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Use of JWKB approximation for single-electron molecules
Volume
3. number
USE
OF
8
CHEMICAL
JWKB
PHYSICS
APPROXIMATION
FOR
S. J. ARNOLD* Departnlent
of Applied
Queen’s
Belfast.
and Theoretical Nwtkem
Physics,
4 July 1969
states of single-electron
mole-
The equation
As is well known the JWKB approximation [ 1: 2] is useful for many quanta1 problems. In the present note we consider a double application of the approximation to single-electron molecular systems. We employ atomic units throughout.
(6) may be cast into a convenient form for applying the phase integral quantum condition of the JWKB approximation [l] by changing the independent variable to t = 21(x)
2. THEORY Let 21 and 22 be the nuclear charges and R be the distance between the nuclei. Using confocal elliptic coordinates we have that the equations to be solved for a state of azimuthal quantum number m are &](x”-l)d$/
+f(X)R
=0
where u(x) is an arbitrary pendent variable to
(7)
function, and the de-
Z = Cp(x)&(_& the prime indicating differentiation transformed equation is
(6) [4]. The
(9)
(1)
and
where
-& j(l43I
d4
+g(p)M
= 0
(2)
with f(A)
= - c
- S2X2
+ (Z~+.Q)RX
-?.?X2-1
(3)
and
y(x) = [2(~~‘)2(2~-P”lP + {3@“)2- 2rr?r”&2 + (9fqv)2] 4(102P2 (10) In the case of a simple potential well the phase integral quantum condition is
(11) g(p)
=c +s2p2+ (Z1-22)Rp
in which C is a separation
- -
E being the electronic
?112
l-,2 constant and
.2 = _$R2E
(4)
(5)
energy [3].
* Now at Defcnce Research .Quebec. Canada.
632
MOLECULES
Ireland
for even the lowest non-s
1. INTRODUCTION
1969
and D. R. B_%TES
Received
The JWB approximation is quite accurate cules. It is lesGsuccessfu1 for &states.
August
SINGLE-ELECTRON
Mathematics
University.
LETTERS
Establishment,
in which N is an integer and x1 and ~2 are the classical turning points. Both equations with which we are concerned mai ?e s~i’Z..ly transformed by taking [4]
(12) Valcartier.
This corresponds to changing the independeft variables in eqs. (I) and (2) to log[(X-l),/(X+l)]Z and
Volume 3, number 8 to lOg[(l+~)/*(l-p)]+ gives
CHEMICAL
respectively.
Formula
Y(X) =f(A)/(Ai)
PHYSICS
(10) (13)
and Y(P) = ‘&N(l-G+
.
(14)
Another suitable transformation is obtained [5] by changing the independent variable in eq. (1) to log(X-1). With this choice of independent variable (13) is replaced by
We shall refer to a transformation by the number of the r-equation. It is apparent that each r(X) has the form corresponding to motion in a simple potential well. Hence condition (11) may be used for the determination of the eigenenergies. If m f 0 the equation for the classical turning points ?-(A) = 0
(16)
has two roots, Xi and X2! in the X 2 1 region; but if ??t= 0 and if R exceeds a critical distance R, it has only one physical root. X3, in which circumstance the lower limit of integration in eq. (11) may be put equal to unity. Gershte’m et al. [4] have proven that transformation (13) yields the exact eigenenergy in the united atom limit if the separation constant C is taken to be (ki)2 there. It may readily be seen that transformation (15) requires C to be the correct value, Z(Z+l), in the united atom limit. The potential well associated with r(p) is not simple. It contains a potential hill. If R exceeds the critical distance RM at which C = ,12
LETTERS
(17)
1969
fect matches the one in transformation (13) so when using the two transformations together we modified neither. However when using transformation (14) with (15) we replaced C in eq. (4) by C + $ to reduce the error. The azimuthal quantum number Z and principal quantum number nu of the united atom may of course be expressed in terms of the integers NcL and Nk appearing in the two phase integral equations; thus [5] Z =Np+
III,
n,, = Nx + Z + 1.
WI
It is a very simple task to find C and, more important, E as functions of R for any particular state by solvtng the relevant pair of phase integral equations by iteration. 3. RESULTS Most of the calculations were carried out on Hi because oi the large number of states for which the exact energies &are available for comparison purposes [5,7]. Some suppLeme?Jary calculations which were carried out on Keii showed closely the same pattern. We shall denote the percentage error in !he calculated value E of an energy by 6 = 100 (E-E_~).&~
(‘:Cj ..
and shall indicate the transformatious used by giving their numbers in brackets after this symbol. Values of 6(14,15) for a few of the states of Table
R
this hill separates the region of allowed classical motion into hvo parts. GershteTn et al. [4] have given an elegant treatment of the problem presented. Using the complex turning point method proposed by Porkovskii and Khalalnikov [6] and invoking the equation of continuity of current [l] they succeeded in taking proper account of reflection at energies above the top of the hill and of transmission at energies below. We were content with carrying out the integration in eq. (11) over all allowed values of p for which g( p) is positive. This approach has the advantage of simplicity. Though it naturally fails at sufficiently large R there is a wide range in which it does not lead to significant loss of accuracy. Transformation (14) gives C to be (Z+f)2 instead of Z(Z+l) in the united atom limit. This de-
August
(units ao)
0 1
2
i
m
0
0
Np = Nx
‘_’
3
0.0
0.0
-0.6 1.8
-0.1 0.3
1 1
1 :!
6ti4. i5)
0.0 -0.7
0.0 -0.2
1.0
0.4
.-._ 1.
9 0
. _.___
0.0 -0.6 0.5
9.u
-0.3 0.4
3
-u
0-G
0.7
0.5
-0.2
4
+1.3
1.0
0.4
0.6
+0_2
0.1
5
0.9
1.5
0.2
0.6
0.5
O.&
6
0.7
-u
0.2
0.2
0.6
0.3
7
0.5
+1.0
0.3
0.L
1.1
0.2
8
0.4
1.1
0.5
-0.0
1.3
0.‘:.
9
0.4
0.8
0.7
+0.1
I.5
0.1
10
eO.5
+0.6
+O.l
+L.7
-1.0
0.5
-0.1
The critical distances Rh are indicated approximately by the positions of horizontal lines: the crisicnl distances Rp are not within the range covered.
633
Volume 3. number 8
CHEMICAL PHYSICS LETTERS
Hi studied are given in table 1. ‘.I? corresponding values of (14.13) are in general rather larger. Their variation with R is similar. As would be expected 161 decrctases as the quantum numbers m. Np and & are increased Provided m + 0 it remains rem?.rkably small (bearing in mind that a dmbb Application of the JWKB approximation is involved) over a wide R range. Thus even in the most extreme case, nl = 1. NF = Nx = 0 16(14,15)[ does not exceed 3% in the range up to 6a. (which is just beyond the critical distance Rcl at which the potential hill first separates the classically allowed region into two parts). If )N = 0 the position is less satisfactory. Unless Np + Nx is 3 or greater 161 may be large in the outer part of the R range (0-lOa,) covered by the table. Moreover in all cases 6 changes sharply when R is near the critical distance Rx at which the inner classical turning point of the
t
634
motion vanishes: it rises side of Rx to a maximum
August 1969 from
a minimum
on the other table 1). This defect is not associated treatment of the p motion and may be the results presented by Gershtein et
on one
side (see with the detected in al. [4].
REFERENCES [I] E. C.Kemble. Phys. Rev. 48 (1935) 549. [2] R. E. Langer. Phys. Rev. 51 (1937) 669. [3] D. Burrau. Kgl. Danske Videnskab. Selskab 7. no. 14 (1927). [a) S. S. Gershteyn. L. I. Ponomarev and T. P. Puzynina. Soviet Phys. JETP 21 (1965) 418. [5] D. R. Bates and R. H. G. Reid, Advan.At. Mol. Phys. 4 (1968) 13. [S] V. L. Pokrovskiy and I. M. Khalatnikov. Soviet Phys. JETP 13 (1961) 1207. [7] D. R. Bates. K. Ledsham and A. L.Stewart. Phil. Trans. Roy. Sot. A246 (1953) 215.
3. number
USE
OF
8
CHEMICAL
JWKB
PHYSICS
APPROXIMATION
FOR
S. J. ARNOLD* Departnlent
of Applied
Queen’s
Belfast.
and Theoretical Nwtkem
Physics,
4 July 1969
states of single-electron
mole-
The equation
As is well known the JWKB approximation [ 1: 2] is useful for many quanta1 problems. In the present note we consider a double application of the approximation to single-electron molecular systems. We employ atomic units throughout.
(6) may be cast into a convenient form for applying the phase integral quantum condition of the JWKB approximation [l] by changing the independent variable to t = 21(x)
2. THEORY Let 21 and 22 be the nuclear charges and R be the distance between the nuclei. Using confocal elliptic coordinates we have that the equations to be solved for a state of azimuthal quantum number m are &](x”-l)d$/
+f(X)R
=0
where u(x) is an arbitrary pendent variable to
(7)
function, and the de-
Z = Cp(x)&(_& the prime indicating differentiation transformed equation is
(6) [4]. The
(9)
(1)
and
where
-& j(l43I
d4
+g(p)M
= 0
(2)
with f(A)
= - c
- S2X2
+ (Z~+.Q)RX
-?.?X2-1
(3)
and
y(x) = [2(~~‘)2(2~-P”lP + {3@“)2- 2rr?r”&2 + (9fqv)2] 4(102P2 (10) In the case of a simple potential well the phase integral quantum condition is
(11) g(p)
=c +s2p2+ (Z1-22)Rp
in which C is a separation
- -
E being the electronic
?112
l-,2 constant and
.2 = _$R2E
(4)
(5)
energy [3].
* Now at Defcnce Research .Quebec. Canada.
632
MOLECULES
Ireland
for even the lowest non-s
1. INTRODUCTION
1969
and D. R. B_%TES
Received
The JWB approximation is quite accurate cules. It is lesGsuccessfu1 for &states.
August
SINGLE-ELECTRON
Mathematics
University.
LETTERS
Establishment,
in which N is an integer and x1 and ~2 are the classical turning points. Both equations with which we are concerned mai ?e s~i’Z..ly transformed by taking [4]
(12) Valcartier.
This corresponds to changing the independeft variables in eqs. (I) and (2) to log[(X-l),/(X+l)]Z and
Volume 3, number 8 to lOg[(l+~)/*(l-p)]+ gives
CHEMICAL
respectively.
Formula
Y(X) =f(A)/(Ai)
PHYSICS
(10) (13)
and Y(P) = ‘&N(l-G+
.
(14)
Another suitable transformation is obtained [5] by changing the independent variable in eq. (1) to log(X-1). With this choice of independent variable (13) is replaced by
We shall refer to a transformation by the number of the r-equation. It is apparent that each r(X) has the form corresponding to motion in a simple potential well. Hence condition (11) may be used for the determination of the eigenenergies. If m f 0 the equation for the classical turning points ?-(A) = 0
(16)
has two roots, Xi and X2! in the X 2 1 region; but if ??t= 0 and if R exceeds a critical distance R, it has only one physical root. X3, in which circumstance the lower limit of integration in eq. (11) may be put equal to unity. Gershte’m et al. [4] have proven that transformation (13) yields the exact eigenenergy in the united atom limit if the separation constant C is taken to be (ki)2 there. It may readily be seen that transformation (15) requires C to be the correct value, Z(Z+l), in the united atom limit. The potential well associated with r(p) is not simple. It contains a potential hill. If R exceeds the critical distance RM at which C = ,12
LETTERS
(17)
1969
fect matches the one in transformation (13) so when using the two transformations together we modified neither. However when using transformation (14) with (15) we replaced C in eq. (4) by C + $ to reduce the error. The azimuthal quantum number Z and principal quantum number nu of the united atom may of course be expressed in terms of the integers NcL and Nk appearing in the two phase integral equations; thus [5] Z =Np+
III,
n,, = Nx + Z + 1.
WI
It is a very simple task to find C and, more important, E as functions of R for any particular state by solvtng the relevant pair of phase integral equations by iteration. 3. RESULTS Most of the calculations were carried out on Hi because oi the large number of states for which the exact energies &are available for comparison purposes [5,7]. Some suppLeme?Jary calculations which were carried out on Keii showed closely the same pattern. We shall denote the percentage error in !he calculated value E of an energy by 6 = 100 (E-E_~).&~
(‘:Cj ..
and shall indicate the transformatious used by giving their numbers in brackets after this symbol. Values of 6(14,15) for a few of the states of Table
R
this hill separates the region of allowed classical motion into hvo parts. GershteTn et al. [4] have given an elegant treatment of the problem presented. Using the complex turning point method proposed by Porkovskii and Khalalnikov [6] and invoking the equation of continuity of current [l] they succeeded in taking proper account of reflection at energies above the top of the hill and of transmission at energies below. We were content with carrying out the integration in eq. (11) over all allowed values of p for which g( p) is positive. This approach has the advantage of simplicity. Though it naturally fails at sufficiently large R there is a wide range in which it does not lead to significant loss of accuracy. Transformation (14) gives C to be (Z+f)2 instead of Z(Z+l) in the united atom limit. This de-
August
(units ao)
0 1
2
i
m
0
0
Np = Nx
‘_’
3
0.0
0.0
-0.6 1.8
-0.1 0.3
1 1
1 :!
6ti4. i5)
0.0 -0.7
0.0 -0.2
1.0
0.4
.-._ 1.
9 0
. _.___
0.0 -0.6 0.5
9.u
-0.3 0.4
3
-u
0-G
0.7
0.5
-0.2
4
+1.3
1.0
0.4
0.6
+0_2
0.1
5
0.9
1.5
0.2
0.6
0.5
O.&
6
0.7
-u
0.2
0.2
0.6
0.3
7
0.5
+1.0
0.3
0.L
1.1
0.2
8
0.4
1.1
0.5
-0.0
1.3
0.‘:.
9
0.4
0.8
0.7
+0.1
I.5
0.1
10
eO.5
+0.6
+O.l
+L.7
-1.0
0.5
-0.1
The critical distances Rh are indicated approximately by the positions of horizontal lines: the crisicnl distances Rp are not within the range covered.
633
Volume 3. number 8
CHEMICAL PHYSICS LETTERS
Hi studied are given in table 1. ‘.I? corresponding values of (14.13) are in general rather larger. Their variation with R is similar. As would be expected 161 decrctases as the quantum numbers m. Np and & are increased Provided m + 0 it remains rem?.rkably small (bearing in mind that a dmbb Application of the JWKB approximation is involved) over a wide R range. Thus even in the most extreme case, nl = 1. NF = Nx = 0 16(14,15)[ does not exceed 3% in the range up to 6a. (which is just beyond the critical distance Rcl at which the potential hill first separates the classically allowed region into two parts). If )N = 0 the position is less satisfactory. Unless Np + Nx is 3 or greater 161 may be large in the outer part of the R range (0-lOa,) covered by the table. Moreover in all cases 6 changes sharply when R is near the critical distance Rx at which the inner classical turning point of the
t
634
motion vanishes: it rises side of Rx to a maximum
August 1969 from
a minimum
on the other table 1). This defect is not associated treatment of the p motion and may be the results presented by Gershtein et
on one
side (see with the detected in al. [4].
REFERENCES [I] E. C.Kemble. Phys. Rev. 48 (1935) 549. [2] R. E. Langer. Phys. Rev. 51 (1937) 669. [3] D. Burrau. Kgl. Danske Videnskab. Selskab 7. no. 14 (1927). [a) S. S. Gershteyn. L. I. Ponomarev and T. P. Puzynina. Soviet Phys. JETP 21 (1965) 418. [5] D. R. Bates and R. H. G. Reid, Advan.At. Mol. Phys. 4 (1968) 13. [S] V. L. Pokrovskiy and I. M. Khalatnikov. Soviet Phys. JETP 13 (1961) 1207. [7] D. R. Bates. K. Ledsham and A. L.Stewart. Phil. Trans. Roy. Sot. A246 (1953) 215.