Extended Townes-Dailey Analysis II: Application to hybridized

orbitals

Columbus, 2010TC 01

Nuclear Quadrupole Coupling ConstantsStewart Novick

Wesleyan University

Quick review of last year’s talk: unhybridized orbitals:

The point here is that the electricfield gradient at a nucleus is dominatedby the p-electrons on that atom.

Electric field gradients at a nucleus due to various hydrogenic electrons

1

HGeBr, L. Kang, F. Sunahori, A.J. Minei, D.J. Clouthier, S.E. Novick, JCP 130, 124317 (2009)

D74Ge79Brχaa(79Br) = 243.246(1) MHzχbb(79Br) = -239.008(3) MHzχcc(79Br) = -4.238(3) MHz

D73Ge79Brχaa(73Ge) = 8.641(16) MHzχbb(73Ge) = 220.(21) MHzχcc(73Ge) = -229.(21) MHz

17 isotopologues of HGeBr have been studied by FTMW spectroscopy. A total of 711 microwave transitions have been measured and assigned.

Some nuclear quadrupole coupling tensor elements.

Notice that the b and celements at bromineare very different!

c

Modeling the hyperfine constants

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

J.G. King, V. Jaccarino, Phys. Rev. 94, 1610 (1954).

t1

t2

t3

t4

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

2s

2px

2py

2pz

f1

f2

f3

f4

1

2

1

2

1

2

1

2

0

1

6

1

6

2

3

0

1

2

1

2

0

3

2

1

2 3

1

2 3

1

2 3

2s

2px

2py

2pz

Generalize to hybridized orbitals, sp3

f1 f2f3 f4

creation of the four Q matrices

These integrals are done in Mathematica

To scale these to the values of the quadrupole coupling constants for a singlep-electron of a nitrogen atom, multiple by 30 a0

3 0, where 0 is an experimentallydetermined constant.

Q1

0

2

0

0

rf1

Vxx

Vxy

Vxz

Vxy

Vyy

Vyz

Vxz

Vyz

Vzz

f1 r2

sin ( )

d

d

d where Vxx3 sin

2 ( ) cos2 ( )

r3

1

r3

etc

Q1

1

80 a03

0

0

0

1

80a03

0

0

0

1

40a03

etc for the other three quadrupole coupling matrices

T D methyl amine

X0 11.2 MHz, A. Schirmacher and H. Winter, Phys Rev A 47, 4891 (1993)

methyl amine quadrupole tensor, M. Kreglewski, W. Stahl, J.-U. Grabow, CPL 196, 155 (1992)

2.813

0

0

0

1.982

0

0

0

4.795

n1 X0

3

8

0

0

0

3

8

0

0

0

3

4

n2 X0

1

8

3

4

1

4 2

3

4

3

8

1

4

3

2

1

4 2

1

4

3

2

1

4

n3 X0

1

8

3

4

1

4 2

3

4

3

8

1

4

3

2

1

4 2

1

4

3

2

1

4

n4 X0

5

8

0

1

2 2

0

3

8

0

1

2 2

0

1

4

X0 = -11.2 MHz, note sign

lone pair bonding to hydrogens bonding to carbon

check: rotation by 120o about z interchanges the last three matrices

2.813

0

0

0

1.982

0

0

0

4.795

n1 X0

3

8

0

0

0

3

8

0

0

0

3

4

n2 X0

1

8

3

4

1

4 2

3

4

3

8

1

4

3

2

1

4 2

1

4

3

2

1

4

n3 X0

1

8

3

4

1

4 2

3

4

3

8

1

4 3

2

1

4 2

1

4 3

2

1

4

n4 X0

5

8

0

1

2 2

0

3

8

0

1

2 2

0

1

4

lone pair bonding to hydrogens bonding to carbonX0 = -11.2 MHz

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.

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

2 2 n1 (lone pair) 2.00 1.90 1.80 1.70 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

5covalent 8ionic 6.29 5.89 5.49 5.09 5.38

nominal max STO-3G ??

total

g1

g2

g3

2pz

1

3

1

3

1

3

0

0

1

2

1

2

0

2

3

1

6

1

6

0

0

0

0

1

2s

2px

2py

2pz

sp2 orbitals

1,2-dihydro-1,2-azaborineA.M. Daly, C. Tanjaroon, A.J.V. Marwitz, S.-Y. Liu,S.G. Kukolich, JACS 132, 5501 (2010)

x

y

z out-of-plane g1 points along the N-H bondg2 points along the N-C bondg3 points along the N-B bond2pz points out of the plane

g1

experimental to H to C to B p z

0.78

0

0

0

0.46

0

0

0

1.25

ng1 X0

1

3

0

0

0

2

3

0

0

0

1

3

ng2 X0

5

12

3

4

0

3

4

1

12

0

0

0

1

3

ng3 X0

5

12

3

4

0

3

4

1

12

0

0

0

1

3

npz X0

1

2

0

0

0

1

2

0

0

0

1

npz 1.00 1.20 1.40 1.50 1.60 2.00 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

Again, two equations and three unknowns. There is no way to distinguishbetween populations of the sp2 orbital to C and to B without xy. Setting npz andcalculating the other populations we obtain:

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.

STO-3G

h1

h2

2px

2py

1

2

1

2

0

0

0

0

1

0

0

0

0

1

1

2

1

2

0

0

2s

2px

2py

2pz

sp hybridization

H3C–C≡N, methyl cyanide, G. Winnewisser & coworkers, JMSp 226,123 (2004)

2.11205

0

0

0

2.11205

0

0

0

4.22410

MHz nh1 X0

1

4

0

0

0

1

4

0

0

0

1

2

nh2 X0

1

4

0

0

0

1

4

0

0

0

1

2

npx X0

1

0

0

0

1

2

0

0

0

1

2

npy X0

1

2

0

0

0

1

0

0

0

1

2

experimental to carbon lone pair px py

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

two equationsthree unknowns

there are no off-diagonal tensor components

Acknowledgements

Stephen Kukolich, University of ArizonaDennis Clouthier, University of KentuckyPete Pringle, Wesleyan UniversityDan Frohman, Wesleyan UniversityBob 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 analysisfor the 73Ge nuclear quadrupole tensor of D73Ge79Br yields

na = 0.71, nb = 1.34, and nc = 0.00

2.00

1.58

1.80

1.34

0.71

0.00

BrGe

np(Ge) = 2.05 np(Br) = 5.38

np(total) = 7.43, “should” be 7

Implies 38% ioniccharacter. “Standard”T-D analysis gives 58%.Electronegativitydifferences estimates 20%ionic character for thisBond.

How does this compare with theory?Townes-Dailey

STO-3G aug-cc-pVTZ

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

# p-electrons 5.38 5.31 5.43

Bottom line: The non-cylindrical symmetry of the χ tensor is a reflection of the p-electron populations