Transcript Extended Townes-Dailey Analysis II a.pptx

Extended Townes-Dailey Analysis
II: Application to hybridized
orbitals
Nuclear Quadrupole Coupling Constants
Stewart Novick
Wesleyan University
Columbus, 2010
TC 01
Quick review of last year’s talk: unhybridized orbitals:
Electric field gradients at a nucleus due to various hydrogenic electrons
The point here is that the electric
field gradient at a nucleus is dominated
by the p-electrons on that atom.
1
HGeBr, L. Kang, F. Sunahori, A.J. Minei, D.J. Clouthier, S.E. Novick, JCP 130, 124317 (2009)
Some nuclear quadrupole coupling tensor elements.
Notice that the b and c
elements at bromine
are very different!
D74Ge79Br
χaa(79Br) = 243.246(1) MHz
χbb(79Br) = -239.008(3) MHz
χcc(79Br) = -4.238(3) MHz
D73Ge79Br
χaa(73Ge) =
8.641(16) MHz
χbb(73Ge) = 220.(21) MHz
χcc(73Ge) = -229.(21) MHz
c
17 isotopologues of HGeBr have been studied by FTMW spectroscopy.
A total of 711 microwave transitions have been measured and assigned.
Modeling the hyperfine constants
J.G. King, V. Jaccarino, Phys. Rev. 94, 1610 (1954).
In a standard Townes-Dailey analysis, nb and nc are set equal to 2 and na is
solved to fit only χaa.
na = 1.58, nb = 2.00, nc = 1.80
 t1 
 
 t2 
 
 t3 
t 
 4










1
1
1
2
2
2
1
2

1
2
1
1
2
2
1
2
f1


1

1
2
2
1
1
2
2
Generalize to hybridized orbitals, sp3
 1 0
1 
 2

2
 2s
 f1   1 1


1 
   

2p
 f2   2
6
2   x

  
  2p 
1
y
 f3    1  1

 
2
2   2p 
6



f
z
4

  
1
1
2
 


2
3
 2
f2
f3

2 

1
1   2s 

2
2 3   2p x 
   2p 
1
1
  y


2
2 3   2p 
 z

1
0 

2 3 
3
0
f4
creation of the four Q matrices
Q1







2 
0








0








 Vxx Vxy Vxz 


2
f1  Vxy Vyy Vyz   f1  r  sin (  ) dr d d
V V V 
 xz yz zz 
where

V
 xx

2
2
3  sin (  )  cos (  )
r
3

1 
r
3
etc

0
 1
0
0

3
80

a

0

1
0

0
Q1  
3

80a0

1
 0
0

3
40a0











etc for the other three quadrupole coupling matrices
These integrals are done in Mathematica
To scale these to the values of the quadrupole coupling constants for a single
p-electron of a nitrogen atom, multiple by 30 a03 0, where 0 is an experimentally
determined constant.
T D methyl amine
X0 = -11.2 MHz, note sign
MHz, A. Schirmacher and H. Winter, Phys Rev A 47, 4891 (1993)
X0  11.2
methyl amine quadrupole tensor, M. Kreglewski, W. Stahl, J.-U. Grabow, CPL 196, 155 (1992)
0 
 2.813 0
 0 1.982 0 


0 4.795 
 0
 3 0
 8

3
n1 X0   0 

8

 0 0

1
3
1 
  1  3

  1
1 
 5


0 


4
4
8
4 2 
4 2 
2 2 
 8
 8



 3
 3
3
3
1 3
3
1 3


0  n2 X0   
0 
 

  n3 X0 
  n4 X0  0  8


4
8
4 2
4
8
4 2






3
1
1 
 1
 1 1 3
1 3
1 
1 

0









 4 2 4 2
 4 2 4 2
4
4 
4 
4 
 2 2


0
lone pair
bonding to hydrogens
bonding to carbon
check: rotation by 120o about z interchanges the last three matrices
X0 = -11.2 MHz lone pair
 2.813
0

 0
0
1.982
0
 3 0
 8

3
n1 X0   0 
8

 0 0



4.795 
0
0
0
0
3
4
bonding to carbon
bonding to hydrogens
  1  3  1

8
4

4 2


3
3
1
3

  n2 X0   4
8
4
2

 1 1 3
1

  4 2 4  2  4


3

  1
8
4


3
  n3 X   3
0
4
8



 1
1

  4 2  4 



1
4 2
1
3
4
2
 
3
2

1
4

 5
0

8

  n4 X   0  3
0
8



1

0

 2 2

1
2 2
0

1
4
Considering only the diagonal elements, there are two independent equations,
but there are three independent unknowns n1, n2 = n3 by symmetry of
methylamine, and n4. The equations are underdetermined. If we assume that
there are 2 electrons in the lone pair (n1 = 2), then n2 + n1 = 2.91, and n4 = 1.38.
nominal max
2
2
n1 (lone pair)
2.00
1.90
1.80
1.70
STO-3G ??
1.65
2
4
(n2+n3) (to H)
2.91
2.71
2.51
2.31
2.48
1
2
n4 (to C)
1.38
1.28
1.18
1.08
1.25
6.29
5.89
5.49
5.09
5.38
total 5covalent
8ionic
xz element (if you believe Xxz = 0) gives n4 = (n2+n3)/2, which is approximately
correct in all cases.
Xxy = 0 and Xyz = 0 are obeyed identically







sp2 orbitals
 g1

 g2

 g3
 2p
 z
x







2
 1

0
0

3
 3
  2s 
 1
  2p 
1
1



0
 3
  x 
2
6

  2py 


1
1
1


0   2p 
 3
 z
2
6


0
0
1
 0
1,2-dihydro-1,2-azaborine
A.M. Daly, C. Tanjaroon, A.J.V. Marwitz, S.-Y. Liu,
S.G. Kukolich, JACS 132, 5501 (2010)
z out-of-plane
g1 points along the N-H bond
g2 points along the N-C bond
g3 points along the N-B bond
2pz points out of the plane
y
g1
experimental
 0.78



0
0
0.46
0
0


1.25 
0
0
to H
 1
 3

ng 1  X0   0

 0

0
2
3
0
to C






0   ng  X  
2 0


1

 
3 

0
5
3
12
4
3
4
0

1
12
0
to B
pz

 5  3 0 
 1 0

 12

4
 2

 3

1

1
0   ng 3  X0   

0   npz  X0 
0

4
12

2




1 
1
 0

0
 
 0 0
3 
3 

0



0

1
0
Again, two equations and three unknowns. There is no way to distinguish
between populations of the sp2 orbital to C and to B without xy. Setting npz and
calculating the other populations we obtain:
npz
1.00
1.20
1.40
1.50
1.60
2.00
STO-3G
1.58
g1
nH
0.91
1.11
1.31
1.41
1.51
1.91
1.28
g2+g3
nC + nB
1.76
2.16
2.56
2.76
2.96
3.76
2.52
ntotal about N
3.67
4.47
5.27
5.67
6.07
7.67
5.38
The two blue columns are the most reasonable (charge on N -0.27 or -0.67).
Kukolich states npz = 1.6 from MP2/6-311+G(d,p) NBO calculation.
sp hybridization
 h1 


 h2 


 2px 
 2p 
 y
 1
 2

 1
 2
 0

 0
1
0 0
2
0 0 
1
2
1 0
0
0 1
0

  2s 
  2p 
  x 
  2py 

  2pz 

H3C–C≡N, methyl cyanide, G. Winnewisser & coworkers, JMSp 226,123 (2004)
experimental
to carbon
lone pair
px
py
 1 0 0 
 1 0 0 
1 0 0 
 1 0 0 
 4

 4



 2

0
0




1
 2.11205




 0
 MHz n  X   0  1 0   n  X   0  1 0   n  X   0  2 0   n  X   0 1 0 
2.11205
0
h1 0 
py 0


4
 h2 0 
4
 px 0 


0
4.22410 
1
1 
 0


0 0  
0 0  
1 
1 


 0 0

 0 0

2 
2 


there are no off-diagonal tensor components 
2 
2 

two equations
three unknowns
nominal
STO-3G
h1+h2
3
2.50
3.00
3.50
3.10
px
1
0.87
1.12
1.37
1.05
py
1
0.87
1.12
1.37
1.05
total
5
4.24
5.24
6.24
5.20
Acknowledgements
Stephen Kukolich, University of Arizona
Dennis Clouthier, University of Kentucky
Pete Pringle, Wesleyan University
Dan Frohman, Wesleyan University
Bob Bohn, University of Connecticut
THE END
Acknowledgements
•
Collaborators and group members
–
–
–
–
–
–
–
–
–
–
–
–
–
–
•
Lu Kang, Southern Polytechnic State University, Marietta, Georgia
Wei Lin, University of Texas at Brownsville, Texas
Pete Pringle, Wesleyan
Andrea Minei, PhD 2009, Wesleyan
Dan Frohman, Graduate Student Wesleyan
Jovan Gayle ‘07, Wesleyan
William Ndugire ‘10, Wesleyan
Ross Firestone ‘12, Wesleyan
Chinh Duong ’13, Wesleyan
Jennifer van Wijngaarden, University of Manitoba
Bob Bohn, University of Connecticut
Karen Peterson, San Diego State University
Tom Blake, Pacific Northwest National Laboratory
Dennis Clouthier, University of Kentucky
Special Thanks
–
–
–
Jens Grabow, University of Hannover, ftmw++
Herb Pickett, Jet Propulsion Laboratory, retired; Visiting Scholar, Wesleyan University,
SPCAT/SPFIT
Mike McCarthy & Pat Thaddeus, Harvard Smithsonian Center for Astrophysics
For 73Ge, χ0 is +224 MHz for one 4p electron. A similar p-population analysis
for the 73Ge nuclear quadrupole tensor of D73Ge79Br yields
na = 0.71, nb = 1.34, and nc = 0.00
Implies 38% ionic
character. “Standard”
T-D analysis gives 58%.
np(Ge) = 2.05
np(Br) = 5.38
Electronegativity
differences estimates 20%
np(total) = 7.43, “should” be 7
ionic character for this
1.34
2.00
Bond.
Ge
0.00
0.71
1.58
1.80
Br
How does this compare with theory?
TownesDailey
STO-3G
aug-ccpVTZ
Ge na
0.71
0.72
0.58
nb
1.34
1.03
0.97
nc
0.00
0.15
0.21
# p-electrons
2.05
1.90
1.76
Br na
1.58
1.44
1.54
nb
2.00
1.99
2.01
nc
1.80
1.88
1.88
5.38
5.31
5.43
# p-electrons
Bottom line: The non-cylindrical symmetry of the χ tensor is a reflection of the p-electron populations