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MS Xα study of energy levels and optical absorption intensities of the laser crystal Ni2+:MgF2
Volume
119. number 2.3
CHEMICAL PHYSICS LEl-I’ERS
MS Xa STUDY OF ENERGY LEVELS AND OF THE LASER CRYSTAL Ni’+ : MgF, Jr-Kang
ZHU
‘, Bing ZHANG
and Song-Ha0
Anlrur Invrrore of Opncr and Fme Mcchonrcs, Acadenm
Rccsived
OPTICAL
30 August
ABSORPTION
1985
INTENSITIES
LIU Smicu, Hefcr, A~rhrrr. Cl~rnu
14 May 1985
NI’+ I hlgF2 ha\e hcsn calculated by The energy levels or NIFZ- in the D2h site symmetry approprrate LOthe larcr cqslal Ihe spm-unreslric[ed MS Xa method. Our calculated value or 10 Dq (7795 cm-’ ) IS m much hetrsr agreement ulth experiment than previous work The magnelic dipole 1rJnsilion intensity 0r ‘AA,,-” T2& has been cJlLuldred using MS Xu waverunctions and the Case-Karplus charge parlition scheme.
l_ Introduction as a laser The system Ni2+ MgF, is of Interest crystal [l] since 1.6-l .8 pm tunable laser output can be obtained from it. The electronic states mvolved in the action are the ground-state 3A2g and the excitedstate 3T2, The value of 1ODq under a hrgh-symmetry (Oh) crystal field has been calculated by the MS I&z method [2], but the result is not in good agreement with experiment. A lower site symmetry for Ni2+(Da) is indicated by the X-ray structure [3] _Under D, symmetry a drfferent calculated value of 1ODq is expected and the energy level 3T2, is split into $ and 3+_ The electric dipole transition
’ Prescnl address- Shanghai Development Software.
601 Yan An Xi Road,
0 009-2614/85/S (North-Holland
Cenler of Computer Shanghai. China.
03.30 0 Elsevier Scrence Publishers B.V. Physics Publishing Division)
tzi spectra; in this case one-electron eigenvalues are sufficienr for analysis of the absorption and lasing spectra [S].
2. Method of calculation The spin-unrestricted MS Xa calculation has been performed for the complex ion NiFzunder Dzh symmetry_ The atomic coordinates are listed in table 1. The exchange scaling parameters crNi and Q~ are taken from Schwarz’s table [6] and the exchange scaling parameters in the intersphere ((u,) and in the ou tersphere (a,, r) are chosen as an average weighted according to the number of valence electrons of the varrous atoms. The radii of atomic spheres are chosen according to Norman [7] with a reduction factor of Table 1 Atomic
coordmatcs
of NC:-
x
(A) I
3’
Ni F,
0.0 0.0
0.0 0.0
F2
0.0
0.0
F3 F4 Fs F6
1.2903 -1-2903 -1.2903 1.2903
1.5242 1.5242 -1.5242 -1.5242
0.0 1.9820 -1.9820 00 0.0 0.0 0.0
141
Volume 119, number 2.3
CHEMICAL PHYSICS LETTERS
0.88 [S]. In the atomic region, the L,, of the spherrcaI harmonic parts of the wavefunctrons are chosen as 2 for Ni and 1 for F. In the outer sphere&, is 4. The oscillator strength of magnetic dipole transitions is expressed as follows f = (8~3mu/3~ze2)l~~~(r~S~)lIMI
J/2(l?*s,)>l*
1
Table 2 One-electron
30 Amst
eigcnvalucs for the cornpiesinn Ne-
Orbital symmetry
(1)
Orbital symmetry C&h>
3e&
7aga
where 2t+
3cgr
with Case-
-1.727
2b3g&
-2.666 -2.693
eg -
aS @ bL, -
The ground state of the complex ion NiFzsymmetry is 3Alg(t&e&)_ It turns out to be
in Oh
3 B,,(a~b&b$,b:,a~)
under the lower symmetry Dzh_ The excited-state 3T,, under 0, symmetry splits into 3As, 3B,, and 3B,, components in D2,,, corresponding to the con~gurations a~b~~b~sb~~a~, a~b&b&bf,al and a~b&b&b&a~, respectrvely The
excited state %T,,
gives me to 3Blg 3Blg;,and 3B,,
in Da,
corresponding
tions
symmetry,
respectively. The transitions b,,
& (t2g)-b,g
-L (es),
b 2z: -I (t2,)--bIE
Jr fe,)
-4.447 -4.474 -4.488
and ag 4 (fZg)-big
be consrdered approximately
J- (ep)
State symmczry
3T2g
3 B3g
3b+
3As 3B 2g
6n$-4b,& 2b2,$-4blg&
7594 7784 8011
3 3J32g
2b=&-7agl
9835
38J33$ ‘S
6agl-7ag* 2bzgt-7ag-C
11105 31360
Transition
3B,s(3A, )-3BagC3Tarr), 3B18<3A,,)-3B2,(3T,,) and 3B,,( 5 A2,)-3Ar,(3T2&. There is similar correspondence for the transitions 3A~-3T,,. Slater transition state [9] calculations for the transitions mentioned above have been performed and the calcuIated results are listed in table 3. It is interesting to compare our work with I_a.rsson’s [2] We obtained an average value for IO& of 7795 cm-l which is rn much better agreement with experiment than busson’s_ Moreover, the fine structures of states 3T.2S and 3Tlg could not be obtained m 0, symmetry @J. In our previous work a radius reduction factorfR was defied, and shown to be very useful in calculations of photoelectron spectra of group IV halides [a]. The present work shows that a carefully chosen reduction factor is also useful in calculation
Energy (Cm--l>
D*h -4b&
can
as the tra~ttlons
and lo&
state symmetry (oh)
3T1g
142
to the configura-
a1b2 bZ b’ a* a2b2 b’ b1 a2 and aib:,b&b&af, g *g 3g 1g P’ g % 3g It? g
Tnblc 3 Transition-stare czkulations
-3 237 -3.305
2bagt 6ngt
The calculated one-electron eigenvalues are listed II-Itabfe 2 The irreducible representations of the higher symmetry group Or, give rise to irreducibie representations of the lower symmetry group D, as followso b3S,
-2.707
7agt
2bsgt
3. Results and discussion
a2S -+ aE o $
63g~
4btgt 2t2g’
-1.659
4blg* 2&&
+1 and $2 are the MS Xcr wavefunctions Karplus charge partition corrections
1985
expt_ [ll]
talc
74 07
9500
12‘1
Yolume
119. number 2.3
CHEMICAL
PHYSICS
method
Table 4 Magnetic dipole o&&tar Statc
state
symmetry
symmetry
(Oh)
(Dzh?
strengths/of
106f (this work)
[lo].
been obtained, talvalueof5.6X
3Azg-3Tzg
expt
30 August
T_ElXERS
An estimated
values=
in good agreement 10s6 111).
6 X 10-6
1985
has
with the experimen-
[ll]
of optical spectra of transition-metal complexes. Calculations of magnetic dipole oscillator strengths of the complex ion hJiF:have not been reported before_ We have carried out the calculations by means of eq. (I) and MS XIY wavefunctions with Case-Karplus charge partition corrections. The calculated magnetic dipojle oscillator strengths of 34s-3T,, are shown in table 4. It is clear that our calculated magnetic dipole oscillator strength is in good agrement with experiment It is known that the electric dipole osci!lator strength is sensitive to the radial charge distribution so that one cannot obtain a reliable value with Case-Karplus charge partioned MS Xa wavefunctions [4] _ The magnetic dipole osctiator strength 1s independent of the radial charge distribution and we expect good results with Case-Karplus charge-partitioned MS Xo wavefunctions. The transition aAaa--aT,, has electric dipole character through coupling to vibrations and its oscillator strength can be estrmated by Tanabe’s
Acknowledgement
We are grateful to Dr. M. Cook and Professor M. Karplus of Harvard University for making the program Xa-VAXIBM available and for helpful discussions.
References [ 11 P.F. Moulton and A. ~~oorad~~, J Opt. Sot_ Am. 68 (1978) 630; Opt Letters 3 (1978) 164. (21 S. LYsxrn and J.W.D. Connofly, J. Chcm. Phhps.60 (1974) 1514. [3] W-H. 2sucr, Acta Cry&. 9 (1956) 515. [4’1 D.k C&c and hi.Xwpius, Chcm. Phys. Letters 39 (I 976) 33. [S] T. Ziegler, A. Rauk and I3.3. l&rends, Thcorct. Chim. ACW 43 (1977) 261. [6] K. Schwaiz, Phyr Rev. B5 (1972) 1466. j7] J.G. Norman Jr.. tlol. Phys. 31 (1976) 1191. [S) J.K. Zhu, D. Li, 3. Limd Y-X; Pan, Theoret Cl&n. Acta 63 (1983) 223. [9] J.C. SIatcr, Advan. Quantum Chem. 6 (1972) 1. [ 101 Y. Tanabe and S. Sugmo, J. Phys Sot. Japan 9 (1954) 766. [ 111 S.X. YWI, L A. Ksppcrs and W.A. Sibley. Phys Rev. 118 (1973) 773.
143
119. number 2.3
CHEMICAL PHYSICS LEl-I’ERS
MS Xa STUDY OF ENERGY LEVELS AND OF THE LASER CRYSTAL Ni’+ : MgF, Jr-Kang
ZHU
‘, Bing ZHANG
and Song-Ha0
Anlrur Invrrore of Opncr and Fme Mcchonrcs, Acadenm
Rccsived
OPTICAL
30 August
ABSORPTION
1985
INTENSITIES
LIU Smicu, Hefcr, A~rhrrr. Cl~rnu
14 May 1985
NI’+ I hlgF2 ha\e hcsn calculated by The energy levels or NIFZ- in the D2h site symmetry approprrate LOthe larcr cqslal Ihe spm-unreslric[ed MS Xa method. Our calculated value or 10 Dq (7795 cm-’ ) IS m much hetrsr agreement ulth experiment than previous work The magnelic dipole 1rJnsilion intensity 0r ‘AA,,-” T2& has been cJlLuldred using MS Xu waverunctions and the Case-Karplus charge parlition scheme.
l_ Introduction as a laser The system Ni2+ MgF, is of Interest crystal [l] since 1.6-l .8 pm tunable laser output can be obtained from it. The electronic states mvolved in the action are the ground-state 3A2g and the excitedstate 3T2, The value of 1ODq under a hrgh-symmetry (Oh) crystal field has been calculated by the MS I&z method [2], but the result is not in good agreement with experiment. A lower site symmetry for Ni2+(Da) is indicated by the X-ray structure [3] _Under D, symmetry a drfferent calculated value of 1ODq is expected and the energy level 3T2, is split into $ and 3+_ The electric dipole transition
’ Prescnl address- Shanghai Development Software.
601 Yan An Xi Road,
0 009-2614/85/S (North-Holland
Cenler of Computer Shanghai. China.
03.30 0 Elsevier Scrence Publishers B.V. Physics Publishing Division)
tzi spectra; in this case one-electron eigenvalues are sufficienr for analysis of the absorption and lasing spectra [S].
2. Method of calculation The spin-unrestricted MS Xa calculation has been performed for the complex ion NiFzunder Dzh symmetry_ The atomic coordinates are listed in table 1. The exchange scaling parameters crNi and Q~ are taken from Schwarz’s table [6] and the exchange scaling parameters in the intersphere ((u,) and in the ou tersphere (a,, r) are chosen as an average weighted according to the number of valence electrons of the varrous atoms. The radii of atomic spheres are chosen according to Norman [7] with a reduction factor of Table 1 Atomic
coordmatcs
of NC:-
x
(A) I
3’
Ni F,
0.0 0.0
0.0 0.0
F2
0.0
0.0
F3 F4 Fs F6
1.2903 -1-2903 -1.2903 1.2903
1.5242 1.5242 -1.5242 -1.5242
0.0 1.9820 -1.9820 00 0.0 0.0 0.0
141
Volume 119, number 2.3
CHEMICAL PHYSICS LETTERS
0.88 [S]. In the atomic region, the L,, of the spherrcaI harmonic parts of the wavefunctrons are chosen as 2 for Ni and 1 for F. In the outer sphere&, is 4. The oscillator strength of magnetic dipole transitions is expressed as follows f = (8~3mu/3~ze2)l~~~(r~S~)lIMI
J/2(l?*s,)>l*
1
Table 2 One-electron
30 Amst
eigcnvalucs for the cornpiesinn Ne-
Orbital symmetry
(1)
Orbital symmetry C&h>
3e&
7aga
where 2t+
3cgr
with Case-
-1.727
2b3g&
-2.666 -2.693
eg -
aS @ bL, -
The ground state of the complex ion NiFzsymmetry is 3Alg(t&e&)_ It turns out to be
in Oh
3 B,,(a~b&b$,b:,a~)
under the lower symmetry Dzh_ The excited-state 3T,, under 0, symmetry splits into 3As, 3B,, and 3B,, components in D2,,, corresponding to the con~gurations a~b~~b~sb~~a~, a~b&b&bf,al and a~b&b&b&a~, respectrvely The
excited state %T,,
gives me to 3Blg 3Blg;,and 3B,,
in Da,
corresponding
tions
symmetry,
respectively. The transitions b,,
& (t2g)-b,g
-L (es),
b 2z: -I (t2,)--bIE
Jr fe,)
-4.447 -4.474 -4.488
and ag 4 (fZg)-big
be consrdered approximately
J- (ep)
State symmczry
3T2g
3 B3g
3b+
3As 3B 2g
6n$-4b,& 2b2,$-4blg&
7594 7784 8011
3 3J32g
2b=&-7agl
9835
38J33$ ‘S
6agl-7ag* 2bzgt-7ag-C
11105 31360
Transition
3B,s(3A, )-3BagC3Tarr), 3B18<3A,,)-3B2,(3T,,) and 3B,,( 5 A2,)-3Ar,(3T2&. There is similar correspondence for the transitions 3A~-3T,,. Slater transition state [9] calculations for the transitions mentioned above have been performed and the calcuIated results are listed in table 3. It is interesting to compare our work with I_a.rsson’s [2] We obtained an average value for IO& of 7795 cm-l which is rn much better agreement with experiment than busson’s_ Moreover, the fine structures of states 3T.2S and 3Tlg could not be obtained m 0, symmetry @J. In our previous work a radius reduction factorfR was defied, and shown to be very useful in calculations of photoelectron spectra of group IV halides [a]. The present work shows that a carefully chosen reduction factor is also useful in calculation
Energy (Cm--l>
D*h -4b&
can
as the tra~ttlons
and lo&
state symmetry (oh)
3T1g
142
to the configura-
a1b2 bZ b’ a* a2b2 b’ b1 a2 and aib:,b&b&af, g *g 3g 1g P’ g % 3g It? g
Tnblc 3 Transition-stare czkulations
-3 237 -3.305
2bagt 6ngt
The calculated one-electron eigenvalues are listed II-Itabfe 2 The irreducible representations of the higher symmetry group Or, give rise to irreducibie representations of the lower symmetry group D, as followso b3S,
-2.707
7agt
2bsgt
3. Results and discussion
a2S -+ aE o $
63g~
4btgt 2t2g’
-1.659
4blg* 2&&
+1 and $2 are the MS Xcr wavefunctions Karplus charge partition corrections
1985
expt_ [ll]
talc
74 07
9500
12‘1
Yolume
119. number 2.3
CHEMICAL
PHYSICS
method
Table 4 Magnetic dipole o&&tar Statc
state
symmetry
symmetry
(Oh)
(Dzh?
strengths/of
106f (this work)
[lo].
been obtained, talvalueof5.6X
3Azg-3Tzg
expt
30 August
T_ElXERS
An estimated
values=
in good agreement 10s6 111).
6 X 10-6
1985
has
with the experimen-
[ll]
of optical spectra of transition-metal complexes. Calculations of magnetic dipole oscillator strengths of the complex ion hJiF:have not been reported before_ We have carried out the calculations by means of eq. (I) and MS XIY wavefunctions with Case-Karplus charge partition corrections. The calculated magnetic dipojle oscillator strengths of 34s-3T,, are shown in table 4. It is clear that our calculated magnetic dipole oscillator strength is in good agrement with experiment It is known that the electric dipole osci!lator strength is sensitive to the radial charge distribution so that one cannot obtain a reliable value with Case-Karplus charge partioned MS Xa wavefunctions [4] _ The magnetic dipole osctiator strength 1s independent of the radial charge distribution and we expect good results with Case-Karplus charge-partitioned MS Xo wavefunctions. The transition aAaa--aT,, has electric dipole character through coupling to vibrations and its oscillator strength can be estrmated by Tanabe’s
Acknowledgement
We are grateful to Dr. M. Cook and Professor M. Karplus of Harvard University for making the program Xa-VAXIBM available and for helpful discussions.
References [ 11 P.F. Moulton and A. ~~oorad~~, J Opt. Sot_ Am. 68 (1978) 630; Opt Letters 3 (1978) 164. (21 S. LYsxrn and J.W.D. Connofly, J. Chcm. Phhps.60 (1974) 1514. [3] W-H. 2sucr, Acta Cry&. 9 (1956) 515. [4’1 D.k C&c and hi.Xwpius, Chcm. Phys. Letters 39 (I 976) 33. [S] T. Ziegler, A. Rauk and I3.3. l&rends, Thcorct. Chim. ACW 43 (1977) 261. [6] K. Schwaiz, Phyr Rev. B5 (1972) 1466. j7] J.G. Norman Jr.. tlol. Phys. 31 (1976) 1191. [S) J.K. Zhu, D. Li, 3. Limd Y-X; Pan, Theoret Cl&n. Acta 63 (1983) 223. [9] J.C. SIatcr, Advan. Quantum Chem. 6 (1972) 1. [ 101 Y. Tanabe and S. Sugmo, J. Phys Sot. Japan 9 (1954) 766. [ 111 S.X. YWI, L A. Ksppcrs and W.A. Sibley. Phys Rev. 118 (1973) 773.
143