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Resonance by the complex coordinate method with hermitean hamiltonian
CHEMICAL
Volume 99, number 4
RESONANCE
BY THE COMPLEX COORDINATE
METHOD WITH HERMITEAN
Nimrod MOISEYEV ** Deparrmenf of chemisny. Technion - Israel Insxitute of Technology, Recked
12 August 1983
PHYSICS LETTERS
Haifa 32000.
HAMILTONIAN
*
Israel
21 April 1983; in final form 6 June 1983
Using the cornpIe.\ coordinate method, a new hermirean hamiltonian is formulated, in which the resonance position and aidth are the “natural” perturbation strength parameters. If the resonance position is known the resonance width obtained by the presented theory is a lower bound to the e?tact value.
1. Introduction The complex coordinate method [l] enables us to calculate by techniques of bound states, atomic [2] and molecular [3] autoionization resonances and predissociation resonances in molecules [4] and van der Waalscomplexes [5]_ However, the resonance position, E,. and width. Ei, are presented as a complex eigenvalue of a non-hermitean hamiltonian_ Thus. in the finite matrix hamiltonian appro_ximation,
[(H, + iHi) - (E, + iEi)l](JI,
•t-iQi)=
0,
(1)
where H, and Hi are constructed of II basis functions_ Since the rotated hamiltonian matrix H, + iHi, is a complex symmetric matrix, then H, and Hi are hermitean matrices. Although the hamiltonian is nonhermitean many theorems that were originally derived for hermitean hamiltonians are still valid if the scalar product is replaced by the complex one [6] _However, the variational solutions are not upper limit to the exact eigenvalues [6,7], and the perturbational theory can be applied only for very specific cases [S] . In the present paper we show that the resonance position and width are vibrational parameters of an hermitean hamiltonian. In this new representation of the complex coordi* Supported in part by the US-Israeli Binational Foundation. ** Yigal Allon Fellow.
364
nate method the variational resonance width is lower bound to the exact value if the resonance position is known from experiment or theory. In addition to the methodical advantage of describing physical phenomena by hermitean hamiltonian as in the presented theory, the perturbational method is the “natural” procedure of calculating resonance widths and positions_
2. Theory By splitting eq. (1) into two equations of which one contains only real terms and the other oniy the imaginary terms, and introducing the notation, H, = Hr -E, one can get the following eigenvahte equation:
where Gr = H, - E, 7 _ Note that theretiis a hermitean hamiltonian. If E, + iEi is not an eigenvalue of eq. (1) the following eigenvalue problem can be considered: %e,Er,Ei)cb=
A+,
(3)
where h is a real non-vanishing parameter being an eigenvahte of H and 4 is the associated eigenvector. Therefore, E, and Ei for any given 13are the variktional parameters of HI, 3f% = x%#B_ 0 009-2614/83/0000-0000/S
(4) 03.00 0 1983 North-Holland
CHEMICALPHYSICSLETTERS
Volume99, number4
12 August1983
Note that X2 is the variance, u, of the approximated complex rotated hamiltonian as defined in eq. (1). By substituting H as defined in eq. (2) into eq. (4) we get that,
iteration Ei and AE, are calculated by eqs. (12) and (13) I and are substituted into eq. (5) for the next step of the calculation. If convergence is achieved then h2 is vanished and
(H(O) + EiHil’ + AErH:1))+=
LY= - [Ei’ -I-(AEr)’ -I-2ErAEr] _
a+ ,
(5)
where or= h2 -E;
(6)
ii:+
H; [Hi, Hr]
H(o) = .
- (AEr)2 - 2AErEr,
--[Hi, HI]
(7)
H,2+H2 > ’
(8)
[jj@, + E_(i-I)g;r) 1 H$l’=
(-2Hr 0
’
(9)
-2H,
and Er is the resonance position obtained by experiment or by other type of calculations. For example, by carrying out the Holien-Midtal stabilization calculations where the internal coordinate of the harniitonian are scaled by a real factor [9]_ If (E, + AE,) + iEi is the eigenvahre of H, + iHi which is defined in eq. (1) then X2 = 0. Therefore, the resonance width and the correction to the resonance position will be the values of Ei and AE, for which, aO’)/aEi
= 0
(10)
and d(XZ)/d(AEr) = 0.
(11)
The resonance width and position can be obtained from eqs. (5)-(11) either by the variational or perturbational methods.
From the Hellmann-Feynman [ 101 theorem one can get that the two stationary conditions given in eq. (10) and eq. (11) are satisfied if
AEr = -$(2E,
, + drTHy)+) _
+ (E-‘i-“)2],$,(1? 1
= X’@(i) ,
(13
where E.ci> = _&@@r)l@(i)) 1
_
(16)
In the first step of the iterations Ejl) is zero. In sub sequent steps Ei(i) -1s calculated according to eq. (16) based on the Hellmann-Feynmann theory, eq. (I 2). Cp(i)is the ground state of the hamiitonian which is defined in eq. (I 5). As convergence is achieved then h’+O
(17)
and EcJ9 1 + -((~exrrti~(o)t~~~ac~)l’~
_
(18)
Since o(l) is the eigenfunction of H(O) whereas eXXt is not then d (X(l))’ = (@(L)IH(0)I#W < c@~Uetlfi’O’]&..,,t>
(19)
and -A(l) is an upper bound to the exact value of Ei, Ei(exact) < -A(‘)
_
(20)
Since Ei = -$r then the resonance width derived from h(l) is a lower bound of the exact value.
2.1. Tile variationalmethod
E_ = -$,$TH$“+ 1
(14)
This iterative variational procedure has the computational advantage that only the ground state of the hamiltonian, W2, should be calculated in order to determine the resonance and width. If the resonance position is known from experiment or theory a lower bound of the corresponding resonance width can be derived: Lower bound to the resonance width. If AEr = 0 and eq. (5) is solved iteratively then
(12) (13)
Therefore eq. (5) can be solved iteratively. In each
2-2. l7re perturbation method In the perturbation approach both Ei and AE, can be considered as the strength perturbational parameters. The eigenvalue a; defined in eq. (5), will be calculated by the double perturbation theory [ 1 I]: n n
a= C
C (Ei)‘(AE,)‘e~g”
k=OI=O
,
(21) 365
CHEMICAL
Volume 99. number 4
where &o are the generalized Rayleigh-Schriidinger perturbation energies [ Ill_ In the zero-order approximation the lowest eigenvalue of H(O) yields the minimal value of X2, H(0)do = eOdo. Therefore, the perturbation energies will be derived for g(o) G do_ By substituting eqs. (6). (10) and (11) into eq_ (21) we get that the resonance width and the correction to the resonance position are the zeros of two n-order polynomials_ The physical zero solutions are those which are converged by increasing II inside the circle of convergence of the perturbation series. Whereas, all the other zeros of the partial sums will converge to every point on the circle [I?]. The errors of the resonance position and width will be estimated by the effect of increasing the size of the basis set on the values of the zeros of those polynomials. The calculations can be much simplified if the resonance position is known and LIE, is negligible. Then the width is the only perturbation parameter and n
&
=
x
(E.)+(k)
k=O
_
(23)
’
The resonance width is determined by using the stationary condition given in eq. (10)
5 bk(Epl=
0.
(23)
k=l
where b, = kc” The lowest which
+ 26 A.2 _ order
of the perturbation
the resonance
second
one,
Ei zz -
E;t)/7(q)
(24) width
+ 1))
expansion
can be estimated
from
PHYSICS
LE?TERS
12 August 1983
or on the basis of the zero-order approximation will taken as the ground state of H(O).
il>
3. Discussion The new representation of the complex coordinate procedure given here shows how the resonance position and width can be calculated from hermitean matrices and by using the common scalar product rather than the complex one [6] *_ The resonance positions and lifetimes which are calculated by the new method have exactly the same value as the eigenvalues of the complex-rotated hamiltonian which are obtained by the conventional procedure. The computational advantage of the presented method is that only the ground state of the hermitean hamiltonian should be calculated as function of the rotational angle. However, it is unclear how this method can be applied to resonance states which possess several open channels [ 141. If the resonance position is known the variational calculations yield a lower bound to the exact I?alueof the resonance width. It seems that in this representation the resonance width is the “natural” strength parameter in the perturbation expansion_ Since Ei is small with respect to the position E, then one may expect that the resonance width will be estimated from the second order perturbational calculations. The applications of the procedure presented here to autoionization and predissociation resonances are presently under investigation.
is the
(25)
* On the incomplete spectrum of the finite complex-scaled hamittonian matrices, see ref. [ 131.
where
[ l] W.P. Reinhardt. Ann. Rev. Phys. Chem. 33 (1982) 223.
and H(‘)d. = ~!‘)d. (28) I I I’ The JZ>state is selected as the one which yields a minimal value of $ 366
[2] The special complex coordinate issue, Intern. J. Quantum Chem. 14 (1978). [ 31 T-N. Rescigno and CW. McCurdy, Phys. Rev. Letters 41 (1978) 1364; N. hloiseyev and CT. Corcoram, Phys Rev. A20 (1979) 814. [4] R. Lefebvre. Chem. Phys Letters 70 (1980) 430; N. hloiseyev, hiol. Phys 42 (1981) 129;lntem. J. Quantum Chem. 29 (1981) 835.
Volume 99, numbfx 4
CHEMICAL PHYSICS LETTERS
[5] S-1 Chu, J, Chem. Phys 72 (1980) 4722. [C] N- Moiseyev, P.R. Certain and Fe Weir&old. Mol. Phys. 36 11978) 1613. f7f N_Mniseyev and F_Weinfrold, Intern. J_Quamum Chem. 17 (1980) 1201. [ 8 f N. Mob*yev and P.R. Certain, Mot Phys 37
12 August 1983
{lo]
H. Hellman, Emfuhrung in die Quantenchemie (Deutiche, Leipz& 1937); A-P. Feynman. Phys. Rev. 56 (1939) 340. f II] J-0. Hlsehfelder, W_Byes Brown and ST_ Epstein, Advan- Quantum Chem. I (1964) 255, and references tlierein. [ 121 E-C. Tit&marsh, The theory af functions, 2nd JZd. (Oxford Univ. Press, London, 1939) p_ 238. [ 131 N, Moiseyev and S. Friedland, Phys. Rev. A22 (1980) 618. 1141 P_ Noro and NS. Taylor. I Phys. B13 (198011377: Z. BaciEand J. Simcms, intern. 1. Quantum Chem. 21 fl982) 727,
367
Volume 99, number 4
RESONANCE
BY THE COMPLEX COORDINATE
METHOD WITH HERMITEAN
Nimrod MOISEYEV ** Deparrmenf of chemisny. Technion - Israel Insxitute of Technology, Recked
12 August 1983
PHYSICS LETTERS
Haifa 32000.
HAMILTONIAN
*
Israel
21 April 1983; in final form 6 June 1983
Using the cornpIe.\ coordinate method, a new hermirean hamiltonian is formulated, in which the resonance position and aidth are the “natural” perturbation strength parameters. If the resonance position is known the resonance width obtained by the presented theory is a lower bound to the e?tact value.
1. Introduction The complex coordinate method [l] enables us to calculate by techniques of bound states, atomic [2] and molecular [3] autoionization resonances and predissociation resonances in molecules [4] and van der Waalscomplexes [5]_ However, the resonance position, E,. and width. Ei, are presented as a complex eigenvalue of a non-hermitean hamiltonian_ Thus. in the finite matrix hamiltonian appro_ximation,
[(H, + iHi) - (E, + iEi)l](JI,
•t-iQi)=
0,
(1)
where H, and Hi are constructed of II basis functions_ Since the rotated hamiltonian matrix H, + iHi, is a complex symmetric matrix, then H, and Hi are hermitean matrices. Although the hamiltonian is nonhermitean many theorems that were originally derived for hermitean hamiltonians are still valid if the scalar product is replaced by the complex one [6] _However, the variational solutions are not upper limit to the exact eigenvalues [6,7], and the perturbational theory can be applied only for very specific cases [S] . In the present paper we show that the resonance position and width are vibrational parameters of an hermitean hamiltonian. In this new representation of the complex coordi* Supported in part by the US-Israeli Binational Foundation. ** Yigal Allon Fellow.
364
nate method the variational resonance width is lower bound to the exact value if the resonance position is known from experiment or theory. In addition to the methodical advantage of describing physical phenomena by hermitean hamiltonian as in the presented theory, the perturbational method is the “natural” procedure of calculating resonance widths and positions_
2. Theory By splitting eq. (1) into two equations of which one contains only real terms and the other oniy the imaginary terms, and introducing the notation, H, = Hr -E, one can get the following eigenvahte equation:
where Gr = H, - E, 7 _ Note that theretiis a hermitean hamiltonian. If E, + iEi is not an eigenvalue of eq. (1) the following eigenvalue problem can be considered: %e,Er,Ei)cb=
A+,
(3)
where h is a real non-vanishing parameter being an eigenvahte of H and 4 is the associated eigenvector. Therefore, E, and Ei for any given 13are the variktional parameters of HI, 3f% = x%#B_ 0 009-2614/83/0000-0000/S
(4) 03.00 0 1983 North-Holland
CHEMICALPHYSICSLETTERS
Volume99, number4
12 August1983
Note that X2 is the variance, u, of the approximated complex rotated hamiltonian as defined in eq. (1). By substituting H as defined in eq. (2) into eq. (4) we get that,
iteration Ei and AE, are calculated by eqs. (12) and (13) I and are substituted into eq. (5) for the next step of the calculation. If convergence is achieved then h2 is vanished and
(H(O) + EiHil’ + AErH:1))+=
LY= - [Ei’ -I-(AEr)’ -I-2ErAEr] _
a+ ,
(5)
where or= h2 -E;
(6)
ii:+
H; [Hi, Hr]
H(o) = .
- (AEr)2 - 2AErEr,
--[Hi, HI]
(7)
H,2+H2 > ’
(8)
[jj@, + E_(i-I)g;r) 1 H$l’=
(-2Hr 0
’
(9)
-2H,
and Er is the resonance position obtained by experiment or by other type of calculations. For example, by carrying out the Holien-Midtal stabilization calculations where the internal coordinate of the harniitonian are scaled by a real factor [9]_ If (E, + AE,) + iEi is the eigenvahre of H, + iHi which is defined in eq. (1) then X2 = 0. Therefore, the resonance width and the correction to the resonance position will be the values of Ei and AE, for which, aO’)/aEi
= 0
(10)
and d(XZ)/d(AEr) = 0.
(11)
The resonance width and position can be obtained from eqs. (5)-(11) either by the variational or perturbational methods.
From the Hellmann-Feynman [ 101 theorem one can get that the two stationary conditions given in eq. (10) and eq. (11) are satisfied if
AEr = -$(2E,
, + drTHy)+) _
+ (E-‘i-“)2],$,(1? 1
= X’@(i) ,
(13
where E.ci> = _&@@r)l@(i)) 1
_
(16)
In the first step of the iterations Ejl) is zero. In sub sequent steps Ei(i) -1s calculated according to eq. (16) based on the Hellmann-Feynmann theory, eq. (I 2). Cp(i)is the ground state of the hamiitonian which is defined in eq. (I 5). As convergence is achieved then h’+O
(17)
and EcJ9 1 + -((~exrrti~(o)t~~~ac~)l’~
_
(18)
Since o(l) is the eigenfunction of H(O) whereas eXXt is not then d (X(l))’ = (@(L)IH(0)I#W < c@~Uetlfi’O’]&..,,t>
(19)
and -A(l) is an upper bound to the exact value of Ei, Ei(exact) < -A(‘)
_
(20)
Since Ei = -$r then the resonance width derived from h(l) is a lower bound of the exact value.
2.1. Tile variationalmethod
E_ = -$,$TH$“+ 1
(14)
This iterative variational procedure has the computational advantage that only the ground state of the hamiltonian, W2, should be calculated in order to determine the resonance and width. If the resonance position is known from experiment or theory a lower bound of the corresponding resonance width can be derived: Lower bound to the resonance width. If AEr = 0 and eq. (5) is solved iteratively then
(12) (13)
Therefore eq. (5) can be solved iteratively. In each
2-2. l7re perturbation method In the perturbation approach both Ei and AE, can be considered as the strength perturbational parameters. The eigenvalue a; defined in eq. (5), will be calculated by the double perturbation theory [ 1 I]: n n
a= C
C (Ei)‘(AE,)‘e~g”
k=OI=O
,
(21) 365
CHEMICAL
Volume 99. number 4
where &o are the generalized Rayleigh-Schriidinger perturbation energies [ Ill_ In the zero-order approximation the lowest eigenvalue of H(O) yields the minimal value of X2, H(0)do = eOdo. Therefore, the perturbation energies will be derived for g(o) G do_ By substituting eqs. (6). (10) and (11) into eq_ (21) we get that the resonance width and the correction to the resonance position are the zeros of two n-order polynomials_ The physical zero solutions are those which are converged by increasing II inside the circle of convergence of the perturbation series. Whereas, all the other zeros of the partial sums will converge to every point on the circle [I?]. The errors of the resonance position and width will be estimated by the effect of increasing the size of the basis set on the values of the zeros of those polynomials. The calculations can be much simplified if the resonance position is known and LIE, is negligible. Then the width is the only perturbation parameter and n
&
=
x
(E.)+(k)
k=O
_
(23)
’
The resonance width is determined by using the stationary condition given in eq. (10)
5 bk(Epl=
0.
(23)
k=l
where b, = kc” The lowest which
+ 26 A.2 _ order
of the perturbation
the resonance
second
one,
Ei zz -
E;t)/7(q)
(24) width
+ 1))
expansion
can be estimated
from
PHYSICS
LE?TERS
12 August 1983
or on the basis of the zero-order approximation will taken as the ground state of H(O).
il>
3. Discussion The new representation of the complex coordinate procedure given here shows how the resonance position and width can be calculated from hermitean matrices and by using the common scalar product rather than the complex one [6] *_ The resonance positions and lifetimes which are calculated by the new method have exactly the same value as the eigenvalues of the complex-rotated hamiltonian which are obtained by the conventional procedure. The computational advantage of the presented method is that only the ground state of the hermitean hamiltonian should be calculated as function of the rotational angle. However, it is unclear how this method can be applied to resonance states which possess several open channels [ 141. If the resonance position is known the variational calculations yield a lower bound to the exact I?alueof the resonance width. It seems that in this representation the resonance width is the “natural” strength parameter in the perturbation expansion_ Since Ei is small with respect to the position E, then one may expect that the resonance width will be estimated from the second order perturbational calculations. The applications of the procedure presented here to autoionization and predissociation resonances are presently under investigation.
is the
(25)
* On the incomplete spectrum of the finite complex-scaled hamittonian matrices, see ref. [ 131.
where
[ l] W.P. Reinhardt. Ann. Rev. Phys. Chem. 33 (1982) 223.
and H(‘)d. = ~!‘)d. (28) I I I’ The JZ>state is selected as the one which yields a minimal value of $ 366
[2] The special complex coordinate issue, Intern. J. Quantum Chem. 14 (1978). [ 31 T-N. Rescigno and CW. McCurdy, Phys. Rev. Letters 41 (1978) 1364; N. hloiseyev and CT. Corcoram, Phys Rev. A20 (1979) 814. [4] R. Lefebvre. Chem. Phys Letters 70 (1980) 430; N. hloiseyev, hiol. Phys 42 (1981) 129;lntem. J. Quantum Chem. 29 (1981) 835.
Volume 99, numbfx 4
CHEMICAL PHYSICS LETTERS
[5] S-1 Chu, J, Chem. Phys 72 (1980) 4722. [C] N- Moiseyev, P.R. Certain and Fe Weir&old. Mol. Phys. 36 11978) 1613. f7f N_Mniseyev and F_Weinfrold, Intern. J_Quamum Chem. 17 (1980) 1201. [ 8 f N. Mob*yev and P.R. Certain, Mot Phys 37
12 August 1983
{lo]
H. Hellman, Emfuhrung in die Quantenchemie (Deutiche, Leipz& 1937); A-P. Feynman. Phys. Rev. 56 (1939) 340. f II] J-0. Hlsehfelder, W_Byes Brown and ST_ Epstein, Advan- Quantum Chem. I (1964) 255, and references tlierein. [ 121 E-C. Tit&marsh, The theory af functions, 2nd JZd. (Oxford Univ. Press, London, 1939) p_ 238. [ 131 N, Moiseyev and S. Friedland, Phys. Rev. A22 (1980) 618. 1141 P_ Noro and NS. Taylor. I Phys. B13 (198011377: Z. BaciEand J. Simcms, intern. 1. Quantum Chem. 21 fl982) 727,
367