Daniel Hrivňk a, Frantiek Karlick a, Ivan Janeček a, Ivana Paidarov b, and Ren Kalus a a...
1
Semiempirical Modelling of He N + Clusters Daniel Hrivňák Daniel Hrivňák a a , , Franti Franti šek Karlický šek Karlický a a , Ivan Janeček , Ivan Janeček a a , , Ivana Paidarová Ivana Paidarová b b , and , and René Kalus René Kalus a a a a Department of Physics, U Department of Physics, U niversity of Ostrava, Ostrava, Czech Republic niversity of Ostrava, Ostrava, Czech Republic b b J. Heyrovsky Institut of Physical Chemistry, Prague, Czech republic J. Heyrovsky Institut of Physical Chemistry, Prague, Czech republic F F inancial support inancial support : : the Grant Agency of the Czech Republic ( the Grant Agency of the Czech Republic ( g g rant rant s s No. 203/02/1204 No. 203/02/1204 and 203/04/2146 and 203/04/2146 ), Ministry of ), Ministry of Education of the Czech Education of the Czech Republic (grant No. 1N04125) Republic (grant No. 1N04125) . . OSTRAVA RESULTS – COMPARISON WITH AB INITIO DATA 2 THEORY I – DIATOMICS-IN-MOLECULES METHOD (DIM) General theory: F. O. Ellison, J. Am. Chem. Soc. 85 (1963), 3540 . Application to He N + : Knowles, P. J., Murrel, J. N., and Hodge, E. J., Mol. Phys. 85 (1995), 243. Ovchinnikov et al., J. Chem. Phys. 108/22 (1998), 9350. Diatomic inputs N N N PQ P P Q P P N DIM DIM 1 1 ˆ ˆ ˆ H H ( 2) H DIM Basis 1 1 2 2 1 (1) (2) (3) (4)... (2 1)... ( 1) ( ), 1,2,..., , ! K K N N a a a a a K a n a n K N n N multielectron wave functions of the form where N is number of He atoms, n=2N-1 is number of electrons, a i is helium 1s-spinorbital with centre on i- th atom (dash over a label denotes opposite spin orientation), || represents Slater determinant (antisymetrizator). K-th wavefunction of the base represents electronic state with the electron hole on K-th helium atom. DIM Hamiltonian .. ˆ H H .. .. PQ PQ DIM P DIM Q F A B F F B A F P-th Q-th P-th Q-th P-th .. ˆ H H .. P P P Q D D C D P-th w here , potentialenergies ofisolated ion andHe atom respectively(setto 0in our calculations). CD He 1 , 2 PQ g PQ u PQ Ar E r E r 1 , 2 PQ g PQ u PQ Br E r E r neut . PQ PQ F r E r Hamilton Matrices a) Overlap neglected b) Overlap included S PQ P Q PQ 11 1 1 1 .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ˆ .. .. .. .. H H .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. P Q N JK PP PQ PQ PQ JK JP JK JQ JK DIM P DIM Q QP QQ JK F S X Y F S F S cc X cc Y cc F S X F S Y F S cc X cc Y cc F S 1 .. .. .. .. .. .. .. N NP NQ NN F S X Y F S P-th Q-th P-th Q-th 1 1 2 2 JP JP JQ X A BS A BS 1 1 2 2 JQ JQ JP Y A BS A BS where and Overlap matrix for He 3 + 2 23 12 13 32 13 12 23 2 13 23 21 13 2 12 1 S .. 1 , .. .. 1 I I I I I I I K cc I I I I cc cc I where 2 2 2 12 23 31 12 23 31 1 2Re K I I I III * JK JK K J I r aad and is overlap integral of atomic orbitals localized on the J-th and K-th atom respectively. For hydrogen-like orbitals it has the form as Thee potential energy curves for He 2 and He 2 + : E neut (r) – ground-state for He 2 [R. A. Aziz, A. R. Janzen, and M. R. Moldover, Phys. Rev. Letters 74 (1995) 1586 ]. E u + (r), E g + (r) – ground state ( 2 u + ) and first excited state ( 2 g + ) of He 2 + [F.X. Gadéa, I. Paidarová, Chem. Phys. 209 (1996) 281. J. Xie, B. Poirier, and G. I. Gellene, J. Chem. Phys. 122 (2005) Art. No. 184310. ] 2 1 3 1 exp JK JK ef JK ef JK ef JK I r Z r Z r Z r where Symmetric (true) configuration results from ab-initio model and DIM with overlap model with Z ef <2. 1.5 1.6 1.7 1.8 1.9 2.0 -3 .0 -2 .9 -2 .8 -2 .7 -2 .6 -2 .5 -2 .4 G ro u n d -sta te p o te n tia le n e rg y o fH e 3 + [eV ] E ffe c tive a to m ic n um b e r Z ef in th e o ve rla p form ula D IM + OVERLAP DIM ABIN ITIO Asymmetric (false) configuration results from standard DIM model and from DIM with overlap model with Z ef >= 2. CONCLUSIONS Median of deviations: DIM – 137 meV Overlap (FIT) – 122 meV Overlap (Z = 1.9) – 65 meV Median of deviations: DIM – 168 meV Overlap (FIT) – 143 meV Overlap (Z = 1.9) – 67 meV Median of deviations: DIM – 117 meV Overlap (FIT) – 96 meV Overlap (Z = 1.9) – 52 meV JK JK JK JK JK I r r r r 2 1 1.323 0.019 exp 1.438 1 Fitted generalized overlap formula: 2 See this poster session, Paidarová a kol., Ab initio calculations on He 3 + of interest for semiempirical modelling of He n + . 0 100 200 300 400 500 0 20 40 60 80 100 120 140 160 180 200 F re q u en cy co unt A b so lu te valu e o f d eviatio n [m eV] DIM O verlap (F IT ) O verlap (Z = 1.9) E n erg y o f th e g ro u n d state o f H e 3 + (statistics o n cca 850 vario u s co n fig u ratio n s) 1 0 100 200 300 400 500 0 50 100 150 200 250 300 350 400 F re q u en cy co unt A b so lu te valu e o f d eviatio n [m eV] DIM O verlap (F IT ) O verlap (Z = 1.9) E n erg y o f th e seco n d excited state o f H e 3 + (statistics o n cca 850 vario u s co n fig u ratio n s) 1 0 100 200 300 400 500 0 20 40 60 80 100 120 140 160 180 200 220 240 260 F re q uen cy co unt A b so lu te valu e o f d eviatio n [m eV] DIM O verlap (F IT ) O verlap (Z = 1.9) E n erg y o f th e first excited state o f H e 3 + (statistics o n cca 850 vario u s co n fig u ratio n s) 1 Standard DIM method gives no satisfactory results for He n + clusters. Stable configuration of He 3 + trimer in DIM approach is a linear asymmetrical instead of the linear symmetrical, for example 3 . Inclusion of the overlap to the DIM method gives relevant changes of results. Parameters of the overlap formula can be set properly to minimize deviation between ab initio data and resulted data from DIM model. The most important parameter is effective atomic number Z ef . Correct (i. e. symmetrical) stable configuration of He 3 + results from model DIM + overlap with Z ef < 2. The best agreement with ab initio data has been achieved for values of Z ef between 1.6 and 1.9, but resulting typical deviation about 60 meV in potential energy is not quite satisfactory. Next possibility to enhance accuracy of the DIM + overlap method is to fit some parameters in the overlap formula. How indicate our first results, this way is not very hopeful. We want to attain really better results by using the so called triatomics-in-molecules method (TRIM). In opposite of the DIM method, the TRIM method organically involves three-body corrections to the diatomic energies. As an input to the TRIM method serve three- atomic potential energy hypersurfaces for three lowest energy levels. Construction of these accurate hypersurfaces is our topical goal. The semiempirical methods mentioned above are based on the semi-classical Born-Oppenheimer approach, whose application to the lightweight helium atoms is quite limited. The main advantage of these methods is their computational inexpensivity. It will be necessary to use some fully quantum THEORY II – TRIATOMICS-IN-MOLECULES METHOD (TRIM) -2 -1 ( ) () TR IM 1 1 1 1 1 3 ˆ ˆ ˆ H H H 2 2 N N N N ABC A A B A C B A N N (123) (123) (123) 11 12 13 (123) (123) (123) (123) 12 22 23 4 (123) (123) (123) 13 23 33 (123) neut 0 0 H (He ) 0 0 0 0 U U U U U U U U U E Coefficients KJ are calculated using the DIM method; in case the three-body correction to the He 3 + interaction energy is a small perturbation, the resulting Hamiltonian matrix is expected to be correct up to 1 st order of perturbation theory. E neut (ABC) … energy of a neutral (ABC) fragment in the electronic ground-state, calculated using semiempirical two- and three-body potentials for helium, E J (ABC) … energy of an ionic (ABC) fragment in the electronic ground (i = 1) and the first two excited (J = 2,3) states, taken from ab initio calculations on He 3 + (see also this poster session: I. Paidarová et al., Ab initio calculations on He 3 + of interest for semiempirical modelling.) TRIM Hamiltonian Triatomic inputs Hamilton Matrix , where ABC ABC AA ABC ABC BB ABC ABC CC ABC ABC DD H U H U H U H E ( ) ( ) 11 ( ) ( ) 22 ( ) ( ) 33 ( ) ( ) neut ABC ABC ABC AB BA ABC ABC ABC AC CA ABC ABC ABC BC CB ABC EF H H U H H U H H U H ( ) ( ) ( ) 12 ( ) ( ) ( ) 13 ( ) ( ) ( ) 23 ( ) 0 3 ( ) ( ) ( ) ( ) 1 , ABC ABC ABC ABC KL J KJ LJ J U E 3 ( ) ( ) 1 . ABC ABC J KJ K K E where is energy of the adiabatic (stationary) state ( ) ABC J E General theory: R. Kalus, Universitas Ostraviensis, Acta Facultatis Rerum Naturalium, Physica-Chemia 8/199/2001. 3 Very good known results, see Knowles, P. J., Murrell, J. N., Mol. Phys. 87 (1996), 827, for example.
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Transcript of Daniel Hrivňk a, Frantiek Karlick a, Ivan Janeček a, Ivana Paidarov b, and Ren Kalus a a...
Semiempirical Modelling of HeN+ Clusters
Daniel HrivňákDaniel Hrivňákaa, , FrantiFrantišek Karlickýšek Karlickýaa, Ivan Janeček, Ivan Janečekaa, , Ivana PaidarováIvana Paidarovább, and , and René KalusRené Kalusaa
aa Department of Physics, UDepartment of Physics, University of Ostrava, Ostrava, Czech Republicniversity of Ostrava, Ostrava, Czech Republicbb J. Heyrovsky Institut of Physical Chemistry, Prague, Czech republicJ. Heyrovsky Institut of Physical Chemistry, Prague, Czech republic
FFinancial supportinancial support:: the Grant Agency of the Czech Republic ( the Grant Agency of the Czech Republic (ggrantrantss No. 203/02/1204 No. 203/02/1204 and 203/04/2146 and 203/04/2146), Ministry of), Ministry of Education of the Czech Republic (grant No. 1N04125)Education of the Czech Republic (grant No. 1N04125)..
OSTRAVA
RESULTS – COMPARISON WITH AB INITIO DATA2
THEORY I – DIATOMICS-IN-MOLECULES METHOD (DIM) General theory: F. O. Ellison, J. Am. Chem. Soc. 85 (1963), 3540. Application to HeN
+: Knowles, P. J., Murrel, J. N., and Hodge, E. J., Mol. Phys. 85 (1995), 243. Ovchinnikov et al., J. Chem. Phys. 108/22 (1998), 9350. Diatomic inputs
N N N
PQ P
P Q P PNDIM DIM
1 1
ˆ ˆ ˆH H ( 2) H
DIM Basis
1 1 2 21 (1) (2) (3) (4)... (2 1)... ( 1) ( ) , 1,2,..., ,
!K K N Na a a a a K a n a n K Nn
N multielectron wave functions of the form
where N is number of He atoms, n=2N-1 is number of electrons, ai is helium 1s-spinorbital with centre on i-th atom (dash over a label denotes opposite spin orientation), || represents Slater determinant (antisymetrizator). K-th wavefunction of the base represents electronic state with the electron hole on K-th helium atom.
DIM Hamiltonian
..
ˆH H..
..
PQ PQDIM P DIM Q
F
A BF
F
B AF
P-th Q-th
P-th
Q-th
P-th ..ˆH H
..
P PP Q
DD
CD
P-th
where , potential energiesof isolated ion and He atomrespectively (set to 0 in ourcalculations).
C DHe
1 ,2PQ g PQ u PQA r E r E r
1 ,2PQ g PQ u PQB r E r E r neut .PQ PQF r E r
Hamilton Matrices a) Overlap neglected b) Overlap included
SPQ P Q PQ 11 1 1 1.. .. .. .. .... .. .. .. .. .. .. ... . .. .. . . .. .. . ... .. .. .. .. .. .. .. ..
ˆ .. .. .. ..H H.. .. .. .. .. .. .. .. ... . .. .. . . .. .. . ... .. .. .. .. ..
P Q N
JK
PP PQ
PQ PQJK JP JK JQ JKDIM P DIM Q
QP QQ
JK
F S X Y F SF S
c c X c c Y c c
F S X F S Y F S
c c X c c Y c cF S
1
.. .... .. .. .. ..N NP NQ NNF S X Y F S
P-th Q-th
P-th
Q-th
1 12 2JP JP JQX A B S A B S 1 1
2 2JQ JQ JPY A B S A B S
where
and
Overlap matrix for He3+
223 12 13 32 13 12 23
213 23 21 13
212
1
S . . 1 ,
. . . . 1
I I I I I I I
K c c I I I I
c c c c I
where
2 2212 23 31 12 23 311 2ReK I I I I I I
*JK JK K JI r a a d
and
is overlap integral of atomic orbitals localized on the J-th and K-th atom respectively. For hydrogen-like orbitals it has the form as
Thee potential energy curves for He2 and He2+:
Eneut(r) – ground-state for He2 [R. A. Aziz, A. R. Janzen, and M. R. Moldover, Phys. Rev. Letters 74 (1995) 1586 ].Eu
+(r), Eg+(r) – ground state (2u
+) and first excited state (2g+) of He2
+ [F.X. Gadéa, I. Paidarová, Chem. Phys. 209 (1996) 281.J. Xie, B. Poirier, and G. I. Gellene, J. Chem. Phys. 122 (2005) Art. No. 184310. ]
2131 expJK JK ef JK ef JK ef JKI r Z r Z r Z r where
Symmetric (true) configuration results from ab-initio model and DIM with overlap model with Zef <2.
1.5 1.6 1.7 1.8 1.9 2.0-3.0
-2.9
-2.8
-2.7
-2.6
-2.5
-2.4
Gro
und-
stat
e po
tent
ial e
nerg
y of
He 3+ [e
V]
Effective atomic number Zef in the overlap formula
DIM + OVERLAP DIM ABINITIO
Asymmetric (false) configuration results from standard DIM model and from DIM with overlap model with Zef >= 2.
CONCLUSIONS
Median of deviations:DIM – 137 meVOverlap (FIT) – 122 meVOverlap (Z = 1.9) – 65 meV
Median of deviations:DIM – 168 meVOverlap (FIT) – 143 meVOverlap (Z = 1.9) – 67 meV
Median of deviations:DIM – 117 meVOverlap (FIT) – 96 meVOverlap (Z = 1.9) – 52 meV
JK JK JK JK JKI r r r r21 1.323 0.019 exp 1.438 1 Fitted generalized overlap formula:
2 See this poster session, Paidarová a kol., Ab initio calculations on He3+ of interest for semiempirical modelling of Hen
+.
0 100 200 300 400 5000
20
40
60
80
100
120
140
160
180
200
Freq
uenc
y co
unt
Absolute value of deviation [meV]
DIM Overlap (FIT) Overlap (Z = 1.9)
Energy of the ground state of He3+
(statistics on cca 850 various configurations)
1
0 100 200 300 400 5000
50
100
150
200
250
300
350
400
Freq
uenc
y co
unt
Absolute value of deviation [meV]
DIM Overlap (FIT) Overlap (Z = 1.9)
Energy of the second excited state of He3+
(statistics on cca 850 various configurations)
1
0 100 200 300 400 5000
20
40
60
80
100
120
140
160
180
200
220
240
260
Freq
uenc
y co
unt
Absolute value of deviation [meV]
DIM Overlap (FIT) Overlap (Z = 1.9)
Energy of the first excited state of He3+
(statistics on cca 850 various configurations)
1
Standard DIM method gives no satisfactory results for Hen+
clusters. Stable configuration of He3+ trimer in DIM approach is
a linear asymmetrical instead of the linear symmetrical, for example3. Inclusion of the overlap to the DIM method gives relevant changes of results. Parameters of the overlap formula can be set properly to minimize deviation between ab initio data and resulted data from DIM model. The most important parameter is effective atomic number Zef. Correct (i. e. symmetrical) stable configuration of He3
+ results from model DIM + overlap with Zef < 2. The best agreement with ab initio data has been achieved for values of Zef between 1.6 and 1.9, but resulting typical deviation about 60 meV in potential energy is not quite satisfactory. Next possibility to enhance accuracy of the DIM + overlap method is to fit some parameters in the overlap formula. How indicate our first results, this way is not very hopeful. We want to attain really better results by using the so called triatomics-in-molecules method (TRIM). In opposite of the DIM method, the TRIM method organically involves three-body corrections to the diatomic energies. As an input to the TRIM method serve three-atomic potential energy hypersurfaces for three lowest energy levels. Construction of these accurate hypersurfaces is our topical goal. The semiempirical methods mentioned above are based on the semi-classical Born-Oppenheimer approach, whose application to the lightweight helium atoms is quite limited. The main advantage of these methods is their computational inexpensivity. It will be necessary to use some fully quantum method for more exact results (Path Integral Monte Carlo, Diffusion Monte Carlo etc.).
THEORY II – TRIATOMICS-IN-MOLECULES METHOD (TRIM)
-2 -1( ) ( )
TRIM1 1 1 1
1 3ˆ ˆ ˆH H H2 2
N N N NABC A
A B A C B A
NN
(123) (123) (123)11 12 13(123) (123) (123)
(123) 12 22 234 (123) (123) (123)
13 23 33(123)neut
00H (He )0
0 0 0
U U UU U UU U U
E
Coefficients KJ are calculated using the DIM method; in case the three-body correction to the He3+ interaction energy is a small perturbation, the resulting Hamiltonian matrix
is expected to be correct up to 1st order of perturbation theory.
Eneut(ABC) … energy of a neutral (ABC) fragment in the electronic ground-state, calculated using
semiempirical two- and three-body potentials for helium,EJ
(ABC) … energy of an ionic (ABC) fragment in the electronic ground (i = 1) and the first twoexcited (J = 2,3) states, taken from ab initio calculations on He3
+ (see also this poster session:I. Paidarová et al., Ab initio calculations on He3
+ of interest for semiempirical modelling.)
TRIM Hamiltonian Triatomic inputs
Hamilton Matrix
, where
ABC ABCAAABC ABC
BBABC ABC
CCABC ABC
DD
H UH UH UH E
( ) ( )11
( ) ( )22
( ) ( )33
( ) ( )neut
ABC ABC ABCAB BAABC ABC ABC
AC CAABC ABC ABC
BC CBABC
EF
H H UH H UH H UH
( ) ( ) ( )12
( ) ( ) ( )13
( ) ( ) ( )23
( ) 0
3( ) ( ) ( ) ( )
1,ABC ABC ABC ABC
KL J KJ LJJ
U E
3( ) ( )
1.ABC ABC
J KJ KK
E
where is energy of the adiabatic (stationary) state
( )ABC
JE
General theory: R. Kalus, Universitas Ostraviensis, Acta Facultatis Rerum Naturalium, Physica-Chemia 8/199/2001.
3 Very good known results, see Knowles, P. J., Murrell, J. N., Mol. Phys. 87 (1996), 827, for example.
