International Symposium on Molecular Spectroscopy
61st Meeting - - June 19-23, 2006
An Accurate ab initio Potential Energy Surface and Calculated
Spectroscopic Constants for BeH2, BeD2 and BeHD
υ
Hui Li, Robert J. Le Roy
Background
• Chemical bond
2
2
Be(1s 2s )
sp hybridization
Test new ab initio methods
• Formation and Stability
Be(g)+H2(g)
59.65 kcal/mol
BeH2(g) (ΔE=-38.35 kcal/mol)
• Solid BeH2
Linear(BeH2)n
in matrix
G. J. Brendel et al. Inorg. Chem. 17, 3589(1978)
X. F. Wang et al. Inorg. Chem. 44, 610(2005)
crystal
structure
G. S. Smith et al. Solid State Communications 67, 491(1988)
Recent New Experiment
• BeH2 detected in argon matrix
T.J.Tague,Jr. et.al, J. Am. Chem. Soc. 115, 12111 (1993)
• Free gaseous BeH2
Infrared emission spectrum of BeH2 and BeD2
P. F. Bernath, et al., Science, 297, 1323 (2002),
A. Shayesteh, et al., J. Chem. Phys., 118, 3622 (2003)
Previous Theoretical Results
Theory
Mode
υ1
BeH2
BeD2
a
Experiment
f
CC /
CCSD(T)/
In Argon
b
a
c
[5s3p2d1f] cc-pCVTZ Matrix
1979.6
1987.0
υ2
716.8
705.2
697.9
υ3
2167.2
2157.8
2159.1
υ1
1420.8
υ2
545.1
531.9
υ3
1676.2
1674.0
Gaseous
d
BeH2
2178.9
1689.7
J. M. L. Martin and Timothy J. Lee, Chem. Phys. Lett. 200, 502 (1992).
b
T. Hrenar, H.-J. Werner & G. Rauhut, Phys. Chem. Chem. Phys. 7 3123 (2005).
c
T. J. Tague, Jr., and L. Andrews. J. Am. Chem. Soc. 115, 12111 (1993).
d
P. F. Bernath, et al., Science, 297, 1323 (2002)
f
A multi-level ab initio Scheme (1D: CCSD(T); 2D: MP4(SDQ), 3D: MP2) & VCI used
Potential Energy Surface
•
•
E(AV5Z): calculated at the icMRCI+Q/aug-cc-pV5Z level.
Δcore: determined at MR-ACPF/cc-pCV5Z level
E(cav5z)=E(av5z)+Δcore
Transition state
r(HH)=2.0250 a0
R= 2.6342 a0
θ=90º
H
Be
H
BeH2
Method
Re(a0)
a
CMRCI/VQZ+K
2.506
2.5147
2.5067
2.5065
AV5Z
AV5Z+Vcore
b
EXP.
a
Linear minimum
2.6342 a0
Re=2.5067 a0
Be+H2
TS. (eV) Be+H2 (eV) BeH+H (eV) H+Be+H (eV)
4.6633
4.2343
4.2493
J. Hinze et al., Mol. Phys. 96, 711 (1999)
b
1.6208
1.6285
1.6628
4.2112
4.2247
6.3797
6.4231
P. F. Bernath, et al., Science, 297, 1323 (2002)
Question: How is the accuracy of calculated vibrational levels
affected by method for interpolating over the grid of PES
Test 1: One by one, omit each grid point, interpolate for it from the rest, and
determine error in the value obtained by different interpolation methods
Test 2: Compare the vibrational levels using different interpolation methods
RMSD (in
-1
cm )
on spline interpolation over
n
V×Ri
Range
No. points
n=0
n=2
n=4
All energies
6864
97.93
5.63
73.53
1857
6.60
0.89
3.53
6864
0.08
reference
0.04
Energies≤20 000
-1
cm
62 lowest pure levels of BeH2
Estimated the RMSD with n=2 interpolation is about 0.01
-1
cm
Ro-vibrational Energy Levels
• The Hamiltonian in Radau coordinates (R1, R2, ) is
ˆ
1

1

1
1

1


J
ˆ
H 


(

)(
sin


)
2
2
2
2
2m1 R1 2m2 R2
2m1 R1 2m2 R2 sin  
 sin 
2
2
2
z
2
ˆJ  2 Jˆ
cot  ˆ
1

ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ


[(
J

i
J
)

(
J

i
J
)]
J

[(
J

i
J
)

(
J

i
J
)]

V
(
R
,
R
,

)
x
y
x
y
z
x
y
x
y
1
2
2
2
2
2m1 R1
2m1 R1
2m1 R1 
2
2
z
R1 and R2 are the radial Radau coordinates
m1 and m2 are the masses of the two H atoms
Jx, Jy, Jz are the components of the total angular momentum J
The z axis of the body-fixed frame lies along the R1 radial Radau vector
• A direct product discrete variable representation (DVR)
• Lanczos recursion used to diagonalize the Hamiltonian
Ro-vibrational Energy Levels and Wave Function
(v1,v2,v3)
(0,0,0)
(0,0,1)
(0,0,2)
(1,0,0)
(1,0,1)
J=0
0.00
2178.79 4323.80 1991.97 4112.25
J=1
9.40
2188.05 4332.93 2001.26 4121.40
J=2
28.21
2206.58 4351.19 2019.84 4139.70
J=3
56.41
2234.37 4378.57 2047.71 4167.14
J=4
94.00
2271.41 4415.07 2084.84 4203.72
J=5
140.97 2317.70 4460.68 2131.25 4249.42
J=6
197.31 2373.21 4515.38 2186.91 4304.24
J=7
263.00 2437.93 4579.16 2251.80 4368.16
J=8
338.02
2511.85 4652.00 2325.92 4441.16
rs = R1,(BeH) + R2,(BeH)
rd = R1, (BeH) - R2,(BeH)
Spectroscopic Constant
• The energy level expression for linear triatomic
E  G (1 , ,3 )  B[ J ( J  1)   ]  D[ J ( J  1)   ]  H [ J ( J  1)   ]

2
0

2
2 2
2 3
1
2
3
  [qJ ( J  1)  qD [ J ( J  1)]  qH [ J ( J  1)]   ,1
2

G: pure vibrational energy level
B: inertial rotational constant
ℓ: vibrational angular momentum quantum
D: centrifugal distortion constant
±: the parity forπ(ℓ=1) levels
Σ state: q=qD=qH=0
• ℓ-type rotational resonances
 E

H 
2
W
20

0

E

E
E

E

2W20

0
1

2
2
2
2
1/ 2
E  W20  [q  qD J ( J  1)  qH J ( J  1)  ( J ( J  1)  2 J ( J  1)) .
2
0

0
 0
0

0
2
G. Amat et al. J. Mol. Spectrosc. 2, 163 (1958) A.G. Maki Jr. et al. J. Chem. Phys. 47, 3206 (1967)
A. Shayesteh and P. F. Bernath J. Chem. Phys. 124, 156101 (2006)
Spectroscopic Constants of BeH2
Gv-ZPE
B
Cal.Obs.
Level
Calc.
0
(0,0 ,0)
0.00
0.00
4.7023 0.0009 1.0528 0.0028
0
(0,0 ,1)
2178.79
-0.08
4.6333 0.0011 1.0360 0.0029
0
(0,0 ,2)
4323.80
0.02
4.5656 0.0014 1.0196 0.0026
0
(0,0 ,3)
6434.32
0.21
4.4989 0.0017 1.0040 0.0055
0
(1,0 ,0)
a
0
(1,0 ,1)
a+2120.28
1
(0,1 ,0)
b
1
(0,1 ,1)
b+2165.80
1
(0,1 ,2)
a+4298.26
1
(1,1 ,0)
c
1
(1,1 ,1)
c+2106.97
a = 711.77
-1
cm
Calc.
Cal.Obs.
10**4 D
Cal.Calc.
Obs.
10**2 q
Cal.Calc.
Obs.
10**6 qD
Cal.Calc.
Obs.
4.6456 0.0012 1.0442 0.0018
0.12
4.5755 0.0014 1.0289 0.0029
4.7130 0.0010 1.0958 0.0050
-9.148 -0.008
8.33
0.21
-0.01
4.6440 0.0012 1.0795 0.0048
-9.107 -0.008
8.25
0.22
0.15
4.5762 0.0014 1.0634 0.0041
-9.069 -0.008
8.17
0.19
4.6558 0.0013 1.0853 0.0034
-9.010
0.012
8.07
-0.57
4.5857 0.0015 1.0698 0.0029
-8.961
0.013
7.96
-0.63
0.18
b = 1991.97
-1
cm
c = 2702.78
-1
cm
BeH2 for levels involving L-type resonance
Gv-ZPE
Level
Calc.
0
(0,2 ,0)
d
2
(0,2 ,0)
d+10.61
B
Cal.Obs.
Calc.
Cal.Obs.
10**4 D
Cal.Calc.
Obs.
10**2 q
Cal.Calc.
Obs.
10**6 qD
Cal.Calc.
Obs.
4.7247 0.0010 1.1442 0.0046
0.05
4.7225 0.0010 1.1378 0.0041
-9.203 -0.011
8.63
0.26
0
(0,2 ,1)
d+2152.83 0.02
4.6557 0.0013 1.1284 0.0052
2
(0,2 ,1)
d+2163.34 0.09
4.6534 0.0012 1.1221 0.0038 -9. 158 -0.010
8.55
0.22
0
(0,2 ,2)
d+4272.76 0.21
4.5878 0.0015 1.1127 0.0023
2
(0,2 ,2)
d+4283.18 0.30
4.5856 0.0015 1.1062 0.0050
8.50
0.19
d = 1417.27
-9.117 -0.011
-1
cm
The RMS of discrepancies for 11 bands 0.15(±0.09)
Relative inertial rotational constants B only 0.028%
-1
cm
Spectroscopic Constants of BeD2
Gv-ZPE
B
Cal.Obs.
Cal.Obs.
Level
Calc.
0
(0,0 ,0)
0.00
0.00
2.36034 -0.0006 2.62478 0.0042
0
(0,0 ,1)
1689.27
-0.40
2.32970 -0.0006 2.57460 -0.0154
0
(0,0 ,2)
3356.06
-0.68
2.29950 -0.0004 2.52583 0.0093
1
(0,1 ,0)
a
1
(0,1 ,1)
a+1680.20
0
(0,2 ,0)
d
2
(0,2 ,0)
d+10.14
-0.36
Calc.
10**4 D
Cal.Calc.
Obs.
-2.985 -0.033
1.31
-0.59
2.33655 -0.0005 2.67047 -0.0006
-2.995 -0.035
1.31
-0.71
-2.988 -0.009
1.36
0.16
-2.996 -0.010
1.36
0.20
2.37421 -0.0007 2.82248 0.0182
2.37370 -0.0005 2.81084 0.0539
0
(0,2 ,1) d+1671.17
-0.33
2.34356 -0.0006 2.77247 0.0142
2
(0,2 ,1)
-0.29
2.34305 -0.0003 2.75870 0.0688
a = 548.36
-1
cm
10**6 qD
Cal.Calc.
Obs.
2.36720 -0.0006 2.72097 0.0276
-0.03
d+1681.26
10**2 q
Cal.Calc.
Obs.
d = 1089.19
-1
cm
The RMS of discrepancies for 5 bands 0.46(±0.19)
Relative inertial rotational constants B only 0.023%
-1
cm
Predicted Spectroscopic Constants of BeHD
ℓ
(v1,v2 ,v3)
Gv-ZPE
B
10**5 D
10**9 H
0
(0,0 ,0)
0.00
3.2324
5.2446
0.95
0
(0,0 ,1)
1537.76
3.1948
5.2112
0.97
0
(0,0 ,2)
3045.66
3.1571
5.1685
1.04
0
(1,0 ,0)
2106.14
3.1943
5.1523
0.83
0
(2,0 ,0)
4153.26
3.1568
5.0297
0.88
1
(0,1 ,0)
635.55
3.2402
5.4226
1
(0,1 ,1)
2173.56
3.2031
1
(1,1 ,0)
2732.42
0
(0,2 ,0)
10**2 q
10**6 qD
10**10 qH
1.10
-4.860
3.124
-2.41
5.3597
1.19
-4.755
3.051
-2.60
3.2017
5.3329
1.00
-4.961
2.929
-3.28
1262.95
3.2483
5.6362
1.23
2
(0,2 ,0)
1275.16
3.2471
5.5912
1.25
-4.867
3.297
-2.82
0
(0,2 ,1)
2800.20
3.2114
5.5641
1.34
2
(0,2 ,1)
2813.07
3.2104
5.5069
1.35
-4.765
3.218
-2.62
0
(1,2 ,0)
3350.67
3.2098
5.5456
1.15
2
(1,2 ,0)
3362.64
3.2085
5.5021
1.14
-4.954
3.130
-3.32
Future work
Difference Vtot-VH-H vs.
rs=R1,(BeH)+R2,(BeH)
&
rd=R1,(BeH)-R2,(BeH)
Trial Analytic Function
V (rs , rd , )  U e {e
2  ( rs ;rd , )( rs  rse )
 2e
  ( rs ;rd , )( rs  rse )
}
Conclusions
• 3D PES constructed from 6864 ab initio points at icMRCI/AV5Z
level and core correlation corrected at MR-ACPF/cc-pCV5Z.
• Calculated spectroscopic constants of BeH2 and BeD2 are in
excellent agreement with recent experiment results validating
the quality of PES.
• The predicted spectroscopic constants of BeHD should provide
a reliable information for the unobserved bands.
• The effect of different interpolation methods has been
examined.
• Future work
Acknowledgments
• P. F. Bernath, A. Shayesteh and M. Nooijen
(University of Waterloo)
• K. A. Peterson (Washington State University)
• NSERCC
Structures for (BeH2)n polymers
G. J. Brendel et al. Inorg. Chem. 17, 3589(1978)
X. F. Wang et al. Inorg. Chem. 44, 610(2005)
Crystal structure of BeH2 showing a
network of corner-sharing BeH4 tetrahedra
G. S. Smith et al. Solid State Communications 67,
491(1988)
Spectroscopic constants (in
a
V
0
(1,0 ,0)
b
V+C
of BeH2 at different levels
d
V+C+F+R
f
V+C+F+R+B
This work
Exp.
1992.07
1992.18
1991.48
1991.97
716.65
716.24
716.36
716.24
716.48
0
(0,0 ,1)
2170.62 2180.47
2179.28
2179.39
2178.70
2178.79 2178.8659
B000
4.67506 4.70681
4.70560
4.70629
4.70407
4.70230
4.70140
4
10 D000
1.04192 1.05404
1.05473
1.05507
1.0543
1.0528
1.0500
1
g
(0,1 ,0)
1984.77 1993.58
c
V+C+F
-1
cm )
713.41
Reference to Jacek Koput and Kirk A. Peterson J. Chem. Phys. (2006)
a
• Determined
using the CCSD(T) method with the cc-Pv6z and aug-cc-Pv6z basis
sets for Be and H, respectively.
b
• Including additional corrections for the core-related correlation effects calculated at
the CCSD(T)/cc-pCV5Z level.
c
• Including additional corrections for the high-order valence correlation effects
calculated at the FCI/ and CCSD(T)/ cc-pCVTZ levels.
d
• Including additional corrections for the scalar relativistic effects calculated at the
CCSD(T)/ cc-pV5Z-DK level.
f
• Including Born-Oppenheimer diagonal correction calculated CISD/ cc-pCVTZ( augcc-pVTZ on H)
f
• J=L=1
DVR and Lanczos algorithm
• A direct product discrete variable representation (DVR)
grid was used in vibrational energy calculation
Each stretching coordinate was represented by 70 PODVR grid, with 200 equidistant
sine-DVR grid on interval [1.6, 5.0]
80 Gauss-Legendre grid points on the interval [60-180] were used for the angular
variable
• Lanczos recursion used to diagonalize the Hamiltonian
ˆ
m1   [ m1m1  (   m )m ]
1
m
ˆ
 m  (m Hm   m1m1 ),
ˆ
ˆ
 m  ( m1m1  (   m )m   m1m1  (   m )m )
1/ 2
0  0
When eigenfunctions are needed, for selected eigenvalue , using inverse iteration method get
the eigenvector and repeat Lanczos recursion get the wavefunction
10000 Lanczos iterations were found adequate to converge the levels within 9000 cm 0.001cm
Remove Spurious Eigenvalues Methods
• The Cullum-Willoughby (CW) method
compare the eigenvalues of its sub-matrix obtained by deleting the first row and
first column. If a Lanczos eigenvalue appears for both matrices, it is regarded
“spurious” and deleted.
•
Different Recursion Steps
Compare the Lanczos eigenvalues in different recursion steps: a true eigenvalue
should not depend on recursion steps. With earliest appearance in the recursion is
regarded as “good”.
Since the “spurious” eigenvalues are shared by both the Lanczos matrix and its submatrix, they have small z1i and can be considered as copies generated from the
round-off errors.
J. Cullum and R. A. Willoughby, J. Comput. Phys. 44, 329 (1981)
R. Chen and H. Guo J. Chem. Phys. 111, 9944 (1999)
Triatomic Radau coordinaties
O: Be atom m1=m2=H
D: Central of mass of two H
C: triatom central of mass
2
B: BD =CD·OD
B. R. Johnson et al. J. Chem. Phys. 85,15(1986)