JOURNAL OF CHEMICAL PHYSICS

VOLUME 108, NUMBER 11

15 MARCH 1998

Static structure factor and electron correlation effects studied by inelastic x-ray scattering spectroscopy Noboru Watanabe, Hisashi Hayashi, and Yasuo Udagawa Research Institute for Scientific Measurements, Tohoku Univ., Katahira 2-1-1, Aoba-ku, Sendai, Japan 980-77

Seiichiro Ten-no and Suehiro Iwata Institute for Molecular Science, Myodaiji, Okazaki, Japan 444

~Received 11 August 1997; accepted 10 December 1997! Inelastic x-ray scattering spectra of methanol, acetonitrile, benzene, and cyclohexane have been measured with 2 eV resolution for momentum transfer q between 0.69 and 2.77 a.u. using synchrotron radiation from the photon factory ~PF! storage ring. By utilizing the Bethe sum rule, the spectra were brought onto an absolute scale, so that the static structure factor S(q) has been obtained. S(q) of these molecules has also been calculated at the single reference configuration interaction ~CI! with several types of basis sets. A new formula is proposed to carry out spherical averaging accurately. It is concluded that the CI singles and doubles ~CISD! treatment is necessary to predict correct S(q), and that an inclusion of polarization function influences S(q) significantly at this level. An addition of f functions also improves the agreement with the experiments. S(q)’s based on CISD wave functions are in good agreement with the experimental ones, in particular at large q. © 1998 American Institute of Physics. @S0021-9606~98!01111-8#

I. INTRODUCTION

while the inelastic part contains information about the electron–electron pair distribution. Except for several reports on N2, 2,3 however, studies on electron correlation effects along this line are few, because the inelastic scattered intensities should accurately be measured separately from the much more intense elastic part. The inelastic x-ray scattering ~IXS! experiment is an alternative to the electron scattering experiment. The cross section contains direct information on the electron–electron pair distribution,1 and is more sensitive to electron correlation than common spectroscopies.4 The double differential cross section of the IXS is expressed by the dynamic structure factor, S(q,E),

How to take electron–electron correlation into consideration has been one of the central issues in theoretical solid state physics and molecular science. Various efforts have been devoted to dealing with the problem. Experimental observables that reflect the correlation effects to assess such theoretical calculations are, however, scarce. The static structure factor S(q) has been known as a sensitive probe for correlated wave functions,1 because it is related to the twoelectron reduced density matrix G(r1 ,r2 ) by Eq. ~1!.

E^ KE E `

S~ q !5

0

5

S ~ q,E ! & V dE

~1!

S D

] 2s E1 5 ~ e0 •e1 ! r 20 S ~ q,E ! . ]V]E E0

G ~ r1 ,r2 ! [email protected] iq• ~ r1 2r2 !# dr1 dr2

2 u F ~ q! u 2

L

V

1N,

Here E 0 ,e0 and E 1 ,e1 are the energies and the polarization vectors of incident and scattered x-ray photons and r 0 5e 2 /mc 2 is the classical electron radius. It has been almost impossible to measure the inelastic x-ray scattering intensities separately from the elastic part because of the small cross sections of the inelastic part, and hence only the total intensities have been studied.5–8 Recent developments of insertion devices at synchrotron radiation facilities have, however, provided a possibility to extract inelastic scatterings from the total scattering. Very recently, Schu¨lke et al.9 measured the inelastic x-ray scattering spectra using single crystal Si with 1.6 eV resolution, and obtained the dynamic structure factor S(q,E) by making use of the f -sum rule. They have derived the static structure factor S(q) and compared it with theoretical ones evaluated with Hartree–Fock ~HF! and random phase approximation ~RPA! methods. We also have carried out a similar experiment on

~2!

where S(q,E)5 ^ S(q,E) & V is the dynamic structure factor which is the function of the momentum and energy transferred, N is the total number of electrons in the target, and ^ 22 & V means the spherical average. The elastic scattering factor F(q) is a Fourier transformation of the one-electron density r (r), F ~ q! 5

Er

~ r! exp~ iq•r! dr.

~3!

S(q,E) can be determined either by high energy electron scattering or the x-ray scattering experiment, of which the former has been employed more frequently in the past. The elastic component in the total scattering cross section bears information about the charge distribution in the system, 0021-9606/98/108(11)/4545/9/$15.00

~4!

4545

© 1998 American Institute of Physics

4546

Watanabe et al.

J. Chem. Phys., Vol. 108, No. 11, 15 March 1998

liquid water over a range 0.69

VOLUME 108, NUMBER 11

15 MARCH 1998

Static structure factor and electron correlation effects studied by inelastic x-ray scattering spectroscopy Noboru Watanabe, Hisashi Hayashi, and Yasuo Udagawa Research Institute for Scientific Measurements, Tohoku Univ., Katahira 2-1-1, Aoba-ku, Sendai, Japan 980-77

Seiichiro Ten-no and Suehiro Iwata Institute for Molecular Science, Myodaiji, Okazaki, Japan 444

~Received 11 August 1997; accepted 10 December 1997! Inelastic x-ray scattering spectra of methanol, acetonitrile, benzene, and cyclohexane have been measured with 2 eV resolution for momentum transfer q between 0.69 and 2.77 a.u. using synchrotron radiation from the photon factory ~PF! storage ring. By utilizing the Bethe sum rule, the spectra were brought onto an absolute scale, so that the static structure factor S(q) has been obtained. S(q) of these molecules has also been calculated at the single reference configuration interaction ~CI! with several types of basis sets. A new formula is proposed to carry out spherical averaging accurately. It is concluded that the CI singles and doubles ~CISD! treatment is necessary to predict correct S(q), and that an inclusion of polarization function influences S(q) significantly at this level. An addition of f functions also improves the agreement with the experiments. S(q)’s based on CISD wave functions are in good agreement with the experimental ones, in particular at large q. © 1998 American Institute of Physics. @S0021-9606~98!01111-8#

I. INTRODUCTION

while the inelastic part contains information about the electron–electron pair distribution. Except for several reports on N2, 2,3 however, studies on electron correlation effects along this line are few, because the inelastic scattered intensities should accurately be measured separately from the much more intense elastic part. The inelastic x-ray scattering ~IXS! experiment is an alternative to the electron scattering experiment. The cross section contains direct information on the electron–electron pair distribution,1 and is more sensitive to electron correlation than common spectroscopies.4 The double differential cross section of the IXS is expressed by the dynamic structure factor, S(q,E),

How to take electron–electron correlation into consideration has been one of the central issues in theoretical solid state physics and molecular science. Various efforts have been devoted to dealing with the problem. Experimental observables that reflect the correlation effects to assess such theoretical calculations are, however, scarce. The static structure factor S(q) has been known as a sensitive probe for correlated wave functions,1 because it is related to the twoelectron reduced density matrix G(r1 ,r2 ) by Eq. ~1!.

E^ KE E `

S~ q !5

0

5

S ~ q,E ! & V dE

~1!

S D

] 2s E1 5 ~ e0 •e1 ! r 20 S ~ q,E ! . ]V]E E0

G ~ r1 ,r2 ! [email protected] iq• ~ r1 2r2 !# dr1 dr2

2 u F ~ q! u 2

L

V

1N,

Here E 0 ,e0 and E 1 ,e1 are the energies and the polarization vectors of incident and scattered x-ray photons and r 0 5e 2 /mc 2 is the classical electron radius. It has been almost impossible to measure the inelastic x-ray scattering intensities separately from the elastic part because of the small cross sections of the inelastic part, and hence only the total intensities have been studied.5–8 Recent developments of insertion devices at synchrotron radiation facilities have, however, provided a possibility to extract inelastic scatterings from the total scattering. Very recently, Schu¨lke et al.9 measured the inelastic x-ray scattering spectra using single crystal Si with 1.6 eV resolution, and obtained the dynamic structure factor S(q,E) by making use of the f -sum rule. They have derived the static structure factor S(q) and compared it with theoretical ones evaluated with Hartree–Fock ~HF! and random phase approximation ~RPA! methods. We also have carried out a similar experiment on

~2!

where S(q,E)5 ^ S(q,E) & V is the dynamic structure factor which is the function of the momentum and energy transferred, N is the total number of electrons in the target, and ^ 22 & V means the spherical average. The elastic scattering factor F(q) is a Fourier transformation of the one-electron density r (r), F ~ q! 5

Er

~ r! exp~ iq•r! dr.

~3!

S(q,E) can be determined either by high energy electron scattering or the x-ray scattering experiment, of which the former has been employed more frequently in the past. The elastic component in the total scattering cross section bears information about the charge distribution in the system, 0021-9606/98/108(11)/4545/9/$15.00

~4!

4545

© 1998 American Institute of Physics

4546

Watanabe et al.

J. Chem. Phys., Vol. 108, No. 11, 15 March 1998

liquid water over a range 0.69