Analytical potential energy functions are of great significance in the study of material science, molecular spectrum,reaction dynamics of atoms and molecules, vibrational and rotational energy-level structures of molecules, interactions between laser and matter, photoionization etc.^{[1- 6]}. So far, several kinds of representative analytical potential energy functions have been proposed, such as Morse potential^[7], Rydberg potential^[8], Murrell-Sorbie(M-S)potential^[9] and Huxley-Murrell-Sorbie(HMS)potential^[10], etc.. The potentials above are valid in describing the behaviors of some individual or classificatory molecules and ions, but none are suitable for all situations. One of the best and most extensively used analytical potential energy functions is M-S potential which is applied to most of the ground-state diatomic molecules. But M-S potential is shown to be unsatisfactory in describing the excited-state diatomic molecules. In this paper, by using a cosine function as a basic potential function and, through derivations, a high precision analytical potential function is obtained, which can describe many kinds of diatomic molecules--the neutral diatomic molecules and the charged diatomic molecular ion etc..

This potential energy function is examined with 18 examples of diatomic molecules and compared with RKR data and M-S potential. It is shown that the computational precision on the vibrational energy level of molecules calculated with this potential function is superior to that of M-S potential.

analytic potential function

Suppose that basic potential function of diatomic molecular satisfies Eq.(1)^[11]

V(r)=Acos[φ+arccos(R_e/r)]+B(1)

where, r is inter-nuclear distance, A and B are undetermined constants, φ is the equivalent phase difference between two interacting atoms, R_e is equilibrium inter-nuclear distance. From Eq.(1)and through theoretical derivations, a universal analy-tic potential function describing diatomic molecules is obtained

V(r)=(D_e{∑ⁿ_i=0H(i)((R_e)/r)²ⁱ+a((R_e)/r)²ⁿ⁺²+b((R_e)/r)²ⁿ⁺⁴+c((R_e)/r)²ⁿ⁺⁶-[ / ∑ⁿ_i=0H(i)(2i)+a(2n+2)+b(2n+4)+c(2n+6)](R_e)/r})/(∑ⁿ_i=0H(i)(2i-1)+a(2n+1)+b(2n+3)+c(2n+5))(2)^[12]

where D_e is the dissociation energy of diatomic molecules; H(i)=(2i)!/[4ⁱ(i!)²(2i-1)]; a, b and c are undetermined parameters which can be determined with the experimental spectroscopic parameters(ω_e, ω_e χ_e, α_e, B_e). For example, when n=1, 3, from Eq.(2), we have

V(r)=(2D_e)/(1+6a+10b+14c){1/2((R_e)/r)²+a((R_e)/r)⁴+b((R_e)/r)⁶+c((R_e)/r)⁸-(1+4a+6b+8c)(R_e)/r},(n=1)(3)

V(r)=(16D_e)/(19+112a+144b+176c){1/2((R_e)/r)²+1/8((R_e)/r)⁴+1/(16)((R_e)/r)⁶+a((R_e)/r)⁸+b((R_e)/r)¹⁰+c((R_e)/r)¹²-((15)/8+8a+10b+12c)(R_e)/r},

(n=3)(4)

Examinations show that Eq.(3)or Eq.(4)can accurately describe the interaction of diatomic molecules on a larger range of equilibrium internuclear distance in calculating the vibrational energy level of molecules but there is a definite deviation between the potential values of the long-range attractive branch of potential curve and RKR data or experimental data. So in order to further ameliorate the qualities of the long-range attractive branch, it is necessary to improve the potential energy function. In this paper, through derivations, an analytical potential function which can describe the whole range of the potential curve is obtained. The potential function is as follows

V(r)=V₃(r)+sgn(r-r₀)V₄(r)(5)

where

V₃(r)=1/2[V₁(a,b,c,r)+V₂(a₀,b₀,c₀,r)]+D_e(6)

V₄(r)=1/2[V₂(a₀,b₀,c₀,r)-V₁(a,b,c,r)](7)

sgn(r-r₀)=(r-r₀)/(|r-r₀|)(8)

We use the exactly same function for the potentials V₁(a, b, c, r)and V₂(a₀,b₀,c₀,r), which can be selected from one of Eq.(3)and Eq.(4). From Eq.(5)to Eq.(8), when n=1, 3 the following potential functions can be given

{V₃(r)=D_eC₀[1/2((R_e)/r)²+C₁((R_e)/r)⁴+C₂((R_e)/r)⁶+

C₃((R_e)/r)⁸-C₄((R_e)/r)]+D_e

V₄(r)=D_eP₀[1/2((R_e)/r)²+P₁((R_e)/r)⁴+P₂((R_e)/r)⁶+

P₃((R_e)/r)⁸-P₄((R_e)/r)]

V(r)=V₃(r)+sgn(r-r₀)V₄(r)

(n=1)(9)

{V₃(r)=8D_eC₀[1/2((R_e)/r)²+1/8((R_e)/r)⁴+1/(16)((R_e)/r)⁶+

C₁((R_e)/r)⁸+C₂((R_e)/r)¹⁰+C₃((R_e)/r)¹²-C₄((R_e)/r)]+D_e

V₄(r)=8D_eP₀[1/2((R_e)/r)²+1/8((R_e)/r)⁴+1/(16)((R_e)/r)⁶+

P₁((R_e)/r)⁸+P₂((R_e)/r)¹⁰+P₃((R_e)/r)¹²-P₄((R_e)/r)]

V(r)=V₃(r)+sgn(r-r₀)V₄(r)

(n=3)(10)

the parameters in Eq.(9)and Eq.(10)can be calculated by the following relations

{C₀=(k₁+k₂)/(k₁k₂), C₁=(a₀k₁+ak₂)/(k₁+k₂),

C₂=(b₀k₁+bk₂)/(k₁+k₂), C₃=(c₀k₁+ck₂)/(k₁+k₂)

P₀=(k₁-k₂)/(k₁k₂), P₁=(a₀k₁-ak₂)/(k₁-k₂),

P₂=(b₀k₁-bk₂)/(k₁-k₂), P₃=(c₀k₁-ck₂)/(k₁-k₂)(n=1,3)(11)

{C₄=1/(k₁+k₂)[(1+4a₀+6b₀+8c₀)k₁+

(1+4a+6b+8c)k₂]

P₄=1/(k₁-k₂)[(1+4a₀+6b₀+8c₀)k₁-

(1+4a+6b+8c)k₂](n=1)(12)

{C₄=1/(k₁+k₂)[((15)/8+8a₀+10b₀+12c₀)k₁+

((15)/8+8a+10b+12c)k₂]

P₄=1/(k₁-k₂)[((15)/8+8a₀+10b₀+12c₀)k₁-

((15)/8+8a+10b+12c)k₂](n=3)(13)

{k₁=1+6a+10b+14c

k₂=1+6a₀+10b₀+14c₀(n=1)(14)

{k₁=19+112a+144b+176c

k₂=19+112a₀+144b₀+176c₀(n=3)(15)

the undetermined parameters a, b, c can be determined with the experimental spectroscopic parameters(ω_e, ω_e χ_e, α_e, B_e)of diatomic molecules. But a₀, b₀, c₀ can be obtained by solving the following equations

{V₁(a,b,c,r₀)=V₂(a₀,b₀,c₀,r₀)

((dV₁)/(dr))_r=r0=((dV₂)/(dr))_r=r0

V₂(a₀,b₀,c₀,r)|_r=8Re=D_e(16)

where

r₀=(R_e+r_m)/2·δ(17)

Here, r_m is the value of inter-nuclear distance when V₁(a, b, c, r)+D_e=D_e. δ is a parameter for adjusting precision.

parameters to determine a, b, c

The undetermined parameters a, b and c can be determined with the experimental spectroscopic parameters(ω_e, ω_e χ_e, α_e, B_e)of diatomic molecules or ions. The principle of this method is, according to the relation between undetermined parameters and force constants, to obtain a, b, c by solving linear equations. The relation between force constants and spectroscopic parameters are as follows

f₂=4π²μω²_ec²=(hcω²_e)/(2R²_eB_e)(18)

f₃=-(3f₂)/(R_e)(1+(α_eω_e)/(6B²_e))(19)

f₄=(f₂)/(R_e²)[15(1+(α_eω_e)/(6B²_e))²-(8ω_e χ_e)/(B_e)](20)

where the force constants at the equilibrium inter-nuclear distance can be given as follows

f_m=((d^mV)/(dr^m))_r=Re(m=2,3,4)(21)

From Eq.(3)and Eq.(4), when n=1, 3, the following linear equations can be obtained

{(1+12a+30b+56c)/(1+6a+10b+14c)=(f₂R²_e)/(2D_e)

(2+32a+100b+224c)/(1+6a+10b+14c)=-(f₃R³_e)/(6D_e)

(3+62a+240b+644c)/(1+6a+10b+14c)=(f₄R⁴_e)/(24D_e)(n=1)(22)

{(35+448a+720b+1 056c)/(19+112a+144b+176c)=(f₂R²_e)/(2D_e)

(98+1 792a+3 360b+5 632c)/(19+112a+144b+176c)=-(f₃R³_e)/(6D_e)(n=3)(23)

(206+5 152a+11 280b+21 648c)/(19+112a+144b+176c)=(f₄R⁴_e)/(24D_e)

The Eq.(22)and Eq.(23)above are all linear equations, which have unique real number solutions for the undetermined parameters a, b and c. In order to compare the potential functions Eq.(9)and Eq.(10)with M-S potential, we also provide it in the following form

V(r)=-D_e[1+a₁(r-R_e)+a₂(r-R_e)²+

a₃(r-R_e)³]exp[-a₁(r-R_e)](24)

The relations between undetermined parameters a₁, a₂, a₃ and force constants are as follows

{D_e(a²₁-2a₂)=f₂

2D_e(3a₁a₂-3a₃-a³₁)=f₃

D_e(3a⁴₁-12a²₁a₂+24a₁a₃)=f₄(25)

Although Rydberg-Klein-Rees(RKR)inverse-method is a pure theoretical method, the values of the vibrational energy level of molecules calculated with this method are extremely and exactly consistent with the experimental data. Hence, the RKR data are usually considered as the experimental data. Under the condition of no experimental data, one of the best method is to use the RKR data to examine a certain potential energy function. The method for calculating RKR data is given as follows^[13]

{Zr_max(U)=(f/g+f ²)^1/2+f

r_min(U)=(f/g+f ²)^1/2-f

f=((B_eR²_e)/(ω_e χ_e))^1/2ln[((ω²_e-4ω_e χ_eU)^1/2)/(ω_e-(4ω_e χ_eU)^1/2)]

g=1/((4R²_eB_e(ω_e χ_e)³)^1/2)[α_e(4ω_e χ_eU)^1/2+ /

(2ω_e χ_eB_e-α_e ω_e)ln[((ω²_e-4ω_e χ_eU)^1/2)/(ω_e-(4ω_e χ_eU)^1/2)]]

U=ω_e(υ+1/2)-ω_e χ_e(υ+1/2)²(26)

Where r_max and r_min are the maximum and minimum classical turning points of the inter-nuclear distance for a molecule vibrating with energy U, U is the value of potential energy. From Eq.(26), f(unit: cm)and g(unit: cm^-1)can be determined when the spectroscopic parameters ω_e, ω_e χ_e, α_e, B_e are given, and then the maximum and minimum classical turning points r_max and r_min can be calculated by using Eq.(26), and the potential curve of U(r)can be plotted out.

Substituting the potential parameters in Table 4 and Table 5 into Eq.(9)or Eq.(10), a specific analytic potential energy function of diatomic molecules can be obtained. For examining potential energy functions of Eq.(9)and Eq.(10), 18 kinds of neutral diatomic molecules and charged diatomic molecular ions have ever been investigated, and the values of the vibrational energy level of molecules calculated by the potential function are compared with RKR data, M-S potential or experimental data. The experimental data on spectroscopic parameters and the potential parameters of part diatomic molecules calculated from Eq.(16)to Eq.(25)are listed in Table 1 to Table 4. The potential parameters r₀, a₀, b₀, c₀ are calculated with Eq.(16). Table 5 is calculated with Eq.(11)to Eq.(15). The vibrational energy levels and classical turning points of heteronuclear excited-state for neutral diatomic molecules AsCl-¹Δ and heternuclear ground-state for charged diatomic molecular ion(CS)⁺-X²Σ⁺ calculated by Eq.(9), Eq.(10), Eq.(24)and Eq.(26)are listed in Table 6 and Table 7.

Table 1 Experimental spectroscopic parameters of diatomic molecules

Table 2 Parameters of part diatomic molecules

Table 3 Parameters a0, b0, c0 of part the electronic states

Table 4 n and r0 parameters of diatomic molecules

Table 5 Parameters of diatomic molecules

Table 6 The vibrational energy levels of heteronuclear excited-state for neutral diatomic molecular ion AsCl-1Δ

Table 7 The vibrational energy levels of heteronuclear ground-state for charged diatomic molecular ion(CS)+-X2Σ+

(Note: the relation between zero-point dissociation energy D_e and ground-state dissociation energy D₀ given by Ref.[14] is D_e=D₀+ ω_e/2- ω_e χ_e/4). For the limitation of the length, the computational values of the vibrational energy level of other diatomic molecules are omitted.

In order to examine the calculation precision of Eq.(9)and Eq.(10), the root-mean-square-errors(RMSE)and relative root-mean-errors between the potential values of 18 kinds of electronic states and RKR data or experimental data are listed in Table 8. For comparison, the calculation precision of M-S are also listed in Table 8. The RMSE and relative root-mean-square-errors(RRMSE)are given in the following formulas

RMSE=(1/N∑^N_i=1(U_RKR(i)-U_CAL(i))²)^1/2(27)

RRMSE=(1/N∑^N_i=1((U_RKR(i)-U_CAL(i))/(U_RKR(i)))²)^1/2(28)

The potential curves of four diatomic molecules are plotted with Origin 6.0 by using potential Eq.(9)and Eq.(10), RKR data and M-S potential are shown in Table 1.

Table 8 The root-mean-square-errors and relative root-mean-errors between the values of potentials and RKR

Fig.1 Energy curves of diatomic molecules(☆=exp.; ○=RKR; —=this work; ┄=M-S)

As shown from Table 6 to Table 8, the potential function given in this paper can describe the behaviors of the whole range of the potential curve. And as for the computational precision, this potential function is superior to M-S potential.

A new analytic potential energy function which is used to describe diatomic molecules and ions is presented with a basic potential function V(r)=Acos[φ+arccos(R_e/r)]+B, and good results are obtained. This potential function has two merits: ① The good universality and high computational precision. This potential is suitable for describing eight kinds of fundamental diatomic molecules——homonuclear ground-state for neutral diatomic molecules,homonuclear excited-state for neutral diatomic molecules, homonuclear ground-state for charged diatomic molecules, homonuclear excited-state for charged diatomic molecules, heteronuclear ground-state for neutral diatomic molecules, heteronuclear excited-state for neutral diatomic molecule, heternuclear ground-state for charged diatomic molecular ions, heteronuclear excited-state for charged diatomic molecular ions. ② As concerning the computational precision, this potential function is superior to M-S potential which is extensively used in atomic and molecular physics at present.