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Relation between broadening and Kramers–Kronig transformation of calculated optical spectra
15 August 1999
Optics Communications 167 Ž1999. 105–109 www.elsevier.comrlocateroptcom
Relation between broadening and Kramers–Kronig transformation of calculated optical spectra Anna Delin Condensed Matter Theory Group, Department of Physics, Uppsala UniÕersity, P. O. Box 530, S-75121 Uppsala, Sweden Received 7 May 1999; received in revised form 16 June 1999; accepted 17 June 1999
Abstract It is shown that, in practice, the Kramers–Kronig transformation and broadening of spectra do not commute. Therefore, broadening of calculated excitation spectra should be performed only on absorptive components. The two operations commute only in the special case when the lifetime of the excited state is taken to be constant, i.e., independent of the excitation energy and of the energies of the electron states involved. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 78.20.Ci
1. Introduction
2. Calculation of the optical conductivity
With modern band-structure methods it has become possible to successfully calculate both magneto-optical and optical properties of materials w1–3x. Broadening of the calculated spectra is an integral part of the calculation, since broadening often is highly relevant for the comparison of the calculated spectra to experimental results. Especially in the calculation of magneto-optical spectra, the broadening is in fact essential in order to obtain results comparable to experiment w4x. In this study the relation between lifetime broadening and the Kramers–Kronig transformation is discussed. In principle the results are valid for any excitation spectra.
The fundamental quantity used for describing excitations caused by a transverse electromagnetic field, i.e., photons, is the optical conductivity s , or, equivalently, the dielectric function. These entities are tensors of the second rank. All components are complex quantities and the real and imaginary part of each component is related by causality. Depending on the convention used in the notation and whether the component is diagonal or off-diagonal, it varies whether the real or the imaginary part is the absorptive component. Therefore, in this paper the discussion will be in terms of absorptive and dispersive components only, and not in terms of real and imaginary parts.
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 3 1 7 - X
A. Delin r Optics Communications 167 (1999) 105–109
106
Within the random phase approximation w5x, neglecting local field effects w6x, two different approaches are often used in the calculation of the interband part of the optical properties. They both start from the following expression w1,7x derived using linear response theory:
sab Ž v . s
ie 2 "m2
q
1
Ýv nnX
X
nn
ž
las derived by Kramers and Kronig w9x. When using the Kramers–Kronig transformation in the calculation of the dispersive component it is often necessary to evaluate the absorption spectrum to high energies in order to achieve a converged result for the dispersive spectrum.
² n < pa < nX : ² nX < pb < n:
v y v nX n q i G
² n < pb < nX : ² nX < pa < n:
v q v nX n q i G
/
,
3. Broadening, general considerations
Ž 1.
In this equation, which is often called the Kubo formula, a , b are Cartesian coordinate indices and e and m are the electron charge and mass, respectively. v is the frequency of the incoming electromagnetic radiation, " v nX n is the energy difference between the initial and final one-particle states < n: and < nX :, Ž p x , p y , pz . s p is the momentum operator, and G is inversely proportional to the lifetime t of the excited state. In the first approach this formula is used directly Ž G then has to be finite. and both the absorptive and dispersive parts of the optical conductivity are calculated from the eigenvalues and eigenfunctions resulting from the band-structure calculation. In the second version the limit t ™ ` is taken in Eq. Ž1. and only the absorptive part is calculated from the eigenvalues and eigenfunctions resulting from the band-structure calculation. In principle, also the dispersive part could be calculated in a similar way in this limit but in practice the dispersive component is calculated from the absorptive one using a causality argument which may be written in a variety of ways. The original expression for calculating the dispersive part from the absorptive was derived by Kramers and Kronig w8,9x. More recently King w10x developed an alternative formula using conjugate Fourier series. Furthermore, Bertie and Zhang w11x have shown that the Kramers–Kronig transformation can be written as two successive fast Fourier transforms, a method which significantly speeds up the calculation of the integral. However, they all amount to the same thing and they are all the same integral operation, if used as intended. In the following this operation will be referred to as the Kramers–Kronig transformation although sometimes this term is used in a more strict sense, referring to the original formu-
At first sight, broadening of spectra appears to be a simple matter. However, as will be proven below, when broadening optical spectra it is important to take into account the fundamental difference between the absorptive and dispersive parts. Failure to do so may result in spurious peaks at low energy in, e.g., reflectivity or Kerr signal, since these spectra consist of a combination of absorptive and dispersive contributions. Broadening of optical spectra serves several distinct goals. One is to simulate a finite experimental resolution, which is usually done by convoluting with a Gaussian. Naturally, the resolution broadening should be performed on the final spectrum, i.e., the reflectivity or the Kerr signal, if that is what has been measured in the experiment one wishes to compare the calculations to. Thermal broadening may also be modeled by a Gaussian. Another goal is to take into account the finite lifetime of excited states. This is called lifetime broadening, and is normally performed by convoluting the spectrum with a Lorentzian. It is found experimentally that lifetime effects play a considerable role in optical transitions. The lifetimes are state dependent and most highly excited states are more unstable and have shorter lifetimes w12x
4. Commutation of broadening and Kramers– Kronig transformation It will be demonstrated here that, in general, lifetime broadening should be performed only on purely absorptive spectra. The fundamental argument is that causality has to hold for the components of the optical conductivity used for calculating the final optical spectrum. This implies that it is always cor-
A. Delin r Optics Communications 167 (1999) 105–109
rect to perform the broadening before the Kramers– Kronig transformation. The opposite order might not be valid, though. Optical spectra are normally thought of as defined on the positive interval w0,`x only. However, they can easily be extended to wy`,`x. In this case the absorptive part becomes odd and the dispersive part even, symmetries which follow from the fact that a real-valued electric field must produce a real-valued polarization of the charges. The extended definition is most useful since it removes all problems connected with the semi-infinite interval and it will be used here. As a matter of fact, Peiponen and Vartiainen w13x have shown that it is even essential in using this extended definition when modeling tails of absorption and extinction curves. Broadening of a function f Ž v ., defined on wy`,`x, is performed by convoluting it with some distribution function B Ž G , v . normalized to unity. The positive parameter G determines the width of the distribution function. The broadened function may then be written `
Bf Ž v . s
X
X
X
Hy` B Ž G , v y v . f Ž v . d v ,
Ž 2.
where B denotes the operation ‘‘convolute with B’’. If the broadening is constant, i.e., G in B Ž G , v . is independent of v , Eq. Ž2. is equivalent to a convolution ‘in the mathematical sense’: `
g) f Ž x. s
Hy` g Ž x y y . f Ž y . dy.
Ž 3.
This stricter type of convolution is to be distinguished from the type of convolution we get if the broadening width is not held constant. Now note that the Hilbert or Kramers–Kronig transformation, which in this paper will be denoted by K Žalthough the standard notation for Hilbert transforms would rather be H ., may be written as a convolution of the type described in Eq. Ž3. of the inverse of the photon energy v , i.e., Kf Ž v . s
1
p
`
f Ž vX .
X
Hy` v y v d v ,
P
X
Ž 4.
where P denotes the Cauchy principal value. A fundamental property of convolutions of the type described in Eq. Ž3. is that they commute, i.e., f ) g s g ) f. It then directly follows that as long as
107
the broadening is constant is does not matter what is done first, broadening or Kramers–Kronig transformation. In this case broadening the dispersive and absorptive parts separately, and then calculating the final optical entity, e.g., the reflectivity or the Kerr signal, is a correct procedure. However, if G is a function of v , i.e., the distribution function in Eq. Ž2. is B Ž G Ž< v <., v ., broadening and Kramers–Kronig transformation in practice no longer commute. This is easily proven by studying a thin slice of an absorption spectrum. Thus, we consider a function which is zero everywhere except at the energy v 0 , and we write it as a Dirac delta function times some constant C. The odd function f Ž v . in Eq. Ž2. becomes C w d Ž v y v 0 . y d Ž v q v 0 .x. The Kramers–Kronig transformation K, followed by broadening B gives ` C BK f Ž v . s P B Ž G Ž < vX <. , v y vX . p y` 1 1 = X y X d vX , Ž 5. v yv0 v qv0
H
ž
/
whereas broadening followed by the Kramers–Kronig transformation gives ` C K Bf Ž v . s P B Ž G Ž < v 0 <. , vX y v 0 . p y` 1 X yB Ž G Ž < v 0 < . , v X q v 0 . X dv . vyv Ž 6.
H
In Fig. 1, Eq. Ž5. and Eq. Ž6. are compared. It is seen that especially for low energies, the two functions deviate from each other. The amplitude K B f has a sharp peak and its derivative goes toward y` as v ™ 0q, whereas the derivative of Eq. Ž6. approaches zero as v ™ 0q. In practice the integration interval has to be finite, and one finds that the larger the integration interval is taken to be, the smaller becomes the difference between the two integrals. However, their derivatives as v ™ 0q remain constant regardless of integration interval. On the other hand, when G is independent of v , no problems occur as a consequence of the finite integration interval and so the commutation is valid also in practice, in that special case. As mentioned, performing the broadening before the Kramers–Kronig transformation is always cor-
108
A. Delin r Optics Communications 167 (1999) 105–109
Fig. 1. Comparison of Eq. Ž5. and Eq. Ž6.. In this example C s 20 and v 0 s 5. B is a Lorentzian with G s 0.7 at < v < s1, increasing linearly with < v <, and with a minimum G value set to 10y5 . The integration is performed on a mesh from y39.9 to 39.9 with step length 0.02.
rect since with this order of the operations causality will always hold for the components of the optical conductivity. Thus, the solid curve corresponding to operation order K B in Fig. 1 is correct. Consequently, the dashed curve, i.e., the operation order BK, must be wrong. This clearly demonstrates that for general broadening schemes it is important to perform lifetime broadening only on the absorptive part of the optical conductivity. 5. Discussion and conclusions One consequence of the noncommutation is that when spectra are calculated using finite G , and the absorptive and dispersive parts are calculated simultaneously, the entire optical calculation will have to be performed for every new lifetime-broadening scheme. At first sight this seems very unfortunate since there appears to exist a great advantage in using the finite G instead of taking the limit t ™ ` and using the Kramers–Kronig technique. With finite G it appears that high-energy bands and wave functions do not enter the calculation, in contrast to the procedure in the Kramers–Kronig method. This would remove the problem with large errors in the high lying bands, errors originating from incomplete
basis sets, linearization in the band-structure calculation, and the independent-particle approximation inherent in the Kohn-Sham equation. However, as elucidated by Gasche w14x, in order to make the integration in Eq. Ž1. converge, these states still have to enter into the calculation. Thus, from a computational point of view there is no intrinsic advantage in using finite G in Eq. Ž1.. On the contrary, the time-consuming optical calculation has to be performed from the beginning for every new broadening scheme one wishes to test. As is already understood, Eq. Ž1. with finite G produces correct results even for excitation energy dependent lifetimes. The reason for this is simple. The finite lifetime enters in the last step of the derivation of Eq. Ž1. as the integral over time t from y` to 0 is carried out. In order to calculate this integral, damping factors of the form expŽ G t . are introduced. G turns out to be the width appearing in the Lorentz broadening and is thus proportional to the inverse of the lifetime of the excited state. Since the excitation energy is a constant of the integration, nothing prevents t to be taken as some function of the excitation energy, and doing this directly in Eq. Ž1. should thus yield absorptive and dispersive spectra which are interconsistent with regard to causality w15x. To conclude, it is shown that lifetime broadening, when the lifetime of the excited state is taken to be dependent on the excitation energy, should not be performed on spectra containing dispersive components. However, it is always correct to include the functional behavior of the lifetime directly in Eq. Ž1.. Acknowledgements Valuable discussions with O. Eriksson, L. Fast, P. Oppeneer, and J. M. Wills are acknowledged. O. Eriksson is acknowledged for a critical reading of the manuscript. This work was financed by the Swedish Research Council for Engineering Sciences. References w1x C.S. Wang, J. Callaway, Phys. Rev. B 9 Ž1974. 4897. w2x H. Ebert, Rep. Prog. Phys. 59 Ž1996. 1665, and references therein. w3x See, e.g., M. Alouani, J.M. Wills, Phys. Rev. B 54 Ž1996.
A. Delin r Optics Communications 167 (1999) 105–109
w4x w5x w6x w7x
w8x
2480; A. Delin, O. Eriksson, R. Ahuja, B. Johansson, M.S.S. Brooks, T. Gasche, S. Auluck, J.M. Wills, Phys. Rev. B 54 Ž1996. 1673. A. Delin, O. Eriksson, B. Johansson, S. Auluck, J.M. Wills, unpublished. P. Nozieres, ` D. Pines, Phys. Rev. 109 Ž1958. 741, 762, 1062; H. Ehrenreich, M.H. Cohen, Phys. Rev. 115 Ž1959. 786. S.L. Adler, Phys. Rev. 126 Ž1962. 413; N. Wiser, Phys. Rev. 129 Ž1963. 62. R. Kubo, J. Phys. Soc. Jpn. 12 Ž1957. 570; J. Callaway, Quantum Theory of the Solid State, Academic Press, New York, 1974, Part B. R. Kronig, J. Opt. Soc. Amer. 12 Ž1926. 547; H.A. Kramers, Atti Congresso Internazionale dei Fisici, Como 2 Ž1927. 545.
109
w9x see, e.g., J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975, pp. 306–311; L.D. Landau, E. M Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1960, p. 256. w10x F.W. King, J. Opt. Soc. Am. 68 Ž1978. 994; J. Phys. C 10 Ž1977. 3199. w11x J.E. Bertie, S.L. Zhang, Can. J. Chem. 70 Ž1992. 520. w12x A. Santoni, F.J. Himpsel, Phys. Rev. B 43 Ž1991. 1305. w13x K.E. Peiponen, E.M. Vartiainen, Phys. Rev. B 44 Ž1991. 8301. w14x T. Gasche, Ph.D. Thesis, Uppsala University, 1993; T. Gasche, M.S.S. Brooks, B. Johansson, Phys. Rev. B 53 Ž1996. 296. w15x E. Corinaldesi, Nuovo Cimento ŽSuppl.. 14 Ž1959. 369.
Optics Communications 167 Ž1999. 105–109 www.elsevier.comrlocateroptcom
Relation between broadening and Kramers–Kronig transformation of calculated optical spectra Anna Delin Condensed Matter Theory Group, Department of Physics, Uppsala UniÕersity, P. O. Box 530, S-75121 Uppsala, Sweden Received 7 May 1999; received in revised form 16 June 1999; accepted 17 June 1999
Abstract It is shown that, in practice, the Kramers–Kronig transformation and broadening of spectra do not commute. Therefore, broadening of calculated excitation spectra should be performed only on absorptive components. The two operations commute only in the special case when the lifetime of the excited state is taken to be constant, i.e., independent of the excitation energy and of the energies of the electron states involved. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 78.20.Ci
1. Introduction
2. Calculation of the optical conductivity
With modern band-structure methods it has become possible to successfully calculate both magneto-optical and optical properties of materials w1–3x. Broadening of the calculated spectra is an integral part of the calculation, since broadening often is highly relevant for the comparison of the calculated spectra to experimental results. Especially in the calculation of magneto-optical spectra, the broadening is in fact essential in order to obtain results comparable to experiment w4x. In this study the relation between lifetime broadening and the Kramers–Kronig transformation is discussed. In principle the results are valid for any excitation spectra.
The fundamental quantity used for describing excitations caused by a transverse electromagnetic field, i.e., photons, is the optical conductivity s , or, equivalently, the dielectric function. These entities are tensors of the second rank. All components are complex quantities and the real and imaginary part of each component is related by causality. Depending on the convention used in the notation and whether the component is diagonal or off-diagonal, it varies whether the real or the imaginary part is the absorptive component. Therefore, in this paper the discussion will be in terms of absorptive and dispersive components only, and not in terms of real and imaginary parts.
0030-4018r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 3 1 7 - X
A. Delin r Optics Communications 167 (1999) 105–109
106
Within the random phase approximation w5x, neglecting local field effects w6x, two different approaches are often used in the calculation of the interband part of the optical properties. They both start from the following expression w1,7x derived using linear response theory:
sab Ž v . s
ie 2 "m2
q
1
Ýv nnX
X
nn
ž
las derived by Kramers and Kronig w9x. When using the Kramers–Kronig transformation in the calculation of the dispersive component it is often necessary to evaluate the absorption spectrum to high energies in order to achieve a converged result for the dispersive spectrum.
² n < pa < nX : ² nX < pb < n:
v y v nX n q i G
² n < pb < nX : ² nX < pa < n:
v q v nX n q i G
/
,
3. Broadening, general considerations
Ž 1.
In this equation, which is often called the Kubo formula, a , b are Cartesian coordinate indices and e and m are the electron charge and mass, respectively. v is the frequency of the incoming electromagnetic radiation, " v nX n is the energy difference between the initial and final one-particle states < n: and < nX :, Ž p x , p y , pz . s p is the momentum operator, and G is inversely proportional to the lifetime t of the excited state. In the first approach this formula is used directly Ž G then has to be finite. and both the absorptive and dispersive parts of the optical conductivity are calculated from the eigenvalues and eigenfunctions resulting from the band-structure calculation. In the second version the limit t ™ ` is taken in Eq. Ž1. and only the absorptive part is calculated from the eigenvalues and eigenfunctions resulting from the band-structure calculation. In principle, also the dispersive part could be calculated in a similar way in this limit but in practice the dispersive component is calculated from the absorptive one using a causality argument which may be written in a variety of ways. The original expression for calculating the dispersive part from the absorptive was derived by Kramers and Kronig w8,9x. More recently King w10x developed an alternative formula using conjugate Fourier series. Furthermore, Bertie and Zhang w11x have shown that the Kramers–Kronig transformation can be written as two successive fast Fourier transforms, a method which significantly speeds up the calculation of the integral. However, they all amount to the same thing and they are all the same integral operation, if used as intended. In the following this operation will be referred to as the Kramers–Kronig transformation although sometimes this term is used in a more strict sense, referring to the original formu-
At first sight, broadening of spectra appears to be a simple matter. However, as will be proven below, when broadening optical spectra it is important to take into account the fundamental difference between the absorptive and dispersive parts. Failure to do so may result in spurious peaks at low energy in, e.g., reflectivity or Kerr signal, since these spectra consist of a combination of absorptive and dispersive contributions. Broadening of optical spectra serves several distinct goals. One is to simulate a finite experimental resolution, which is usually done by convoluting with a Gaussian. Naturally, the resolution broadening should be performed on the final spectrum, i.e., the reflectivity or the Kerr signal, if that is what has been measured in the experiment one wishes to compare the calculations to. Thermal broadening may also be modeled by a Gaussian. Another goal is to take into account the finite lifetime of excited states. This is called lifetime broadening, and is normally performed by convoluting the spectrum with a Lorentzian. It is found experimentally that lifetime effects play a considerable role in optical transitions. The lifetimes are state dependent and most highly excited states are more unstable and have shorter lifetimes w12x
4. Commutation of broadening and Kramers– Kronig transformation It will be demonstrated here that, in general, lifetime broadening should be performed only on purely absorptive spectra. The fundamental argument is that causality has to hold for the components of the optical conductivity used for calculating the final optical spectrum. This implies that it is always cor-
A. Delin r Optics Communications 167 (1999) 105–109
rect to perform the broadening before the Kramers– Kronig transformation. The opposite order might not be valid, though. Optical spectra are normally thought of as defined on the positive interval w0,`x only. However, they can easily be extended to wy`,`x. In this case the absorptive part becomes odd and the dispersive part even, symmetries which follow from the fact that a real-valued electric field must produce a real-valued polarization of the charges. The extended definition is most useful since it removes all problems connected with the semi-infinite interval and it will be used here. As a matter of fact, Peiponen and Vartiainen w13x have shown that it is even essential in using this extended definition when modeling tails of absorption and extinction curves. Broadening of a function f Ž v ., defined on wy`,`x, is performed by convoluting it with some distribution function B Ž G , v . normalized to unity. The positive parameter G determines the width of the distribution function. The broadened function may then be written `
Bf Ž v . s
X
X
X
Hy` B Ž G , v y v . f Ž v . d v ,
Ž 2.
where B denotes the operation ‘‘convolute with B’’. If the broadening is constant, i.e., G in B Ž G , v . is independent of v , Eq. Ž2. is equivalent to a convolution ‘in the mathematical sense’: `
g) f Ž x. s
Hy` g Ž x y y . f Ž y . dy.
Ž 3.
This stricter type of convolution is to be distinguished from the type of convolution we get if the broadening width is not held constant. Now note that the Hilbert or Kramers–Kronig transformation, which in this paper will be denoted by K Žalthough the standard notation for Hilbert transforms would rather be H ., may be written as a convolution of the type described in Eq. Ž3. of the inverse of the photon energy v , i.e., Kf Ž v . s
1
p
`
f Ž vX .
X
Hy` v y v d v ,
P
X
Ž 4.
where P denotes the Cauchy principal value. A fundamental property of convolutions of the type described in Eq. Ž3. is that they commute, i.e., f ) g s g ) f. It then directly follows that as long as
107
the broadening is constant is does not matter what is done first, broadening or Kramers–Kronig transformation. In this case broadening the dispersive and absorptive parts separately, and then calculating the final optical entity, e.g., the reflectivity or the Kerr signal, is a correct procedure. However, if G is a function of v , i.e., the distribution function in Eq. Ž2. is B Ž G Ž< v <., v ., broadening and Kramers–Kronig transformation in practice no longer commute. This is easily proven by studying a thin slice of an absorption spectrum. Thus, we consider a function which is zero everywhere except at the energy v 0 , and we write it as a Dirac delta function times some constant C. The odd function f Ž v . in Eq. Ž2. becomes C w d Ž v y v 0 . y d Ž v q v 0 .x. The Kramers–Kronig transformation K, followed by broadening B gives ` C BK f Ž v . s P B Ž G Ž < vX <. , v y vX . p y` 1 1 = X y X d vX , Ž 5. v yv0 v qv0
H
ž
/
whereas broadening followed by the Kramers–Kronig transformation gives ` C K Bf Ž v . s P B Ž G Ž < v 0 <. , vX y v 0 . p y` 1 X yB Ž G Ž < v 0 < . , v X q v 0 . X dv . vyv Ž 6.
H
In Fig. 1, Eq. Ž5. and Eq. Ž6. are compared. It is seen that especially for low energies, the two functions deviate from each other. The amplitude K B f has a sharp peak and its derivative goes toward y` as v ™ 0q, whereas the derivative of Eq. Ž6. approaches zero as v ™ 0q. In practice the integration interval has to be finite, and one finds that the larger the integration interval is taken to be, the smaller becomes the difference between the two integrals. However, their derivatives as v ™ 0q remain constant regardless of integration interval. On the other hand, when G is independent of v , no problems occur as a consequence of the finite integration interval and so the commutation is valid also in practice, in that special case. As mentioned, performing the broadening before the Kramers–Kronig transformation is always cor-
108
A. Delin r Optics Communications 167 (1999) 105–109
Fig. 1. Comparison of Eq. Ž5. and Eq. Ž6.. In this example C s 20 and v 0 s 5. B is a Lorentzian with G s 0.7 at < v < s1, increasing linearly with < v <, and with a minimum G value set to 10y5 . The integration is performed on a mesh from y39.9 to 39.9 with step length 0.02.
rect since with this order of the operations causality will always hold for the components of the optical conductivity. Thus, the solid curve corresponding to operation order K B in Fig. 1 is correct. Consequently, the dashed curve, i.e., the operation order BK, must be wrong. This clearly demonstrates that for general broadening schemes it is important to perform lifetime broadening only on the absorptive part of the optical conductivity. 5. Discussion and conclusions One consequence of the noncommutation is that when spectra are calculated using finite G , and the absorptive and dispersive parts are calculated simultaneously, the entire optical calculation will have to be performed for every new lifetime-broadening scheme. At first sight this seems very unfortunate since there appears to exist a great advantage in using the finite G instead of taking the limit t ™ ` and using the Kramers–Kronig technique. With finite G it appears that high-energy bands and wave functions do not enter the calculation, in contrast to the procedure in the Kramers–Kronig method. This would remove the problem with large errors in the high lying bands, errors originating from incomplete
basis sets, linearization in the band-structure calculation, and the independent-particle approximation inherent in the Kohn-Sham equation. However, as elucidated by Gasche w14x, in order to make the integration in Eq. Ž1. converge, these states still have to enter into the calculation. Thus, from a computational point of view there is no intrinsic advantage in using finite G in Eq. Ž1.. On the contrary, the time-consuming optical calculation has to be performed from the beginning for every new broadening scheme one wishes to test. As is already understood, Eq. Ž1. with finite G produces correct results even for excitation energy dependent lifetimes. The reason for this is simple. The finite lifetime enters in the last step of the derivation of Eq. Ž1. as the integral over time t from y` to 0 is carried out. In order to calculate this integral, damping factors of the form expŽ G t . are introduced. G turns out to be the width appearing in the Lorentz broadening and is thus proportional to the inverse of the lifetime of the excited state. Since the excitation energy is a constant of the integration, nothing prevents t to be taken as some function of the excitation energy, and doing this directly in Eq. Ž1. should thus yield absorptive and dispersive spectra which are interconsistent with regard to causality w15x. To conclude, it is shown that lifetime broadening, when the lifetime of the excited state is taken to be dependent on the excitation energy, should not be performed on spectra containing dispersive components. However, it is always correct to include the functional behavior of the lifetime directly in Eq. Ž1.. Acknowledgements Valuable discussions with O. Eriksson, L. Fast, P. Oppeneer, and J. M. Wills are acknowledged. O. Eriksson is acknowledged for a critical reading of the manuscript. This work was financed by the Swedish Research Council for Engineering Sciences. References w1x C.S. Wang, J. Callaway, Phys. Rev. B 9 Ž1974. 4897. w2x H. Ebert, Rep. Prog. Phys. 59 Ž1996. 1665, and references therein. w3x See, e.g., M. Alouani, J.M. Wills, Phys. Rev. B 54 Ž1996.
A. Delin r Optics Communications 167 (1999) 105–109
w4x w5x w6x w7x
w8x
2480; A. Delin, O. Eriksson, R. Ahuja, B. Johansson, M.S.S. Brooks, T. Gasche, S. Auluck, J.M. Wills, Phys. Rev. B 54 Ž1996. 1673. A. Delin, O. Eriksson, B. Johansson, S. Auluck, J.M. Wills, unpublished. P. Nozieres, ` D. Pines, Phys. Rev. 109 Ž1958. 741, 762, 1062; H. Ehrenreich, M.H. Cohen, Phys. Rev. 115 Ž1959. 786. S.L. Adler, Phys. Rev. 126 Ž1962. 413; N. Wiser, Phys. Rev. 129 Ž1963. 62. R. Kubo, J. Phys. Soc. Jpn. 12 Ž1957. 570; J. Callaway, Quantum Theory of the Solid State, Academic Press, New York, 1974, Part B. R. Kronig, J. Opt. Soc. Amer. 12 Ž1926. 547; H.A. Kramers, Atti Congresso Internazionale dei Fisici, Como 2 Ž1927. 545.
109
w9x see, e.g., J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975, pp. 306–311; L.D. Landau, E. M Lifshitz, Electrodynamics of Continuous Media, Pergamon, Oxford, 1960, p. 256. w10x F.W. King, J. Opt. Soc. Am. 68 Ž1978. 994; J. Phys. C 10 Ž1977. 3199. w11x J.E. Bertie, S.L. Zhang, Can. J. Chem. 70 Ž1992. 520. w12x A. Santoni, F.J. Himpsel, Phys. Rev. B 43 Ž1991. 1305. w13x K.E. Peiponen, E.M. Vartiainen, Phys. Rev. B 44 Ž1991. 8301. w14x T. Gasche, Ph.D. Thesis, Uppsala University, 1993; T. Gasche, M.S.S. Brooks, B. Johansson, Phys. Rev. B 53 Ž1996. 296. w15x E. Corinaldesi, Nuovo Cimento ŽSuppl.. 14 Ž1959. 369.