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Modified electronic charge density of some alkali fluoride crystals
MODIFIED ELECTRONIC CHARGE DENSITY OF SOME ALKALI FLUORIDE CRYSTALS A. K. GHOSH Birla Indus~~ and Tecbnolo~c~ Museum,C.S.I.R.,Calcutta-19,India.
A. MANNA Departmentof Physics, University of Jadavpur,Calcutta-32,India
S. K. DEB Departmentof Physics, University of Kalyani, Kalyani, Indiat (Received21 may 1976;accepted in revisedfonn
15 October 1976)
Ab&nct--The free ion wavefunctions of alkali fluorideshave been modifiedby consideringthe effect of crystal lattice.It has been postulatedthatonly the outermostelectronsof the ions are affectedand the ~~~~~ ions have been ~p~~rna~ as pointchatges. A H~~F~~~la~r scheme has been utihsedto determinethe wavefunc~n througha numericalprocedure.The co~es~~i~ modifiedelectronic charge densities of the ions indicate the extension of the positive ion radii and contractionof the negative ion radii in the crystal.
1. lNTRonucrloN
Kim and Gordon[l] have recently demonstrated that the free-ion model for alkali halides is good enough to predict equilibrium separation, cohesive energy and bulk modulus. There is, however, both theoretical as well as experimental evidence that this kind of treatment, even for the simplest ionic crystals, ought to be to some extent oversimplified. Korhonen et al. [2,3] have shown through an analysis of the experimental structure ampli~des of alkali halide crystals that the outer shells of negative ions are appreciably deformed. emetically, Froman and Lowdin[4] have demonstrated that the virial theorem AT=-E,.,(T=T,,-Tf) AV= 2E,,,(V=
V,,- V,)
(1)
is not satisfied at all at the crystal equilibrium in the case of free ion model. (T,, V, are crystal K. E. and P. E., T,, V, are free ion K.E. and P.E.) Consequently, to overcome the diRiculty, they scaled uniformly all the nuclear and electron c~rdinates based on a v~ational principle and f~l~g the virial theorem. YamashitalS) has shown earlier that the diamagnetic susceptibility, x, of LiF was lowered by about 5% if only 2p statefunctions of F ion were modified for the crystalline state by a variational procedure. And it is known the experimental x values of ionic crystals is lower than those calculated from free ion Hartree-Fock(HF) wavefunctions. Now, the overlap of the ions in the crystal is mainly due to the overlap occuring between the charge densities of the outershells of neighbouring ions. In addition the outershell electrons are loosely bound as compared to the inner shell electrons. Natu~lly, t~refore, charge clouds
contributed by outer shell electrons are more easily deformable. In other words, the deformation of ions in the ionic crystal would mainly be owing to the deformation of outer shell wavefunctions. This picture is corroborated by experiments mentioned above. Thus, Mansikka and Bystrand calculated the cohesive energy, lattice parameter and compressibility of LiF crystals by deforming the radial wavefunc~ns in the 2p shell of F ion by means of the scaling method in the Heifer-~ndon scheme. The limitations of the method by Mansikka and Bystrand[6] are obvious, the calculations become too cumbersome for ions having greater number of electrons. In this paper, we report the results of our calculations of revision of wavefunctions of outershell electrons by a numerical procedure. From the wavefunctions the corresponding modified electronic charge densities are represented graphically. Electronic charge densities of the inner shell electrons are derived from the clementi[‘l] wavefunctions. We have examined three guorides, viz. LiF, NaF and KF, in the hrst instance. RbF could not be examined due to the dearth of HF-wavefunctions for free Rb’ ion. The method of calculation and results are presented below. 2. METHOD OF CALCULATION A point ion lattice model which incorporates the Hartree-Fock-Slater (HFS) procedure to compute outershell electron orbitals has been adopted. In addition, the model contains estimates of the correlation energy for the same electrons. The latter formalism is analogous to the one by which SlaterIIl estimated the exchange energy of the free electron gas. Neglecting spin dependent terms, we write the total crystal Hamiltonian as H = T, + TN + U(r, x)
tAddress for correspondence. 701
(2)
where U(r. x) = CL(r)+ &(r. x)t WNN(x).Here T, is harmonic function and a, is the spin function. We shall the electronic kinetic energy, TN is the nuclear kinetic use the more restrictive function. energy, U, is the electron electron PAL, L&Nis the electron-nucleus P.E. and UNN is the nucleus-nucleus P.E. The quantity r denotes the set of electronic co-ordinates and x denotes the set of nuclear coordinates. We define, From the ~arn~~ton~, we see that the total wavefunction Iy is a fun&on of ah the electronic and nu&ar ~o~rd~~a~~s. But because the motion of beavy nuclei +zaG be serrated from those of faster moving electrons, we may write through horn-~penhe~mer approximations
Thus, following equations for electronic and nuclear systems can be written
spherically averaged tatal electronic charge density for both spins is defined as, p,,(r) f2:- e[a( r)/4a?l o(r) = s WnlP,?(r)
Xtiswe~~k~o~ that the partinent physics is contained in the zeroeth order solution of the electronic equation (eqn. 9 because the amplitude of the lattice vibration is small. Thus eqn (4) is written as. I?+,+
U%, x0)1&, x0)= E!xxoM%r, x0).
(6)
We consider the ions as point charges Z and we write the electron point ion interaction operator in the form
where the prime means that the site Y= 0 is not in&&d in the summation, r is the position vector for one.of the outermost shell electrons and rv is the location of the yth ion. The Madelung constant is defined by, a, =
r.
uw,0) Z,
(14)
Now, we write the HFS variationaI equation for an eIectron in the outermost shell as,
Vdr)
J
+ W(r) Pdr) = EnrPdr)
(13)
where, the coulomb potential, V,(r) has the form,
V&r) has the form v,,(r) 3 *-onl r0
(17)
I
the sign depending on the cation or anion site. We have also introduced in the HFS equations an where vois the harmonic nearest neighbonr ~~~~~~~at~o~) additional central “correlation” potential W(r) to which pairs of e%e&ronsof opposite spin are subject. The form distartce for the N&t structure. of the potentizd is t&t of Wiirfstf Tile cor~es~n~~ HF vials equations, to the ~br~dj~~er eqns @$-@]may be solved in principle by n~rn~r~~ methods of iteration. ~nfort~~te~y~ such a procedure involves excessive ~omputa~on time. To reduce the computer time, we have used Slater’s[8l version of the HF equations in which the exebange part is where a0 is the Bohr radius and given as a potential, t
V~&r) = -3e2[(3/8n)e-‘/l~,,(~)~~“3
(9)
Now, the most general central field representation of the
HF wavefunction U is,
The numerical procedure adopted by us was an improved version of cooley’s and it could give the energy eigenvalue to an accuracy of (~MZ]IE)=0.001. They also give the se&c.ons&ent pote~ti~ which appears in the HF equation to an accuT$cy of (~SVVv) = O.OL.The q~n~~s &E and hV are respective&, the changes in the trial eigenvahre and the self consistent potential which occur
Mailed
electronic charge density of some alkali fluoridecrystals
between two successive iterations in the numerical integration procedure.
3
I
3. RESULTS
We have drawn the electronic charge densities of the constituent ions of the alkali fluorides in the manner as shown in the graph. The following observations are made. (A) As suggested in our previous paper[lO], the free ionic charge density of Li is modified considerably in a crystalline con~guration. For all the other ions, the charge densities are also modified. The negative ion charge densities are contracted in all the cases while the positive ion charge densities are extended in all the cases. In all the cases, the positive ion radii extend and the negative ion radii decrease in the crystal field. This effect is very prominent in the case of LiF. The change of ionic radii in crystalline fields was also reported by many workers [4, 61. (B) The overlap height increases slightly when the ions are placed in the crystal field. For LiF, the increase in overlap height(h) is considerable but for NaF and KF, the overlap height increase is small. The overlap spread
Interionic
Fig. 1. Free ion and crystai field electronic charge density of LiF. Continuous (W) curve indicates free ion values and dashed (X-X) curve indicates crystal field values.
equilibrium
separation-+
Fig. 2. Free ion and.crystal field electronic charge density of NaF. Continuous (O-O) curve indicates free ion values and dashed (X--X) curve indicates crystal field values.
Table 1. Ionic radii as obtained from the modified charge density
Crystal
Values of r+ (in a.u.) from free ion crystal j. catculation calculation
LiF
1.06
NaF KF
1.75 2.51
1.53 2.06 2.a4
Pauling data [ill
Values of r_ (in a.u.) from TosiFumi free ion crystal F. data [12] calculation calculation
1.13
1.63
1.80 2.51
2.25 2.85
2.70 2.59 2.47
2.23 2.28 2.14
Pauling data 1111 2.51 2.51 2.51
Tosi Fumi data [12] 2.17 2.14 2.21
A. K. Gi-rosuet af.
I;
Fig. 3. Free ion and crystaf field values of electronic charge density of KF. Continuous (a-0, curve indicates free ion values and dashed (X--X) curve indicates crystal field values.
parameter (s) is also slightly increased when the ions are considered in the crystal field. (C) Another interesting observation results from a close analysis of the values of r, and r- in the free ion and in the crystaliine fields. The free ion radii of the ions as obtained from the graphf%] resemble ciosefy to the Pauling. radii while the crystal fieki radii of the ions obtained from the present work, agree to the Tosi-Fumi radii of the ions. The agreement is, however, restricted due to the reason that the ~~~~i~g ions are considered only as point charges and their detailed structure is neglected. In a table, the free ion radii as well as the crystal field radii are compared with the Pauling radii and the Tosi-Fumi radii. With these changed s and h values, one can redetermine the Born-Mayer repulsive parameters as was done in our previous paper[lOl. This will be done after the charge densities of the other afkah haiides in the crystal geld are‘ obtained.
to complete this work. The numerical integration was done in a CDC 3600-160A computer at the Tata InstiMe of Fundamental Research, Bombay, The authors are also thankful to the authorities of the University of Kalyani and in particular to Prof. D. C. Sarkar, Head of the Dspartment of Physics, University of Kalyani for theii many he$ and encouragement during the progress of the work,
~c~o~~g~t~~ of us (SKD) is t~u1 to Prof. R. Y~jay~~van, Head, Sotid State Physics Group, TIFR, Bombay. for providing computationai facilities which enabled him
Wiley, New York (I%?). 12, Fumi F. G. and ‘I’osi hf. P.. L Phys. &hem. Sotids 28, 31 (E%4).
1. Kim Y. S. and Gordon R G.. Phvs. Km. BB. 3548 (1974). 2. Korhonen U., Annals Aced. $ti. *Fmr. Af, 223 (19iS). 3. Korhonen U. and Vihinen S., Anafs Acad. Sci. Fenn, AYI, 8 (1961). 4. Froman A. and Lowdin P. O., J. Phys. Chem. Solids 23.75 (1%2). 5. Yamashita J., J. Phys. Sot. (Japan) 7, 284 (1952). 6. Mansikka K. and Bvstrand F.. J. Phvs. Chem. Solids 27,1073 (1966). * . 7. Clementi E., IBM. .L Res. LhwlopmmtrSuppi.9, 2 (1965). 8. Slater 3. C., Phys. Reu. 81, 385 (1951). 9. Wigner E., Phys. Rev. 46, 1082(1934); Trans. Faruday. Sot, 34,678 <193@. la. Deb S. K, and Ghosh A. K., Ind. L Pkys. 4% 528 (197% 1f. Kittel C., in f~tro~ctj~ to So@ State Physics, 2nd Ed, p. 82.
A. MANNA Departmentof Physics, University of Jadavpur,Calcutta-32,India
S. K. DEB Departmentof Physics, University of Kalyani, Kalyani, Indiat (Received21 may 1976;accepted in revisedfonn
15 October 1976)
Ab&nct--The free ion wavefunctions of alkali fluorideshave been modifiedby consideringthe effect of crystal lattice.It has been postulatedthatonly the outermostelectronsof the ions are affectedand the ~~~~~ ions have been ~p~~rna~ as pointchatges. A H~~F~~~la~r scheme has been utihsedto determinethe wavefunc~n througha numericalprocedure.The co~es~~i~ modifiedelectronic charge densities of the ions indicate the extension of the positive ion radii and contractionof the negative ion radii in the crystal.
1. lNTRonucrloN
Kim and Gordon[l] have recently demonstrated that the free-ion model for alkali halides is good enough to predict equilibrium separation, cohesive energy and bulk modulus. There is, however, both theoretical as well as experimental evidence that this kind of treatment, even for the simplest ionic crystals, ought to be to some extent oversimplified. Korhonen et al. [2,3] have shown through an analysis of the experimental structure ampli~des of alkali halide crystals that the outer shells of negative ions are appreciably deformed. emetically, Froman and Lowdin[4] have demonstrated that the virial theorem AT=-E,.,(T=T,,-Tf) AV= 2E,,,(V=
V,,- V,)
(1)
is not satisfied at all at the crystal equilibrium in the case of free ion model. (T,, V, are crystal K. E. and P. E., T,, V, are free ion K.E. and P.E.) Consequently, to overcome the diRiculty, they scaled uniformly all the nuclear and electron c~rdinates based on a v~ational principle and f~l~g the virial theorem. YamashitalS) has shown earlier that the diamagnetic susceptibility, x, of LiF was lowered by about 5% if only 2p statefunctions of F ion were modified for the crystalline state by a variational procedure. And it is known the experimental x values of ionic crystals is lower than those calculated from free ion Hartree-Fock(HF) wavefunctions. Now, the overlap of the ions in the crystal is mainly due to the overlap occuring between the charge densities of the outershells of neighbouring ions. In addition the outershell electrons are loosely bound as compared to the inner shell electrons. Natu~lly, t~refore, charge clouds
contributed by outer shell electrons are more easily deformable. In other words, the deformation of ions in the ionic crystal would mainly be owing to the deformation of outer shell wavefunctions. This picture is corroborated by experiments mentioned above. Thus, Mansikka and Bystrand calculated the cohesive energy, lattice parameter and compressibility of LiF crystals by deforming the radial wavefunc~ns in the 2p shell of F ion by means of the scaling method in the Heifer-~ndon scheme. The limitations of the method by Mansikka and Bystrand[6] are obvious, the calculations become too cumbersome for ions having greater number of electrons. In this paper, we report the results of our calculations of revision of wavefunctions of outershell electrons by a numerical procedure. From the wavefunctions the corresponding modified electronic charge densities are represented graphically. Electronic charge densities of the inner shell electrons are derived from the clementi[‘l] wavefunctions. We have examined three guorides, viz. LiF, NaF and KF, in the hrst instance. RbF could not be examined due to the dearth of HF-wavefunctions for free Rb’ ion. The method of calculation and results are presented below. 2. METHOD OF CALCULATION A point ion lattice model which incorporates the Hartree-Fock-Slater (HFS) procedure to compute outershell electron orbitals has been adopted. In addition, the model contains estimates of the correlation energy for the same electrons. The latter formalism is analogous to the one by which SlaterIIl estimated the exchange energy of the free electron gas. Neglecting spin dependent terms, we write the total crystal Hamiltonian as H = T, + TN + U(r, x)
tAddress for correspondence. 701
(2)
where U(r. x) = CL(r)+ &(r. x)t WNN(x).Here T, is harmonic function and a, is the spin function. We shall the electronic kinetic energy, TN is the nuclear kinetic use the more restrictive function. energy, U, is the electron electron PAL, L&Nis the electron-nucleus P.E. and UNN is the nucleus-nucleus P.E. The quantity r denotes the set of electronic co-ordinates and x denotes the set of nuclear coordinates. We define, From the ~arn~~ton~, we see that the total wavefunction Iy is a fun&on of ah the electronic and nu&ar ~o~rd~~a~~s. But because the motion of beavy nuclei +zaG be serrated from those of faster moving electrons, we may write through horn-~penhe~mer approximations
Thus, following equations for electronic and nuclear systems can be written
spherically averaged tatal electronic charge density for both spins is defined as, p,,(r) f2:- e[a( r)/4a?l o(r) = s WnlP,?(r)
Xtiswe~~k~o~ that the partinent physics is contained in the zeroeth order solution of the electronic equation (eqn. 9 because the amplitude of the lattice vibration is small. Thus eqn (4) is written as. I?+,+
U%, x0)1&, x0)= E!xxoM%r, x0).
(6)
We consider the ions as point charges Z and we write the electron point ion interaction operator in the form
where the prime means that the site Y= 0 is not in&&d in the summation, r is the position vector for one.of the outermost shell electrons and rv is the location of the yth ion. The Madelung constant is defined by, a, =
r.
uw,0) Z,
(14)
Now, we write the HFS variationaI equation for an eIectron in the outermost shell as,
Vdr)
J
+ W(r) Pdr) = EnrPdr)
(13)
where, the coulomb potential, V,(r) has the form,
V&r) has the form v,,(r) 3 *-onl r0
(17)
I
the sign depending on the cation or anion site. We have also introduced in the HFS equations an where vois the harmonic nearest neighbonr ~~~~~~~at~o~) additional central “correlation” potential W(r) to which pairs of e%e&ronsof opposite spin are subject. The form distartce for the N&t structure. of the potentizd is t&t of Wiirfstf Tile cor~es~n~~ HF vials equations, to the ~br~dj~~er eqns @$-@]may be solved in principle by n~rn~r~~ methods of iteration. ~nfort~~te~y~ such a procedure involves excessive ~omputa~on time. To reduce the computer time, we have used Slater’s[8l version of the HF equations in which the exebange part is where a0 is the Bohr radius and given as a potential, t
V~&r) = -3e2[(3/8n)e-‘/l~,,(~)~~“3
(9)
Now, the most general central field representation of the
HF wavefunction U is,
The numerical procedure adopted by us was an improved version of cooley’s and it could give the energy eigenvalue to an accuracy of (~MZ]IE)=0.001. They also give the se&c.ons&ent pote~ti~ which appears in the HF equation to an accuT$cy of (~SVVv) = O.OL.The q~n~~s &E and hV are respective&, the changes in the trial eigenvahre and the self consistent potential which occur
Mailed
electronic charge density of some alkali fluoridecrystals
between two successive iterations in the numerical integration procedure.
3
I
3. RESULTS
We have drawn the electronic charge densities of the constituent ions of the alkali fluorides in the manner as shown in the graph. The following observations are made. (A) As suggested in our previous paper[lO], the free ionic charge density of Li is modified considerably in a crystalline con~guration. For all the other ions, the charge densities are also modified. The negative ion charge densities are contracted in all the cases while the positive ion charge densities are extended in all the cases. In all the cases, the positive ion radii extend and the negative ion radii decrease in the crystal field. This effect is very prominent in the case of LiF. The change of ionic radii in crystalline fields was also reported by many workers [4, 61. (B) The overlap height increases slightly when the ions are placed in the crystal field. For LiF, the increase in overlap height(h) is considerable but for NaF and KF, the overlap height increase is small. The overlap spread
Interionic
Fig. 1. Free ion and crystai field electronic charge density of LiF. Continuous (W) curve indicates free ion values and dashed (X-X) curve indicates crystal field values.
equilibrium
separation-+
Fig. 2. Free ion and.crystal field electronic charge density of NaF. Continuous (O-O) curve indicates free ion values and dashed (X--X) curve indicates crystal field values.
Table 1. Ionic radii as obtained from the modified charge density
Crystal
Values of r+ (in a.u.) from free ion crystal j. catculation calculation
LiF
1.06
NaF KF
1.75 2.51
1.53 2.06 2.a4
Pauling data [ill
Values of r_ (in a.u.) from TosiFumi free ion crystal F. data [12] calculation calculation
1.13
1.63
1.80 2.51
2.25 2.85
2.70 2.59 2.47
2.23 2.28 2.14
Pauling data 1111 2.51 2.51 2.51
Tosi Fumi data [12] 2.17 2.14 2.21
A. K. Gi-rosuet af.
I;
Fig. 3. Free ion and crystaf field values of electronic charge density of KF. Continuous (a-0, curve indicates free ion values and dashed (X--X) curve indicates crystal field values.
parameter (s) is also slightly increased when the ions are considered in the crystal field. (C) Another interesting observation results from a close analysis of the values of r, and r- in the free ion and in the crystaliine fields. The free ion radii of the ions as obtained from the graphf%] resemble ciosefy to the Pauling. radii while the crystal fieki radii of the ions obtained from the present work, agree to the Tosi-Fumi radii of the ions. The agreement is, however, restricted due to the reason that the ~~~~i~g ions are considered only as point charges and their detailed structure is neglected. In a table, the free ion radii as well as the crystal field radii are compared with the Pauling radii and the Tosi-Fumi radii. With these changed s and h values, one can redetermine the Born-Mayer repulsive parameters as was done in our previous paper[lOl. This will be done after the charge densities of the other afkah haiides in the crystal geld are‘ obtained.
to complete this work. The numerical integration was done in a CDC 3600-160A computer at the Tata InstiMe of Fundamental Research, Bombay, The authors are also thankful to the authorities of the University of Kalyani and in particular to Prof. D. C. Sarkar, Head of the Dspartment of Physics, University of Kalyani for theii many he$ and encouragement during the progress of the work,
~c~o~~g~t~~ of us (SKD) is t~u1 to Prof. R. Y~jay~~van, Head, Sotid State Physics Group, TIFR, Bombay. for providing computationai facilities which enabled him
Wiley, New York (I%?). 12, Fumi F. G. and ‘I’osi hf. P.. L Phys. &hem. Sotids 28, 31 (E%4).
1. Kim Y. S. and Gordon R G.. Phvs. Km. BB. 3548 (1974). 2. Korhonen U., Annals Aced. $ti. *Fmr. Af, 223 (19iS). 3. Korhonen U. and Vihinen S., Anafs Acad. Sci. Fenn, AYI, 8 (1961). 4. Froman A. and Lowdin P. O., J. Phys. Chem. Solids 23.75 (1%2). 5. Yamashita J., J. Phys. Sot. (Japan) 7, 284 (1952). 6. Mansikka K. and Bvstrand F.. J. Phvs. Chem. Solids 27,1073 (1966). * . 7. Clementi E., IBM. .L Res. LhwlopmmtrSuppi.9, 2 (1965). 8. Slater 3. C., Phys. Reu. 81, 385 (1951). 9. Wigner E., Phys. Rev. 46, 1082(1934); Trans. Faruday. Sot, 34,678 <193@. la. Deb S. K, and Ghosh A. K., Ind. L Pkys. 4% 528 (197% 1f. Kittel C., in f~tro~ctj~ to So@ State Physics, 2nd Ed, p. 82.