- Email: [email protected]
The lifetime of the koopmans-theorem state in core photoelectron spectroscopy. An estimate based upon the sudden approximation
Volume 104, number 4
CHEWCAL-PHYSICS LmERS
“I%IELIFETIME OF THE KOOPM&VSTHEOREM STATE IN CO=
10 Februwy 1984
PHOTOELECTRON SPECTROSCOPY _
_ANESTIMATE BASED UPON THE SUDDEN ~~PROX~ATION Rolf MANNE Department of Chemistry
Universi@ of Bergen, ALSOi?OBeeen, Norway
Received 6 December 1983
A lifetime for the unrelaxed or Koapmaxwtheorem state can be estimated from the second moment of the spectral proBe of a core photoelectron spectrum assuming validity of the sudden approximation. For Ne 1s the second moment 28509 eV2 leads to a lifetime of 1.0 X 10-l' s. This time is two orders of magnitude shorter than the lifetime of individual electronic states in the spectrum but is of the same magnitude as times considered to be zero in the sudden approximation. Within the validity of this approximation the decay of the Koopmans-theorem state is therefore instantaneous.
1. Introduction The Koopmans-theorem.( or unrelaxed state is obtained by removal of one orbital from a many-electron wavefunction without changing it in other ways. This state, which is not an eigenstate of the hamiltonian, is normally considered only as a theoretical construct but can nevertheless be associated with averages of observable quantities. Hayes [i] has considered one such average, the second moment of a core photoelectron spectrum and expressed it in terms of relaxation energies from a potent&I model. Numerical agreement between calculation and the experimental moment from Ne Is ionization, however, was moderate. This note, which is inspired by the work of Hayes, presents a different interpretation of the second moment, leading to a time estimate for the decay of the Koopmans-theorem state formed according to the sudden approximation. The practical utility of such an estimate is by necessity limited but it may nevertheless have conceptual interest. The theory presented is elementary and similar in approach to a paper from 1947 by Fock and Krylov
121. 2. The sudden approximation and spectral moments The sudden approximation may be described as follows [3,4] I At times t < 0 the system is an eigenstate of the hamiltonian. At time t = 0 the hamiltonian is suddenly changed. In photoefectron spectroscopy this change is effected by the sudden removal of a core electron, leaving the system in a state $R which is not an eigenstate of the new hamiitonian. On the SCF level of approximation $n is the frozen-orbital wavefunction used, e.g. for the calculation of ionization energies using Koopmans’ theorem. In a more rigorous description 3/R may be obtained
by applying the appropriate annihilation operator to a correlated wavefunction of the neutral ground state, i.e. as WOWOne may expand @R in eigenstates of the new hamiltonian as
378
0 Ok-2614~~/$03.00 0 Elsevier Science Publishers B-V, (North-Hoiland Physics Publishing Division)
10 February 1984
CHEMICAL PHYSICS LETTERS
Volume 104, number 4
with
0)
ci=($‘jl@R>. The summations
in (1) and in the following include the normal relaxed core-hole state as well as shake-up and
shake-off states. In the latter a second excitation has occurred leading to discrete and continuum tively. The ~robab~ity for the system to be in the eigenstate JII- is given by
states, respec-
Ii = lCi12 .
(3)
A more complete account of the many-eIectron aspects of photoionization leading to the sudden approximation is given in ref. [S] _ The zeroth-moment sum-rule gives the normalization of the relative intensities: MO=
Cfi= i
1
(4)
and has been discussed by Fadfey [63 in connection with core-level ionization. tained from the expectation value of the hamiltonian with sespect to $R or ~R=(&!‘R/&]~R)= which, together
The fast-moment
sum-rule
is ob-
xEiiCi12, i
C%
with (4), gives
Ml = Cli(Ei i
- ER) = 0 _
(66)
Since Ii and Ei (or Ei - Eg, the ionization energy) are observables, Ml gives an experimental estimate of the unrelaxed energy ER, or ER - Eo, the ionization energy with a frozen or Koopmans-theorem wavefunction [4,7]. The second moment may be written M2=
~li(~~-~~2=~~~I(;Z).-~~)2~~~~=~l(~-~~)~R~2d~, i
(7)
and is known as the mew-squ~e-devia~on (AE)2 of the energy for the state $Q_ T&e square-root AE of this quantity is the root-mean-square (rms) deviation occurring in, e.g. Heiscnberg?s uncertainty relation. It is the purpose of this note to point out the connection betweenM2 and the uncertainty principle and to estimate the lifetime of $R for Is-ionized Ne. There are several difficulties in obtaining a time estimate from the energy uncertainty AZ?. An uncertainty relation involving energy and time may be written formally AEAt 2812
,
(8)
but since time is not a qu~~rn-rnech~~ observable in ordinary non-relativistic theory this relation has a different status than the uncertainty of, e.g. position and momentum f2] _ The lifetime T is defined as the time by which the population of a state is reduced by a factor I/e [8]. For an exponentially decaying state this population is given by exp(-I’tlfi) which leads to I%=fi and the lorentzian
(9) lineshape
f(E - EJ=r($lr)l[(E
-Ei)2+$IP]
,
with I as the full width at half maximum (fwbm) [2]. We now consider the influence of spectral broadening the moment analysis and let each line be convoluted by a broadening function fit;:(E) which may be different
(10) upon for 379
CHEhiiCALPHYSICSLJXTERS
Volume 104, number 4
10 February 1984
different lines. We assumefi(f?) to be-symmetric and normalized and write the full intensity distribution as IQ=
+fi(E--Ei),
(11)
with Ii given according-to (3). The broadening does not affect MOor Ml_ For M2 one obtains fW
The additional term in (12) may be seen as an intensity-weighted average of the second moments of the individual broadening functions. For lorentzians, however, this term is infinite. Therefore, in order to obtain a lifetime estimate we consider expkitly the time evolution of the state $R for times t < 0.
3. Time evolution of the Koopmans-theorem state The probability to find the system in the original state 3/R(O) at time r is given by [2,8] w(t)=
(13)
I{~R(o~l~R~?~}i2,
where the overlap integral ($R@) i @R(t)> is C’tOSely related to propagators or Green’s functions [9]. we may write $R(t) in the Schradinger representation as *R(Z) = C $$j i
eXp(-iEit/?i
-
Tit/m)
,
(14)
where $q and q are ~dependent of time and where we have introduced exponential factors describing the decay of the individual states $Q into a continuum of states orthogonal to *R(t). The main contribution to these decay processes comes from the Auger effect. A Taylor expansion of the exponent& in (14) gives [Z] (3/n(O) I $~(t))
=exPt-~Rfltf)
= eXP(-&?Rt/fi)
Cl~iP[l i
c i
ICi12 expE--i(Ei - ER) t/B - Fitlui]
-i(Ei-E~)flfi-_Ei-E~)2?~2fr2+---](f_ri~~~+..-)
For the lifetime T one obtains
from w&h T could be obtained. We have to assume that Hayes’ estimate of the experiment& second moment for the Ne Is photoelectron spectrum disregards lorentzian broadening and corresponds to the fust term in (12). His value is 2850.9 eV2 which gives A/I?= 53.4 eV. In comparison, the fwhm of individual core-ionized states of Ne is measured to 0.27 + 0.10 eV [lo] _ In the time evolution (16) we may thus ignore terms in I” and write ‘?@i$
=ti2(t
- lie)
07)
Volume 104, number 4
CHEMICALPHYSICS LETTERS
10 February 1984
or AET = 0.8 n )
08)
which is of the same magnitude as eq. (9) I nsertion of numerical values gives r = 1.0 X 10 --I7 s.
4. Discussion It is interesting to contemplate the magnitude of the lifetime T and its relation to the description of the ionization event. First we consider, classically, the excitation time as the time it takes for the photon to pass through the Ne Is orbital. The distance is O-1 X IO-lo m and the speed of light 3 X 10’ m/s. The excitation time 3 X lO-2o s is considerably shorter than the KT state lifetime. Photoelectrons expelled with Al KOLradiation from the normal Ne 1s state have kinetic energy 6 17 eV which gives the velocity 1.5 X lo7 m/s. During the KT state lifetime they have moved only 1 .S A - a distance of atomic dimensions. A classical electron leaving the atom will initially have somewhat larger kinetic energy in order to compensate for the attractive force from the nucleus. This does not change the order of magnitude, however. The sudden approximation assumes that the change of the hamiltonian, i.e. the withdrawal of the photoelectron, is instantaneous. The present analysis shows that the decay of the KT state is an event on the same timescale, i.e. it is also instantaneous. One may even ask whether the use of the sudden approximation for the study of such an event is permissible_ The approximation is certainly too crude if anything more than the order of magnf tude is desired. On the other hand, the use of the sudden approximation for calculations of relative photoionization cross sections is justified under less stringent conditions, i.e. if the transition moment between the core and the continuum orbital varies slowly with the ener,y of the photoelectron, or in other words, if the spread of photoelectron energies is small compared with the average photoelectron energy. This condition is definitely fulfilled in the present case. We have here considered the Koopmans-theorem state as a quasi-stationary state with a short but non-zero lifetime_ In the orthodox picture it is the individual peaks in the photoelectron spectrum which represent quasi-stationary states with lifetimes of the order of 1O-15 s. In that picture, which we normally adhere to, the KT state is a theoretical construct in the evaluation of relative probab~~es of (shake-up) ionization. The difference between the two points of view lies in the precision of the definition of a quasi-stationary state. There is no a priori way to tell that one point of view is more correct than the other. With still higher precision one may consider Auger emission as part of the same process as photoionization [ 111, and in the extreme one may consider as non-stationary an optically active molecule which converts into its mirror image (enantiomer) once during the lifetime of the universe.
Acknowledgement The initial part of this work was done while the author visited the Chemistry Department, University of Washington, Seattle, Washington in 1982 under the faculty exchange program between the University of Bergen and the University of Washington. Thanks are due to E.R. Davidson for support and hospitality. The comments by T. Aberg, H. I&en, and C. Nordling are gratefully arknowledged. Referemxs [l] R.G. Hayes, 3. Electron Spectry. 22 (1981) 365. [2] V. Fock and N. Krylav, J- Phys, (USSR) 11 (1947) 112.
381
V&me
104;ntimber
4
CHEMICAL-PHYSICS
LE’l-l-ERS
131 T, Aber& Arm. Acad. St5 Fenn A VI 308 (1969) 1. r41 R. Manne and T. Aberg, Chem. Phys. Letters 7 (1970) 282. 1st R. Arneberg, J. Miiller and RiManne, C&em. Phys 64 (1982) 249. [61 C.S. Fadley, Chem.Phy+ Letters 2511974) 225. 3.1. Lundquist, Phys. Kondens. Mater. 9 (1969) 236. ;;: AS, Davydov, Quantum mechanics, 2nd Ed. @%gamon Press, Oxford, 1976) pp. 52 ff., pp. 413 ff. 191 L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 (1977) 205. llO1 H. Agree, J. Nordgren, L. &lander, C. Nordling rend K. Siegbabn. 3. Electron Spectxy. 14 (1978) 27, [Ill T. Aberg and G. Howat, Handbuch der Physik. Vol. 31 (Springer, Berlin, 1982) p. 469.
382
10 February 1984
CHEWCAL-PHYSICS LmERS
“I%IELIFETIME OF THE KOOPM&VSTHEOREM STATE IN CO=
10 Februwy 1984
PHOTOELECTRON SPECTROSCOPY _
_ANESTIMATE BASED UPON THE SUDDEN ~~PROX~ATION Rolf MANNE Department of Chemistry
Universi@ of Bergen, ALSOi?OBeeen, Norway
Received 6 December 1983
A lifetime for the unrelaxed or Koapmaxwtheorem state can be estimated from the second moment of the spectral proBe of a core photoelectron spectrum assuming validity of the sudden approximation. For Ne 1s the second moment 28509 eV2 leads to a lifetime of 1.0 X 10-l' s. This time is two orders of magnitude shorter than the lifetime of individual electronic states in the spectrum but is of the same magnitude as times considered to be zero in the sudden approximation. Within the validity of this approximation the decay of the Koopmans-theorem state is therefore instantaneous.
1. Introduction The Koopmans-theorem.( or unrelaxed state is obtained by removal of one orbital from a many-electron wavefunction without changing it in other ways. This state, which is not an eigenstate of the hamiltonian, is normally considered only as a theoretical construct but can nevertheless be associated with averages of observable quantities. Hayes [i] has considered one such average, the second moment of a core photoelectron spectrum and expressed it in terms of relaxation energies from a potent&I model. Numerical agreement between calculation and the experimental moment from Ne Is ionization, however, was moderate. This note, which is inspired by the work of Hayes, presents a different interpretation of the second moment, leading to a time estimate for the decay of the Koopmans-theorem state formed according to the sudden approximation. The practical utility of such an estimate is by necessity limited but it may nevertheless have conceptual interest. The theory presented is elementary and similar in approach to a paper from 1947 by Fock and Krylov
121. 2. The sudden approximation and spectral moments The sudden approximation may be described as follows [3,4] I At times t < 0 the system is an eigenstate of the hamiltonian. At time t = 0 the hamiltonian is suddenly changed. In photoefectron spectroscopy this change is effected by the sudden removal of a core electron, leaving the system in a state $R which is not an eigenstate of the new hamiitonian. On the SCF level of approximation $n is the frozen-orbital wavefunction used, e.g. for the calculation of ionization energies using Koopmans’ theorem. In a more rigorous description 3/R may be obtained
by applying the appropriate annihilation operator to a correlated wavefunction of the neutral ground state, i.e. as WOWOne may expand @R in eigenstates of the new hamiltonian as
378
0 Ok-2614~~/$03.00 0 Elsevier Science Publishers B-V, (North-Hoiland Physics Publishing Division)
10 February 1984
CHEMICAL PHYSICS LETTERS
Volume 104, number 4
with
0)
ci=($‘jl@R>. The summations
in (1) and in the following include the normal relaxed core-hole state as well as shake-up and
shake-off states. In the latter a second excitation has occurred leading to discrete and continuum tively. The ~robab~ity for the system to be in the eigenstate JII- is given by
states, respec-
Ii = lCi12 .
(3)
A more complete account of the many-eIectron aspects of photoionization leading to the sudden approximation is given in ref. [S] _ The zeroth-moment sum-rule gives the normalization of the relative intensities: MO=
Cfi= i
1
(4)
and has been discussed by Fadfey [63 in connection with core-level ionization. tained from the expectation value of the hamiltonian with sespect to $R or ~R=(&!‘R/&]~R)= which, together
The fast-moment
sum-rule
is ob-
xEiiCi12, i
C%
with (4), gives
Ml = Cli(Ei i
- ER) = 0 _
(66)
Since Ii and Ei (or Ei - Eg, the ionization energy) are observables, Ml gives an experimental estimate of the unrelaxed energy ER, or ER - Eo, the ionization energy with a frozen or Koopmans-theorem wavefunction [4,7]. The second moment may be written M2=
~li(~~-~~2=~~~I(;Z).-~~)2~~~~=~l(~-~~)~R~2d~, i
(7)
and is known as the mew-squ~e-devia~on (AE)2 of the energy for the state $Q_ T&e square-root AE of this quantity is the root-mean-square (rms) deviation occurring in, e.g. Heiscnberg?s uncertainty relation. It is the purpose of this note to point out the connection betweenM2 and the uncertainty principle and to estimate the lifetime of $R for Is-ionized Ne. There are several difficulties in obtaining a time estimate from the energy uncertainty AZ?. An uncertainty relation involving energy and time may be written formally AEAt 2812
,
(8)
but since time is not a qu~~rn-rnech~~ observable in ordinary non-relativistic theory this relation has a different status than the uncertainty of, e.g. position and momentum f2] _ The lifetime T is defined as the time by which the population of a state is reduced by a factor I/e [8]. For an exponentially decaying state this population is given by exp(-I’tlfi) which leads to I%=fi and the lorentzian
(9) lineshape
f(E - EJ=r($lr)l[(E
-Ei)2+$IP]
,
with I as the full width at half maximum (fwbm) [2]. We now consider the influence of spectral broadening the moment analysis and let each line be convoluted by a broadening function fit;:(E) which may be different
(10) upon for 379
CHEhiiCALPHYSICSLJXTERS
Volume 104, number 4
10 February 1984
different lines. We assumefi(f?) to be-symmetric and normalized and write the full intensity distribution as IQ=
+fi(E--Ei),
(11)
with Ii given according-to (3). The broadening does not affect MOor Ml_ For M2 one obtains fW
The additional term in (12) may be seen as an intensity-weighted average of the second moments of the individual broadening functions. For lorentzians, however, this term is infinite. Therefore, in order to obtain a lifetime estimate we consider expkitly the time evolution of the state $R for times t < 0.
3. Time evolution of the Koopmans-theorem state The probability to find the system in the original state 3/R(O) at time r is given by [2,8] w(t)=
(13)
I{~R(o~l~R~?~}i2,
where the overlap integral ($R@) i @R(t)> is C’tOSely related to propagators or Green’s functions [9]. we may write $R(t) in the Schradinger representation as *R(Z) = C $$j i
eXp(-iEit/?i
-
Tit/m)
,
(14)
where $q and q are ~dependent of time and where we have introduced exponential factors describing the decay of the individual states $Q into a continuum of states orthogonal to *R(t). The main contribution to these decay processes comes from the Auger effect. A Taylor expansion of the exponent& in (14) gives [Z] (3/n(O) I $~(t))
=exPt-~Rfltf)
= eXP(-&?Rt/fi)
Cl~iP[l i
c i
ICi12 expE--i(Ei - ER) t/B - Fitlui]
-i(Ei-E~)flfi-_Ei-E~)2?~2fr2+---](f_ri~~~+..-)
For the lifetime T one obtains
from w&h T could be obtained. We have to assume that Hayes’ estimate of the experiment& second moment for the Ne Is photoelectron spectrum disregards lorentzian broadening and corresponds to the fust term in (12). His value is 2850.9 eV2 which gives A/I?= 53.4 eV. In comparison, the fwhm of individual core-ionized states of Ne is measured to 0.27 + 0.10 eV [lo] _ In the time evolution (16) we may thus ignore terms in I” and write ‘?@i$
=ti2(t
- lie)
07)
Volume 104, number 4
CHEMICALPHYSICS LETTERS
10 February 1984
or AET = 0.8 n )
08)
which is of the same magnitude as eq. (9) I nsertion of numerical values gives r = 1.0 X 10 --I7 s.
4. Discussion It is interesting to contemplate the magnitude of the lifetime T and its relation to the description of the ionization event. First we consider, classically, the excitation time as the time it takes for the photon to pass through the Ne Is orbital. The distance is O-1 X IO-lo m and the speed of light 3 X 10’ m/s. The excitation time 3 X lO-2o s is considerably shorter than the KT state lifetime. Photoelectrons expelled with Al KOLradiation from the normal Ne 1s state have kinetic energy 6 17 eV which gives the velocity 1.5 X lo7 m/s. During the KT state lifetime they have moved only 1 .S A - a distance of atomic dimensions. A classical electron leaving the atom will initially have somewhat larger kinetic energy in order to compensate for the attractive force from the nucleus. This does not change the order of magnitude, however. The sudden approximation assumes that the change of the hamiltonian, i.e. the withdrawal of the photoelectron, is instantaneous. The present analysis shows that the decay of the KT state is an event on the same timescale, i.e. it is also instantaneous. One may even ask whether the use of the sudden approximation for the study of such an event is permissible_ The approximation is certainly too crude if anything more than the order of magnf tude is desired. On the other hand, the use of the sudden approximation for calculations of relative photoionization cross sections is justified under less stringent conditions, i.e. if the transition moment between the core and the continuum orbital varies slowly with the ener,y of the photoelectron, or in other words, if the spread of photoelectron energies is small compared with the average photoelectron energy. This condition is definitely fulfilled in the present case. We have here considered the Koopmans-theorem state as a quasi-stationary state with a short but non-zero lifetime_ In the orthodox picture it is the individual peaks in the photoelectron spectrum which represent quasi-stationary states with lifetimes of the order of 1O-15 s. In that picture, which we normally adhere to, the KT state is a theoretical construct in the evaluation of relative probab~~es of (shake-up) ionization. The difference between the two points of view lies in the precision of the definition of a quasi-stationary state. There is no a priori way to tell that one point of view is more correct than the other. With still higher precision one may consider Auger emission as part of the same process as photoionization [ 111, and in the extreme one may consider as non-stationary an optically active molecule which converts into its mirror image (enantiomer) once during the lifetime of the universe.
Acknowledgement The initial part of this work was done while the author visited the Chemistry Department, University of Washington, Seattle, Washington in 1982 under the faculty exchange program between the University of Bergen and the University of Washington. Thanks are due to E.R. Davidson for support and hospitality. The comments by T. Aberg, H. I&en, and C. Nordling are gratefully arknowledged. Referemxs [l] R.G. Hayes, 3. Electron Spectry. 22 (1981) 365. [2] V. Fock and N. Krylav, J- Phys, (USSR) 11 (1947) 112.
381
V&me
104;ntimber
4
CHEMICAL-PHYSICS
LE’l-l-ERS
131 T, Aber& Arm. Acad. St5 Fenn A VI 308 (1969) 1. r41 R. Manne and T. Aberg, Chem. Phys. Letters 7 (1970) 282. 1st R. Arneberg, J. Miiller and RiManne, C&em. Phys 64 (1982) 249. [61 C.S. Fadley, Chem.Phy+ Letters 2511974) 225. 3.1. Lundquist, Phys. Kondens. Mater. 9 (1969) 236. ;;: AS, Davydov, Quantum mechanics, 2nd Ed. @%gamon Press, Oxford, 1976) pp. 52 ff., pp. 413 ff. 191 L.S. Cederbaum and W. Domcke, Advan. Chem. Phys. 36 (1977) 205. llO1 H. Agree, J. Nordgren, L. &lander, C. Nordling rend K. Siegbabn. 3. Electron Spectxy. 14 (1978) 27, [Ill T. Aberg and G. Howat, Handbuch der Physik. Vol. 31 (Springer, Berlin, 1982) p. 469.
382
10 February 1984