Coupled channel formulation of the perturbed finite-temperature atomic random phase approximation: single channel approximation

Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

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Coupled channel formulation of the perturbed +nite-temperature atomic random phase approximation: single channel approximation G. Csanaka;∗ , D.P. Kilcreaseb , G.D. Menesesb a

Atomic & Optical Theory, Theoretical Division, Los Alamos National Laboratory, P.O. Box 1663 MS B212, Los Alamos, NM 87545, USA b Instituto de Fisica “Gleb Wataghin”, UNICAMP, CP6165, 13081 Campinas, SP, Brazil Accepted 15 February 2003

Abstract We present the spin and angular momentum analysis of the inhomogeneous coupled partial integro-di:erential equation formulation of the +nite-temperature random phase approximation of an atom in a plasma. We demonstrate the cancellation of the direct and exchange self-interaction terms in the three-dimensional form of the equations. The direct self-interaction terms are also shown to cancel for each partial wave in the radial equations obtained from the spin and angular momentum analysis. We discuss how this inhomogeneous formulation avoids normalization di
1. Introduction In an earlier work [1], hereafter called CKII, we have formulated the +nite-temperature random phase approximation as an inhomogeneous coupled partial integro-di:erential equation problem. In order to perform calculations that system requires spin and angular momentum analysis. Further, in a manner similar to the case for the homogeneous +nite-temperature RPA equations [2] approximations need to be introduced. As discussed earlier [1], the advantage of using the inhomogeneous equations is that the normalization problem can be dispensed with. In the case of solving the homogeneous (eigenvalue) problem, a separate approximation needed to be introduced [2] for the normalization. ∗

Corresponding author. E-mail address: [email protected] (G. Csanak).

0022-4073/03/$ - see front matter ? 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-4073(03)00077-3

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G. Csanak et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

However, it is di
(we shall follow the notation of CKII where  = x; y; z referred to the Cartesian components of a vector but  = 3 will refer to the z component and we use x as a variable to refer to both the spatial, ˜r, and spin, , coordinates of the electron) where ˜ij (z) was de+ned by CKII Eq. (28) as
ijn
n∗  kl Dkl (2) ˜ij (z) = ! − z n n k;l

and the ’j (x) refer to the eigenfunctions of the Hartree–Fock Hamiltonian. It is shown in CKII that the last factor in Eq. (2) can be written in the form (CKII Eq. (29))

 n∗  d xln∗ (x)D ’l (x); kl Dkl = (3) k;l

l

where ln (x) is the function
ln (x) = iln ’i (x) l

de+ned by CKII Eq. (27) as (4)

and integration over x signi+es integration in Cartesian space for the variable ˜r and summation in spin space for . Thus, we obtain the expression
jn (x)
 d x in∗ (x )D ’i (x ): ˜j (z; x) = (5) ! − z n n i In order to perform an angular momentum and spin analysis on the inhomogeneous system, we will need an angular momentum and spin analysis of the ˜j (z; x) functions. Due to symmetry we need to do this only for the  = 3 case. Eqs. (1)–(3) show that in order to accomplish this we +rst need an angular momentum representation of jn (x) (in the following we shall not use l as a general index but use it for the angular momentum quantum number) that can be obtained from the formula  (6) jn (x) = d x n (x; x )’j (x ): This latter equation can be immediately obtained from the de+nition of the jn (x) function as given above by Eq. (4) and the equation given by CKII Eq. (7),
n (x; x ) = ijn ’i (x)’∗j (x ): (7) i; j

G. Csanak et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

Using Eq. (4) in Eq. (6) we obtain
n (x; x ) = jn (x)’∗j (x ):

249

(8)

j

Multiplying Eq. (8) by ’j (x ) and using the orthonormality of the ’j (x ) functions gives Eq. (6). Csanak and Meneses [2], hereafter referred to as CM, wrote n (x; x ) in the following angular momentum and spin analyzed form (Eq. (3) of CM), n (x; x ) ≡ n (˜r ;˜r   ) = nLM (˜r;˜r  )SMS ( ;  )
MSMS  LM n;l; ˆ rˆ )SMS ( ;  ); = l
(r; r )Yl; l
(r;

(9)

l;l


where Yl;LM ˆ rˆ ) = l
(r;




(lm; l m |ll LM )Ylm (r)Y ˆ l
m
(rˆ )

(10a)

m; m


and SMS ( ;  ) =


ms ; m
s

(−1)1=2+ms ( 12 − ms ; 12 ms | 12


1 2

SMS )m
s ( )ms (  ):

(10b)

Using now Eqs. (9) – (10b) in Eq. (6) we obtain the following angular momentum–spin expansion of the jn (x) function: S jn (x) ≡ nLMSM (˜r; ) j lj m j m s

=




j

nLj lj l (r)(lM + mj lj − mj |llj LM )(−1)mj YlM +mj (r) ˆ

l

·(−1)1=2+msj −MS ( 12 MS − msj

1 2

msj | 12

1 2

SMS )ms −MS ( ):

(11)

Using this representation in Eq. (5), we obtain the following angular momentum and spin representation of ˜3j (z; x) where we have taken  = 3
S=0 ˜3j (z; x) = (−1)mj msj ( ) (lmj lj − mj |llj 10)Ylmj (r)f ˆ L=1; (z; r): (12) j lj l l=|lj −1|;lj +1 L=1;S=0

Using now this representation of ˜3j (z; x) in the fundamental equation for that function (Eqs. (32)  and (33) in CKII), where HF (x; x ) = i ni ’∗i (x)’i (x ), we obtain  3 3 (hHF − j − z)˜j (z; x) = −nj D ’j (x) + d x1 HF (x1 ; x)D3 ’j (x1 ) + nj




d x1 ’j (x1 )V (˜r − ˜r 1 )’∗i (x1 )˜3i (z; x)

i

− nj


 i

d x1 ˜3i (z; x1 )V (˜r − ˜r 1 )’∗i (x1 )’j (x)

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G. Csanak et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

+


 

d x1 d x2 HF (x2 ; x)V (˜r 2 − ˜r 1 )’j (x2 )’∗i (x1 )˜3i (z; x1 )

i

−


 

d x1 d x2 HF (x2 ; x)’j (x1 )V (˜r 2 − ˜r 1 )’∗i (x1 )˜3i (z; x2 ):

(13)

i

After considerable algebra (and using the identity given by Eq. (33) of CKII) we obtain our fundamental equations for the radial functions gj lj l (z; r), that are de+ned as S=0 gj lj l (z; r) = rfL=1; (z; r); j lj l

(14)

in the following form (where l = |lj − 1| or lj + 1 for dipole transitions), d 2 gj lj l (z; r) l(l + 1) Z − gj lj l (z; r) − gj lj l (z; r) 2 2 dr r r  ∞
ni li (2li + 1) dr1 #0 (r; r1 )|Pi li (r1 )|2 gj lj l (z; r) +2 0

i li

−




 (2l + 1)

%;li

2 

lli % 000

∞

0

dr1 #% (r; r1 ) li (r; r1 )gj lj l (z; r1 )

− j lj gj lj l (z; r) − zgj lj l (z; r)    (2lj + 1)(2l + 1) llj 1 = 4& 000     ∞
4& dr1 r1 ni l P∗i l (r1 )Pi l (r)Pj lj (r1 ) rPj lj (r) − × nj lj 3 0  + n j lj  ×

%lj li



000

 ×



(−1)l +lj (2lj + 1)(2li + 1)(2l + 1)(2l + 1)

%l l









llj 1




∞

0

dr1 Pj lj (r1 )#% (r; r1 )P∗i li (r1 )gi li l
(z; r)

2 (2lj + 1)(2li + 1)(2l + 1)(2l + 1) 3



000

nn l Pn l (r)

1lj l



%l li

i li l
=|li −1|;li +1

l i l 1

%



000




000

n

i




i li l
=|li −1|;li +1

+ n  j lj

+




0

∞

dr1 gi li l
(z; r1 )#1 (r; r1 )P∗i li (r1 )Pj lj (r)




i li l
=|li −1|;li +1

2 3



lj l1 000



l i l 1 000



G. Csanak et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

251

× (2lj + 1)(2li + 1)(2l + 1)(2l + 1)  ∞ ∞ dr1 dr2 P∗n l (r2 )Pj lj (r2 )#1 (r1 ; r2 )P∗i li (r1 )gi li l
(z; r1 ) × 0

−




0

nn l Pn l (r)

n







i li l
=|li −1|;li +1




 (−1)

%

%

%l l 000



%li lj



000

× (2lj + 1)(2li + 1)(2l + 1)(2l + 1)  ∞ ∞ dr1 dr2 P∗n l (r2 )Pj lj (r1 )#% (r1 ; r2 )P∗i li (r1 )gi li l
(z; r2 ); × 0

0

1lj l



%l li

(15)

% %+1 where l can only have the values (see Eq. (12)) l = |lj − 1| and lj + 1, #% (r1 ; r2 ) = r¡ =r¿ and Z is the nuclear charge. The radial density matrix is de+ned by 2li + 1
ni li Pi li (r)P∗i li (r  ); (16) li (r; r  ) = 4&  i

where ni li refers to the occupation number of orbital Pi li of energy i li and is given by the Fermi– Dirac distribution function ni li = 1=(e)(i −*) + 1)

(17)

with ) = 1=kT where T is the temperature, k is Boltzmann’s constant and * is the chemical potential of the electron gas. From Eq. (15) one can see the cancellation of the self-interaction, an important physical e:ect discussed in the papers of Csanak and Kilcrease [3] and CM. If we consider the i li ≡ j lj term with % = 0 in the second term of the right-hand side of Eq. (15) and evaluate the 3-j and 6-j symbols we get the expression, 
l+l
+1 −nj lj (−1) ,ll
dr1 |Pj lj (r1 )|2 #0 (r; r1 )gj lj l (z; r) l
=|lj −1|;lj +1



= n j lj

dr1 |Pj lj (r1 )|2 #0 (r; r1 )gj lj l (z; r)

(18)

for the two values l = |lj − 1|; lj + 1. This can be combined with the i li = j lj term of the second line of the left-hand side of Eq. (15). This results in removal of the one orbital fractionally occupied by nj lj electrons and the occupation number becoming 2(2lj + 1)nj lj − nj lj ;

(19)

which when multiplied by the +rst integral on the left-hand side of Eq. (15) provides the physically meaningful potential for the electron. An attempt to do this heuristically was made by Carson et al. [4] and Shalitin et al. [5]. In addition to this charge cancellation in the direct term, an analogous charge cancellation also occurs for the self-exchange interaction but not in each partial wave separately in Eq. (15). However, the cancellation is readily apparent in Eq. (13).

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G. Csanak et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

Since the Hartree–Fock (HF) one-electron Hamiltonian [6] is given by the formula ˝2 2 ∇ + -HF 2m where the HF potential, -HF ; is given by, hHF = −

(20)


Z   -HF (x; x ) = − ,(x − x ) + ,(x − x ) d x1 V (˜r − ˜r 1 ) ni |’i (x1 )|2 r i
− V (˜r − ˜r  ) ni ’i (x)’∗i (x ); 

(21)

i

it is immediately obvious that if the i = j term from the third term on the right-hand side of Eq. (13) is moved to the left-hand side, it would remove the i = j term from the second term in -HF , i.e. the self-interaction is removed from the electrostatic potential. We now want to point out how that analogous cancellation occurs for the self-exchange term also. If the i = j term from the fourth term on the right-hand side of Eq. (13) is moved to the left-hand side, it will remove the i = j term from the summation in the third term of the expression for -HF , i.e. it removes the self-exchange interaction. Thus, if we de+ne the operator N − nj

hHF

=−

˝2 2 N −n ∇ + -HF j 2m

(22)

with N − nj

-HF

Z (x; x ) = − ,(x − x ) + ,(x − x ) r 

− V (˜r − ˜r  )




 d x1 V (˜r − ˜r 1 )

ni ’i (x)’∗i (x );




ni |’i (x1 )|2

i=j

(23)

i=j

then the left-hand side of Eq. (13) becomes N − nj

(hHF

− j − z)˜3j (z; x)

(24)

and in the third and fourth terms, on the right-hand side, the restriction i = j must be used. The various values of i li were called “channels” by CM and thus Eq. (15) can be termed a “coupled-channel” system of equations (the additional index l can be called a component index). Following CM we introduce the single-channel approximation, by neglecting all terms on the right-hand side of Eq. (15) with i li = j lj . This amounts to neglecting channel coupling. This approximation will retain the important physical e:ect of creating the physically meaningful potential for the excited electron as discussed above. Introducing the quantities      l l l %l %l 1l j j j A%ll
lj ≡ (2lj + 1) (2l + 1)(2l + 1) ; 000 000 %l lj     lj l1 lj l 1 2 Bll
lj ≡ (2lj + 1) (2l + 1)(2l + 1) 3 000 000

G. Csanak et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

253

and
Ijnlllj (z) ≡ Bll
lj



∞ ∞

0




−

0

A%ll
lj

dr1 dr2 P∗n l (r2 )Pj lj (r2 )#1 (r1 ; r2 )P∗j lj (r1 )gj lj l
(z; r1 )



∞ ∞

0

%=even

0

dr1 dr2 P∗n l (r2 )|Pj lj (r1 )|2 #% (r1 ; r2 )gj lj l
(z; r2 );

we obtain d 2 gj lj l (z; r) l(l + 1) Z − gj lj l (z; r) − gj lj l (z; r) 2 2 dr r r  ∞
ni li (2li + 1) dr1 #0 (r; r1 )|Pi li (r1 )|2 gj lj l (z; r) +2 0

i ;li

− (2l + 1)






%;li

%lli

2 

000

∞

0

dr1 #% (r; r1 ) li (r; r1 )gj lj l (z; r1 )

− j lj gj lj l (z; r) − zgj lj l (z; r)    (2lj + 1)(2l + 1) llj 1 = 4& 000     ∞
4& rPj lj (r) − ni l Pi l (r) dr1 r1 P∗j l (r1 )Pj lj (r1 ) × nj lj 3 0  i

− n j lj

lj +1





A%ll
lj

 0

l
=|lj −1| %=even

+ n  j lj

lj +1


 B

ll
l

j

Pj lj (r)

l
=| l j − 1 |

+


n

∞

nn l Pn l (r)

lj +1
l
=

0




dr1 |Pj lj (r1 )|2 #% (r; r1 )gj lj l
(z; r) ∞

dr1 gj lj l
(z; r1 )#1 (r; r1 )P∗j lj (r1 )

Ijnlllj (z):

(25)

| lj − 1|

If we go further in approximation and neglect the coupling between the l = |lj − 1| and l = lj + 1 components, i.e. keeping only the l = l terms on the right-hand side of Eq. (25), then we obtain equations that are very similar to a dipole perturbed version of the homogeneous single-channel single-component equation obtained by CM. Once gj lj l (z; r) is obtained for given (j lj ) and z values, Eqs. (12) and (13) are used to obtain the ˜3j (z; x) function for given j at a +xed z value. Performing this calculation for a whole host of

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G. Csanak et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 81 (2003) 247 – 254

j values, Eq. (40) of CKII can be used to obtain the atomic polarizability in the form
 zz d x’∗j (x)D3 ˜3j (z; x);  (z) =

(26)

j

where z = ! + i, and the photoabsorption cross section at ! = lim,→0 (! + i,) is obtained from the formula (see Eq. (41) of CKII), 4&! lim Im zz (z): (27) (!) = c ,→0+ 3. Summary and conclusions In summary, we note that we have performed angular momentum and spin analysis on the +nite-temperature, inhomogeneous atomic random phase approximation where we considered a dipole perturbation. We have obtained a coupled-channel and component integro-di:erential system. We have shown that the self-interaction term cancels in each partial wave while the self-exchange interaction term cancels in the three-dimensional coupled equation system but not in each partial wave separately. Further work will be needed to introduce the local exchange approximation in a consistent manner. Finally, we emphasize that the solution of the inhomogeneous system has the advantage over the homogeneous system in that one does not need to deal with the problem of normalization and the quantities of primary interest such as polarizability, photoabsorption coe
Csanak G, Kilcrease DP. JQSRT 2001;71:273. Csanak G, Meneses GD. JQSRT 2001;71:281. Csanak G, Kilcrease DP. JQSRT 1997;58:537. Carson TR, Mayers DF, Stibbs DWN. Mon Not R Astronom Soc 1968;140:483. Shalitin D, Yin RY, Pratt RH. JQSRT 1982;27:219. Csanak G, Daughton W. JQSRT 2003;in press.