oh16$(29)

## 4.6 Algorithm Produced in [23]

$LTE4−St−2S−12−1D=−45469339472265932800h14[(3ξx2φx+31ξ″xφx+6ξ′x2φ′x)Γ5+⋯]+Oh16$(30)

## 4.7 Algorithm Produced in [25]

$LTE4−St−2S−12−2D=−4546921217016620800h14[(ξ″xφx)Γ5+⋯]+Oh16$(31)

## 4.8 Algorithm Produced in Section 3

$LTE4−St−2S−12−5D=−1106085083104000h14[(−1773291ξ(6)xφx−636566ξ(5)xφ′x−1909698ξ(4)xφxξx−4774245ξ(3)xφxξ′x−3182830ξ(2)x2φx)Γ3+⋯]+Oh16$(32)

We have the following theorem:

For the four symmetric two-step algorithms investigated in this section we have the following conclusions:

• The algebraic order of all algorithms under comparison is twelve.

• For the Classical algorithm, the error is dependent from the seventh power of Γ.

• For the algorithm developed in [23], the error is dependent from the fifth power of Γ.

• For the algorithm developed in [25], the error is dependent from the fifth power of Γ.

• For the algorithm produced in Section 3, the error is dependent from the third power of Γ.

The above conclusions lead to the following: For the approximation of the problem (23) the new produced in Section 3 algorithm is the most effective one in the cases of very big values of \Γ\.

## 5 Stability Analysis

Our study is based on the following chart:

• The scalar test equation for the stability analysis is given by:

$φ″=−ω2φ.$(33)

It is easy for one to see that ω ≠ ϕ.

• The difference equation mentioned below is produced by applying the algorithm to the problem (33):

$A1s,vφn+1+φn−1+A0s,vφn=0$(34)

where $A1s,v=T7Tdenom5,A0s,v=T8Tdenom6$(35)

where s = ω h and v = ϕ h and Tj, j = 7, 8, Tdenom5 and Tdenom6 are given in Supplement Material B. The s-v domain for the new algorithm is presented in Figure 2.

Figure 2

s - v domain of the new algorithm

Observing the s-v domain we have the following remarks:

- The algorithm is stable within the shadowed area of the s-v domain.

- The algorithm is unstable within the white area of the s-v domain.

In the cases of problems where where their mathematical models have only one frequency per differential equation, we have to observe on s - v domain the area where ω = ϕ or s = v.

In the cases where s = v, the interval of periodicity for the new algorithm is equal to: (0, ∞), i.e. it is P-stable.

We have the following theorem:

The new algorithm obtained in Section 3:

• is of twelfth algebraic order,

• has eliminated the phase-lag and its derivatives up to order four,

• has a s - v region presented in Figure 2

• is P-stable i.e. its interval of periodicity is equal to: (0, ∞) (in the cases where s = v).

## 6 Numerical Results

In this section we will present the application of the new developed method to the numerical solution of the coupled differential equations arising from the Schrödinger equation.

## 6.1 Error Estimation

The numerical example contains a variable-step algorithm and therefore an error estimation procedure.

We mention here that in the literature the last decades there are several variable step procedures for the approximation of the solution for systems of Schrödinger type equations [1–14].

Basic methodologies for the local truncation error estimation (see for example [15–27]) are based on:

• the order q in hq in the LTE expressions of the algorithms,

• the maximum order p in the expressions xp expw x) or in the expressions xp cos (±w x) and/or xp sin (±w x) which are the algorithm integrates exactly,

• order of the phase-lag of the algorithms (i.e. zero order of the phase-lag is for eliminated phase-lag algorithms, first order of the phase-lag is for the algorithms with eliminated the phase-lag the derivative of the first order, n-th order of the phase-lag order is for the algorithms with eliminated phase-lag and its derivatives up to order n.

We will use the first of the above mentioned methodologies. Our embedded pair is based

• on the order q in hq in the LTE expressions of the algorithms

• on the fact that in order to obtain highly accurate numerical solutions, the maximum algebraic order q of an algorithm must be used.

The local truncation error in ${y}_{n+1}^{L}$ is estimated by $LTE=∣yn+1H−yn+1L∣$(36)

where ${y}_{n+1}^{L}$ is the low order approximation and we use for this approximation the algorithm obtained in [24] and ${y}_{n+1}^{H}$ is the high order approximation and we use for this approximation the algorithm produced in Section 3.

For our numerical example we reduced the changes of the step sizes on its duplication . More specifically:

• if LTE < acc then the step size is duplicated, i.e. hn+1 = 2hn.

• if accLTE ≤ 100 acc then the step size remains stable, i.e. hn+1 = hn.

• if 100 acc < LTE then the step size is halved and the step is repeated, i.e. ${h}_{n+1}=\frac{1}{2}\phantom{\rule{thinmathspace}{0ex}}{h}_{n}$

where hn the size of the step used for the nth step of integration and acc is the accuracy defined by the user.

For our numerical tests we used the local extrapolation technique, i.e. while we use the lower algebraic order solution ${y}_{n+1}^{L}$ for an estimation of the local truncation error less than acc, we accept at each point of integration the higher algebraic order solution ${y}_{n+1}^{H}$ as approximation of the solution.

## 6.2 Coupled differential equations

The coupled differential equations of the Schrödinger type can be found in several scientific areas. The general form of the close-coupling differential equations arising from the Schrödinger equation is of the form: $d2dx2+ki2−li(li+1)x2−Viiφij=∑m=1NVimφmj$(37)

for 1 ≤ iN and mi.

There are two cases for the above system of differential equations: (1) the open channels case and (2) the close channels case. We will study the first case with conditions on boundaries [16] (we note here that the new algorithm can be applied to close channels case also): $φij=0atx=0$(38) $φij∼kixjli(kix)δij+kikj1/2Kijkixnli(kix)$(39)

where j1(x) are spherical Bessel functions and n1(x) are spherical Neumann functions.

For the details on the model of the problem one can see [16]. Based on the analysis presented in [16], the following matrix K′ and diagonal matrices M, N are defined as: $Kij′=kikj1/2KijMij=kixjli(kix)δijNij=kixnli(kix)δij$

and the boundary condition (39) now is given by: $y∼M+NK′$

The close-coupling differential equations arising from the Schrödinger equation of the our numerical test describe, under environment of neutral particle impact, the phenomenon of the rotational excitation of a diatomic molecule. We use the same notations, as in [16]:

• for the entrance channel we use the quantum numbers (j, l),

• for the exit channels we use the quantum numbers (j′ , l′) and

• for the total angular momentum we use the notation J =j+l = j′ + l′.

Based on the above notations we have the following form of the problem: $d2dx2+kj′j2−l′(l′+1)x2φj′l′Jjl(x)=2μℏ2∑j″∑l″φj″l″Jjl(x)$(40)

where $kj′j2=2μℏ2[E+ℏ22I{j(j+1)−j′(j′+1)}]$(41)

E is the energy of the corresponding system, I is the moment of inertia of the rotator, and µ is the mass of the corresponding system.

The function V is given by ([16]): $V(x,k^j′jk^jj)=V0(x)P0(k^j′jk^jj)+V2(x)P2(k^j′jk^jj),$(42)

The coupling matrix element can be written as $=δj′j″δl′l″V0(x)+f2(j′l′,j″l″;J)V2(x)$(43)

where the f2 coefficients can are given via the formulae mentioned in Bernstein et al. [18] and ${\stackrel{\mathbf{^}}{\mathbf{k}}}_{{j}^{\prime }j}$ is a vector parallel to the vector kj’j and Pi , i = 0, 2 are Legendre polynomials (see for details [19]). Now we have the following form for the boundary conditions : $φj′l′Jjl(x)=0atx=0$(44) $φj′l′Jjl(x)∼δjj′δll′exp⁡[−i(kjjx−1/2lπ)]−kikj1/2SJ(jl;j′l′)exp⁡[i(kj′jx−1/2l′π)]$(45)

where: $S=(I+iK)(I−iK)−1$(46)

A program was developed for the numerical solution of this problem. With this program we calculated cross sections for rotational excitation of molecular hydrogen by impact of various heavy particles ([16], [17]). We included also a subroutine for the numerical integration from the initial value to the matching points and we applied the variable step method described in the previous subsection.

We made the computations using the following parameters: $2μℏ2=1000.0,μI=2.351,E=1.1,V0(x)=1x12−21x6,V2(x)=0.2283V0(x).$

As we mentioned previously we followed the procedure described in [16]. We taken the values J = 6 and from j = 0 to j′ = 2, 4 and 6 and therefore, we have sets of four, nine and sixteen coupled differential equations, respectively. We used the methodology obtained by Bernstein [19] and Allison [16] and consequently, for x less than some x0, the potential tends to infinite. For this case we have for x0: $yj′l′Jjl(x0)=0$(47)

We solved the above described problems using the algorithms:

• the Iterative Numerov algorithm of Allison [16] which is symbolized as Algorithm I,

• the variable-step algorithm of Raptis and Cash [15] which is symbolized as Algorithm II,

• the embedded Runge-Kutta Dormand and Prince algorithm 5(4) [13] which is symbolized as Algorithm III,

• the embedded Runge-Kutta algorithm ERK4(2) introduced in Simos [20] which is symbolized as Algorithm IV,

• the introduced embedded symmetric two-step algorithm introduced in [22] which is symbolized as Algorithm V

• the introduced embedded symmetric two-step algorithm introduced in [23] which is symbolized as Algorithm VI

• the introduced embedded symmetric two-step algorithm introduced in [23] which is symbolized as Algorithm VII

• the introduced embedded symmetric two-step algorithm introduced in [25] which is symbolized as Algorithm VIII

• the introduced embedded symmetric two-step algorithm introduced in [26] which is symbolized as Algorithm IX

• the introduced embedded symmetric two-step algorithm introduced in this paper which is symbolized as Algorithm X

Table 1 shows the computational cost (in seconds) requested by the algorithms I-X mentioned above for the computation of | S |2 for the system of N differential equations. The same Table shows also the maximum error of the computation of | S |2. N {4, 9, 16} symbolizes the number of equations of the system of the Schrödinger type differential equations.

Table 1

System of Differential Equations of the Schrödinger type. Computational cost (in seconds) (CC) and maximum value of the absolute error (MErr) of the computation of | S |2 for the Algorithm I -Algorithm X. acc=10−6. hmax: maximum step length

## 7 Conclusions

A new algorithm with eliminated phase-lag and its derivatives up to order five is produced, for the first time in the literature, in the present paper. More specifically for the new algorithm we presented:

1. its construction,

2. the analysis of its error,

3. the analysis of its stability,

4. the analysis of its effectiveness by applying it to the numerical solution of systems of Schrödinger type equations.

The theoretical and numerical achievements lead to the conclusion that the new algorithm is more effective than other known ones or recently constructed for the approximation of systems of equations of the Schrödinger type.

We used for our computations a IBM PC-AT compatible 80486 using arithmetic with 16 significant digits accuracy (IEEE standard).

## Appendix A Expressions of the Derivatives of φn

Here we present expressions of the derivatives of φn using the test equation (23): $φn(2)=V(x)−Vc+Γφ(x)φn(3)=ddxξxφx+ξx+Γddxφxφn(4)=d2dx2ξxφx+2ddxξxddxφx+ξx+Γ2φxφn(5)=d3dx3ξxφx+3d2dx2ξxddxφx+4ξx+Γφxddxξx+ξx+Γ2ddxφxφn(6)=d4dx4ξxφx+4d3dx3ξxddxφx+7ξx+Γφxd2dx2ξx+4ddxξx2φx+6ξx+Γddxφxddxξx+ξx+Γ3φxφn(7)=d5dx5ξxφx+5d4dx4ξxddxφx+11ξx+Γφxd3dx3ξx+15ddxξxφxd2dx2ξx+13ξx+Γddxφxd2dx2ξx+10ddxξx2ddxφx+9ξx+Γ2φxddxξx+ξx+Γ3ddxφxφn(8)=d6dx6ξxφx+6d5dx5ξxddxφx+16ξx+Γφxd4dx4ξx+26ddxξxφxd3dx3ξx+24ξx+Γddxφxd3dx3ξx+15d2dx2ξx2φx48ddxξxddxφxd2dx2ξx+22ξx+Γ2φxd2dx2ξx+28ξx+Γφxddxξx2+12ξx+Γ2ddxφxddxξx+ξx+Γ4φx…$

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## SUPPLEMENT MATERIAL A

T1 = 193505760 – 86002560v2 – 193505760 cos (v)

+ 17200512v4 – 5786844 v6 + 7112 v10 + 50800 v8

− 28448 cos (v) sin (v) v9 + 26670 (cos (v))3 sin (v) v5

− 11430 (cos (v))3 sin (v) v7 – 4645772 (cos (v))2 sin (v) v7

+ 11413668 cos (v) sin (v) v5 + 43001280 cos (v) sin (v) v3

− 45720 cos (v) sin (v) v7 – 1016 (cos (v))3 sin (v) v9

− 265670370 (cos (v))2 sin (v) v3 + 96752880 (cos (v))2 sin (v) v

− 387011520 cos (v) sin (v) v + 179426 (cos (v))2 sin (v) v9

− 48798386 (cos (v))2 sin (v) v5 + 193505760 (cos (v))3

− 193505760 (cos(v))2 + 290258640v sin (v)

− 5023928 v9 sin(v) – 14704804 v7 sin(v)

− 104763652v5 sin(v) + 319421970 v3 sin(v)

+ 127 (cos (v))4 v10 + 3810 (cos (v))4 v8

+ 2150064 (cos (v))3 v8 + 25800768 (cos (v))3 v6

+ 25600743 (cos (v))2 v4 + 279508320 (cos (v))2 v2

− 397761840 cos (v) v2 + 66675 (cos(v))4 v6

+ 200025 (cos (v))4 v4 + 156954672 (cos (v))3 v4

+ 204256080 (cos (v))3 v2 + 8636 (cos (v))2 v10

+ 91440 (cos (v))2 v8 – 20067264 cos (v) v8

− 1220016 (cos (v))2 v6 – 109958268 cos (v) v6

− 103203072 cos (v) v4

T2 = −387011520 + 494514720 v2 + 387011520 cos(v)

+ 223606656v4 – 160538112 v6 + 5733504v8

− 1475921 (cos (v))2 sin (v) v7

− 151937856 cos (v) sin (v) v5

− 731021760 cos (v) sin (v) v3

− 194305860 (cos (v))2 sin (v) v3

− 193505760 (cos (v))2 sin (v) v

+ 774023040 cos (v) sin (v) v – 4826 (cos (v))2 sin (v) v9

− 26018573 (cos (v))2 sin (v) v5

− 127 (cos (v))2 sin (v) v11

− 387011520 (cos(v))3 + 387011520 (cos(v))2

− 580517280v sin (v) + 35814v9 sin (v)

+ 40079918v7 sin (v)

+ 220210949v5 sin(v) – 558216540 v3 sin(v)

+ 33020 (cos (v))3 v8 – 8413566 (cos (v))3 v6

+ 12900384 (cos (v))2 v4 – 881526240 (cos (v))2 v2

+ 1118033280 cos(v) v2 – 120003534 (cos(v))3 v4

− 731021760 (cos (v))3 v2 + 1433376 (cos (v))2 v8

+ 148590 cos (v) v8 + 45868032 (cos (v))2 v6

+ 80429076 cos (v) v6 + 335009934 cos(v)v4

+ 762 (cos (v))3 v10 – 7112 cos (v) v10

+ 3556 v11 sin (v)

T3 = 96752880 – 43001280v2 – 96752880 cos(v)

+ 8600256v4 – 2866752 v6 + 145129320vsin(v)

− 3556 v9 sin (v) – 10039982 v7 sin (v)

− 55069406v5 sin(v) + 139554135 v3 sin(v)

+ 2150064 (cos (v))3 v6 + 12900384 (cos (v))2 v4

+ 139754160 (cos (v))2 v2 – 279508320 cos(v)v2

+ 30100896 (cos (v))3 v4

+ 182755440 (cos (v))3 v2

− 716688 (cos (v))2 v6 – 20040594 cos(v)v6

+ 21500640 cos (v) sin (v) v3

− 193505760 cos (v) sin (v) v

+ 364694 (cos (v))2 sin (v) v7

+ 48376440 (cos (v))2 sin (v) v

+ 5733504 cos (v) sin (v) v5

+ 48576465 (cos (v))2 sin (v) v3

+ 127 (cos (v))2 sin (v) v9

+ 6507977 (cos (v))2 sin (v) v5

+ 96752880 (cos (v))3 – 96752880 (cos (v))2

− 83852496 cos (v) v4

T4 = 2032 sin (v) (cos (v))3 v6

+ 56896 cos (v) sin (v) v6

− 360630 (cos (v))2 sin (v) v6

− 48393585 (cos (v))2 sin (v) v2

− 127 (cos (v))2 sin (v) v8

+ 30480 cos (v) sin (v) v4

+ 7620 sin (v) (cos (v))3 v4

− 68580 sin (v) (cos (v))3 v2

− 6467337 (cos (v))2 sin (v) v4

+ 137160 v2 sin (v) cos (v)

− 48376440 (cos (v))2 sin (v)

− 96804315 v2 sin(v)

+ 3556 sin (v) v8 + 10053190 sin (v) v6

+ 6400662 sin (v) v4 + 96851940 v3 cos (v)

+ 34290 cos (v) v + 243840 (cos (v))2 v3

− 34290 (cos (v))2 v + 34290 (cos (v))4 v

− 4572 (cos (v))4 v5 – 49530 (cos (v))4 v3

− 34290 (cos (v))3 v + 254 (cos (v))4 v7

− 10668 (cos (v))3 v5 – 64770 (cos (v))3 v3

− 762 (cos (v))3 v7 + 7112 cos (v) v7

+ 17272 (cos (v))2 v7 – 18288 (cos (v))2 v5

+ 29718 cos (v) v5 + 14224v7

+ 60960v5 – 228600 v3 + 48376440 sin(v)

T5 = (cos (v))2 sin (v) v7 + 6 (cos (v))3 v6

+ 18 (cos (v))2 sin (v) v5 – 28 v7 sin (v)

+ 84 (cos (v))3 v4 – 56 cos (v) v6

+ 135 (cos (v))2 sin (v) v3 – 154 v5 sin (v)

+ 510 (cos (v))3 v2 – 234 cos (v) v4

+ 135 (cos (v))2 sin (v) v + 390 v3 sin (v)

+ 270 (cos (v))3 – 780 cos (v) v2

+ 405 v sin (v) – 270 cos (v)

T6 = 72564660 cos(v) + 22860v4

− 60960v6 + 60470550v sin (v)

+ 2508408v7 sin (v) – 47838924v5 sin (v)

+ 72564660 v3 sin (v) – 1612548 (cos (v))3 v6

+ 76200 (cos (v))2 v4 – 120015 (cos (v))2 v2

− 48376440 cos (v) v2 + 33325992 (cos (v))3 v4

+ 120941100 (cos (v))3 v2 + 71628 (cos (v))2 v6

+ 15050448 cos (v) v6 – 8636 (cos (v))2 v8

− 127 (cos (v))4 v8 + 8382 (cos (v))4 v6

+ 55245 (cos (v))4 v4 + 120015 (cos (v))4 v2

− 7112 v8 – 68580 cos (v) sin (v) v3

− 89586 (cos (v))2 sin (v) v7

− 205599870 (cos (v))2 sin (v) v

+ 15240 cos (v) sin (v) v5

− 12094110 (cos (v))2 sin (v) v3

+ 11287836 (cos (v))2 sin (v) v5

− 72564660 (cos (v))3 – 105890652 cos (v) v4 + 34290 (cos (v))3 sin (v) v3

+ 56896 cos (v) sin (v) v7

+ 2032 (cos (v))3 sin (v) v7

+ 3810 (cos (v))3 sin (v) v5

Tdenom1 = v4 -290258640 – 387011520v2

+ 197805888v4 + 8600256 v6

+ 3810 (cos (v))2 sin (v) v7 + 17200512 cos (v) sin (v) v5

+ 193505760 cos (v) sin (v) v3

+ 257175 (cos (v))2 sin (v) v3

+ 580517280 cos (v) sin (v) v

− 127 (cos (v))2 sin (v) v9

+ 165735 (cos (v))2 sin (v) v5 + 290258640 (cos (v))2

+ 3556 v9 sin (v) + 65278 v7 sin (v)

+ 300990 v5 sin (v) – 120015 v3 sin (v)

+ 1524 (cos (v))3 v8 + 27432 (cos (v))3 v6

− 4300128 (cos (v))2 v4

+ 96752880 (cos(v))2 v2 + 480060 cos(v)v2

− 175260 (cos (v))3 v4 – 480060 (cos (v))3 v2

− 14224 cos (v) v8 + 2150064 (cos (v))2 v6)

− 297942 cos(v)v6 – 441960 cos (v) v4

Tdenom2 = 2150064 v (cos (v))2 sin (v) v7

+ 6 (cos (v))3 v6 + 18 (cos (v))2 sin (v) v5

− 28 v7 sin (v) + 84 (cos (v))3 v4

− 56 cos (v) v6 + 135 (cos (v))2 sin (v) v3

− 154 v5 sin (v) + 510 (cos (v))3 v2

− 234 cos (v) v4 + 135 (cos (v))2 sin (v) v

+ 390 v3 sin (v) + 270 (cos (v))3

− 780 cos (v) v2 + 405 v sin (v))

− 270 cos (v)

Tdenom3 = 96752880 – 43001280v2 – 96752880 cos(v)

+ 8600256 v4 – 2866752v6 + 145129320 vsin(v)

− 3556 v9 sin (v) – 10039982 v7 sin (v)

− 55069406v5 sin (v) + 139554135 v3 sin(v)

+ 2150064 (cos (v))3 v6

+ 12900384 (cos (v))2 v4

+ 139754160 (cos (v))2 v2

− 279508320 cos (v) v2 + 30100896 (cos (v))3 v4

+ 182755440 (cos (v))3 v2

− 716688 (cos(v))2 v6 – 20040594 cos(v)v6

+ 21500640 cos (v) sin (v) v3

− 193505760 cos (v) sin (v) v

+ 364694 (cos (v))2 sin (v) v7

+ 48376440 (cos (v))2 sin (v) v

+ 5733504 cos (v) sin (v) v5

+ 48576465 (cos (v))2 sin (v) v3

+ 127 (cos (v))2 sin (v) v9

+ 6507977 (cos (v))2 sin (v) v5

+ 96752880 (cos (v))3 – 96752880 (cos (v))2

− 83852496 cos(v)v4

Tdenom4 = 290258640 + 387011520v2 – 197805888 v4

− 8600256 v6 – 3556 v9 sin(v)

− 65278 v7 sin (v) – 300990 v5 sin (v)

+ 120015 v3 sin (v) – 27432 (cos (v))3 v6

+ 4300128 (cos (v))2 v4

− 96752880 (cos (v))2 v2

− 480060 cos (v) v2 + 175260 (cos (v))3 v4

+ 480060 (cos (v))3 v2

− 2150064 (cos (v))2 v6

+ 297942 cos (v) v6 – 1524 (cos (v))3 v8

+ 14224 cos (v) v8 – 193505760 cos (v) sin (v) v3

− 580517280 cos (v) sin (v) v

− 3810 (cos (v))2 sin (v) v7

− 17200512 cos (v) sin (v) v5

− 257175 (cos (v))2 sin (v) v3

+ 127 (cos (v))2 sin (v) v9

− 165735 (cos (v))2 sin (v) v5

− 290258640 (cos(v))2 + 441960 cos(v)v4

## SUPPLEMENT MATERIAL B

T7 = 580517280 v4 + 1354540320v6

− 1011555 sin(v)s2v7

+ 201930 cos(v)s6v2 + 2293620 cos(v) v8

− 480060 cos(v)v6 – 1369695 v9 sin(v)

+ 222885 v7 sin (v) + 40005 s2v5 sin (3 v)

+ 600075s4v3 sin (3 v) – 1935057600s2v2 cos(2v)

− 1161034560 v5 sin(2v) + 175260v8 cos(3 v)

+ 8600256 s2v8 cos (2v) + 52324 cos(v) v12

− 580517280v4 cos(2v) – 257175v7 sin (3v)

− 51435 s2v7 sin (3 v) + 99060 s2v8 cos (3 v)

− 10668 s6v4 cos (3 v) – 12954 s2v9 sin (3 v)

− 3810 v11 sin (3 v) – 1524 v12 cos (3 v)

− 38701152v10 – 782623296 v8 + 320040 cos (v) s4v6

+ 2080260 cos (v) s2v8 – 387011520v7 sin (2 v)

− 38701152s4v6 + 180605376 s4v4

+ 322509600s4v2 – 1161034560s4vsin(2 v)

+ 2286 s2v10 cos (3 v) – 127 s6v7 sin (3 v)

− 4300128 s4v6 cos (2v) + 77402304s4v4 cos(2v)

+ 4880610 cos (v) s2v6 + 1560195 sin(v) s4v5

− 17145 s6v sin (3 v) + 26162 cos (v) s6v6

+ 14097 sin (v) s6v7 – 1800225 sin (v) s4v3

− 381 s2v11 sin (3 v) + 34290 cos (v) s6

+ 34401024s4v5 sin(2 v) + 77402304 s2v8

− 14097 sin (v) v13 – 42291 sin (v) s4v9

+ 42291 sin (v) s2v11 – 64770 s6v2 cos (3 v)

− 193505760 v6 cos (2 v) + 480060 v6 cos(3v)

+ 19050 s4v7 sin (3 v) – 264922 v11 sin (v)

+ 1109472 cos(v)v10 – 17145 s6v3 sin (3v)

+ 127v13 sin (3 v) – 4300128v10 cos(2 v)

+ 75946 sin (v) s6v5 + 246126 sin (v) s2v9

− 774023040 s2v5 sin (2v) + 129003840 s4v3 sin(2v)

+ 381 s4v9 sin (3 v) – 3870115200 s2v3 sin (2 v)

− 165735 v9 sin(3 v) – 78486 cos(v)s2v10

− 1806053760s2v6 + 3483103680s2v4

+ 1935057600s2v2

− 57150 sin (v) s4v7 + 86868 cos (v) s6v4

+ 580517280 s4 + 8600256 v8 cos (2 v)

− 215265 sin (v) s6v3 – 34290 s6 cos (3 v)

− 762s6v6 cos(3v) + 387011520s2v4 cos(2v)

− 580517280 s4 cos(2 v) – 760095 sin (v)s2v5

− 2286 s6v5 sin(3v) + 258007680s2v6 cos(2v)

+ 560070 s2v6 cos (3v) – 34401024v9 sin (2 v)

+ 173355 s4v5 sin (3 v) – 222885 sin (v) s6v

+ 1200150s2v4 cos(3 v) + 838524960s4v2 cos(2v)

− 27432v10 cos(3v) – 1200150 cos(v)s2v4

T8 = −1161034560 s2v5 cos (3 v) + 254s6v7 cos(4v)

− 220980 v9 cos(4v) + 290258640 sin(v)s6

− 4572 s6v5 cos (4 v) – 2613660 v9 + 480060v7

+ 79552368 sin(v)s6v6 + 183642s6v7

− 550926 s2v11 + 400812s6v5 – 400050s2v7 cos(4v)

− 480060 v7 cos(4v) – 258007680 s2v7 cos (3v)

+ 3773362320 sin(v) s4v2 – 2103120 v9 cos (2v)

+ 64501920s2v8 sin(3 v) + 777240 s6v3 cos (2 v)

− 442913184 cos(v) v11 + 508 v13 cos(4 v)

+ 140208 v13 cos (2v) – 79552368 sin (v) v12

+ 180605376 s4v5 cos(3 v) + 34290s6v cos(4v)

− 91440s6v5 cos(2v) – 4701540s2v9

− 1002030s6v3 – 3760470 s2v7

+ 2588677056 cos(v)v9

+ 741772080 s2v6 sin (3v) – 96752880s6 sin(3 v)

+ 367284 v13 + 704088v11 – 49530 s6v3 cos(4 v)

− 15240v10 sin(4 v) – 926592 v12 sin(2v)

− 12900384 s6v4 sin(3v) + 411480s6v2 sin (2 v)

− 580517280 cos(v)s4v + 1440542880 sin (v) v10

+ 290258640s4v4 sin (3 v) – 290258640 sin (v) v6

− 1354540320 cos (v) v7 + 231648 s6v6 sin(2v)

− 8128v12 sin (4 v) + 2418822000 sin (v) s2v4

− 6720840s2v7 cos(2v) – 762 s2v11 cos(4 v)

− 580517280 cos(v)v5 – 2225316240 sin (v) v8

− 210312 s2v11 cos(2 v) + 3644358480 sin(v)s4v4

− 137160 v8 sin(4v) – 967528800v7 cos(3v)

− 1200150s2v5 cos(4v) + 1096532640 s4v3 cos (3 v)

+ 38701152 s4v6 sin(3v) + 70104 s6v7 cos (2 v)

+ 12900384 s4v7 cos (3 v) – 22860s2v9 cos(4 v)

− 33528v11 cos (4v) – 1280160 v11 cos (2 v)

+ 137160 s6v4 sin (2v) + 1234440 s2v8 sin(2v)

+ 68580s2v8 sin(4 v) – 96752880 s6v2 sin(3v)

+ 7620 s6v4 sin (4 v) + 2150064 s4v8 sin (3 v)

− 2150064s2v10 sin(3 v) + 1935057600 s2v3 cos (3v)

+ 38701152 sin(v)s6v4 – 1935057600 cos (v)s2v3

− 238657104 sin (v) s4v8 + 960120s2v6 sin (2 v)

+ 822960 v8 sin(2v) + 716688v12 sin (3 v)

− 90302688 v10 sin(3v) – 1285738272 sin (v) s4v6

− 25800768 s2v9 cos (3 v) – 160020 s2v6 sin(4v)

− 68580s6v2 sin(4 v) – 34290s6v + 1200150s2v5

− 716688 s6v6 sin (3v) – 266607936 v9 cos(3v)

+ 12900384 v11 cos (3 v) – 442913184 cos(v) s4v7

+ 6096s2v10 sin (4 v) + 885826368 cos(v) s2v9

− 3418601760 cos (v) s4v3 + 774023040 cos(v)s6v3

− 1470643776 cos (v) s4v5 + 2580076800 cos (v) s2v7

+ 2032s6v6 sin (4 v) + 4353879600s2v4 sin (3 v)

+ 238657104 sin (v) s2v10 – 6579195840 cos(v)s2v5

+ 580517280s4v cos(3v) + 483764400 sin (v) s2v6

+ 580517280v5 cos(3 v) – 2286000s2v9 cos(2v)

− 193505760 sin (v) s2v8 + 290258640s4v2 sin(3 v)

+ 694944 s2v10 sin (2 v) – 274320v10 sin(2 v)

+ 96752880 v8 sin(3v) + 1644798960 v6 sin(3v)

− 870775920 sin (v) s6v2

Tdenom5 = 580517280 v4 + 1354540320 v6

+ 2293620 cos(v)v8

− 480060 cos(v) v6 – 1369695 v9 sin(v)

+ 222885v7 sin (v) – 1161034560v5 sin (2 v)

+ 175260v8 cos(3v) + 52324 cos (v) v12

− 580517280 v4 cos (2v) – 257175v7 sin (3 v)

− 3810 v11 sin (3 v) – 1524 v12 cos (3 v)

− 38701152v10 – 782623296 v8 – 387011520v7 sin (2 v)

− 14097 sin (v) v13 – 193505760v6 cos(2 v)

+ 480060v6 cos(3v) – 264922 v11 sin (v)

+ 1109472 cos (v) v10 + 127 v13 sin(3v)

− 4300128 v10 cos (2 v) – 165735 v9 sin(3 v)

+ 8600256 v8 cos (2 v) – 34401024v9 sin (2 v)

− 27432 v10 cos(3 v)

Tdenom6 = −782623296v9 + 1354540320v7

+ 580517280 v5 + 8600256v9 cos(2 v) + 52324 cos(v)v13

− 1524 cos (3v) v13 – 580517280v5 cos(2 v)

+ 1109472 cos (v) v11 – 264922 sin (v) v12

+ 2293620 cos (v) v9 – 38701152 v11 – 1369695 sin(v)v10

− 480060 cos(v)v7 – 1161034560v6 sin (2 v)

− 14097 sin (v) v14 + 127 sin (3 v) v14

+ 222885 sin(v) v8 + 480060v7 cos(3 v)

− 4300128 v11 cos (2 v) – 387011520 v8 sin(2v)

− 3810v12 sin(3 v) – 165735 v10 sin(3v)

+ 175260v9 cos(3v) – 27432 v11 cos (3v)

− 34401024v10 sin (2 v) – 193505760 v7 cos (2 v)

− 257175v8 sin (3 v)

Accepted: 2016-08-04

Published Online: 2016-12-30

Published in Print: 2016-01-01

Citation Information: Open Physics, Volume 14, Issue 1, Pages 628–642, ISSN (Online) 2391-5471,

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## Citing Articles

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[2]
Fei Hui and T. E. Simos
Journal of Mathematical Chemistry, 2019, Volume 57, Number 4, Page 1088
[3]
[4]
Zhong Chen, Chenglian Liu, and T. E. Simos
Journal of Mathematical Chemistry, 2018, Volume 56, Number 9, Page 2591
[5]
Ke Yan and T. E. Simos
Journal of Mathematical Chemistry, 2018, Volume 56, Number 8, Page 2454
[6]
Nan Yang and T. E. Simos
Journal of Mathematical Chemistry, 2019, Volume 57, Number 3, Page 895
[7]
V. N. Kovalnogov, R. V. Fedorov, A. A. Bondarenko, and T. E. Simos
Journal of Mathematical Chemistry, 2018, Volume 56, Number 8, Page 2302
[8]
Junfeng Yao and T. E. Simos
Journal of Mathematical Chemistry, 2018, Volume 56, Number 6, Page 1567
[9]
V. N. Kovalnogov, R. V. Fedorov, D. V. Suranov, and T. E. Simos
Journal of Mathematical Chemistry, 2019, Volume 57, Number 1, Page 232
[10]
Guo-Hua Qiu, Chenglian Liu, and T. E. Simos
Journal of Mathematical Chemistry, 2019, Volume 57, Number 1, Page 119
[11]
Jie Fang, Chenglian Liu, and T. E. Simos
Journal of Mathematical Chemistry, 2018, Volume 56, Number 2, Page 423