LIST OF FIGURES
Figure 1 : Shadow representation of the solid angle of two spheres A and B
Figure 2 : Projection of two atoms onto the plane perpendicular to vector OG:
a - OG defines smallest enveloping cone, used in White et al.[3]
b - OG constructed through region of overlap
Figure 3 : a - Ellipse projection used by White et al.[3]
b - Ellipse projection with G in region of overlap
Figure 4 : a - CH3 with only C-H overlap
b - CH3 with all pairs of atoms overlapping
Figure 5 : CH3 with successive addition and subtraction of regions of overlap
Figure 6 : Types of overlap regions:
a - Overlap enclosed by single atom
b - Single atom forms more than one segment
c - Normal overlap region
Figure 7 : Region of triple overlap projected onto the plane perpendicular to G
Figure 8 : White et al.[3] and Multiple overlap solid angles compared to the Tolman cone angle[1]
Figure 9 : Solid angle radial profile of PPh2Me
LIST OF TABLES
Table I : CH3 calculation using double overlap only, for all permutations
Table II : CH3 calculation using double overlap only, for select permutations (White et al.)
Table III : CH3 calculation using all orders of multiple overlap
Table IV : Solid Angles and Cone Angles for 12 selected molecular fragments
Table I : CH3 calculation using double overlap only, for all permutations
Calcul running total in steradians
ation
detail
s
Additi
on of
all
two
atom
solid
angles
:
+ (CH1):[1.460833] = 1.460833
+ (CH2):[1.460861] = 2.921694
+ (CH3):[1.460859] = 4.382553
+ (H1H2):[1.919629] = 6.302182
+ (H1H3):[1.919634] = 8.221816
+ (H2H3):[1.926091] = 10.147907
Remova
l of
all
extra
single
atom
solid
angles
:
2* (C): [0.841767] = 8.464373
- 2* (H1): [1.055183] = 6.354007
- 2* (H2): [1.055183] = 4.243640
- 2* (H3): [1.055180] = 2.133280
Total solid angle for CH3 = 2.133280sr (97.327deg or 17%
sphere)
Table II : CH3 calculation using double overlap only, for select permutations (White et al.)
Calcul running total in steradians
ation
detail
s
Additi
on of
all
two
atom
solid
angles
:
+ (CH1):[1.460833] = 1.460833
+ (CH2):[1.460861] = 2.921694
+ (CH3):[1.460859] = 4.382553
Remova
l of
extra
single
atom
solid
angles
:
2* (C): [0.841767] = 2.699020
Total solid angle for CH3 = 2.69902sr (110.44deg or 21%
sphere)
Table III : CH3 calculation using all orders of multiple overlap
Calcul running total in steradians
ation
detail
s
Calcul
ation
of
all
single
atom
solid
angles
+ (C): [0.841767] = 0.841767
+ (H1): [1.055183] = 1.896950
+ (H2): [1.055183] = 2.952133
+ (H3): [1.055180] = 4.007313
Remova
l of
all
double
overla
ps
(CH1):[0.436117] = 3.571196
(CH2):[0.436090] = 3.135106
(CH3):[0.436088] = 2.699018
(H1H2):[0.190737] = 2.508281
(H1H3):[0.190729] = 2.317552
(H2H3):[0.184272] = 2.133280
Additi
on of
all
triple
overla
ps
+ (H2CH1) [0.170066] = 2.303345
+ (H3CH1) [0.170061] = 2.473407
+ (H3CH2) [0.166343] = 2.639749
+ (H3H1H2) [0.039941] = 2.679690
Remova
l of
all
quadru
ple
overla
ps
(H3H2CH1) [0.039941] = 2.639749
Total solid angle for CH3 = 2.63975sr (109.12deg or 21%
sphere)
Table IV : Solid Angles and Cone Angles for 12 molecular fragments
Cone angle in Solid angles in
radians steradians
Fragment Name Tolman procedure White et al. Multiple Overlap
Imidazole 1.491077 1.590351 1.415416
NH2OH 1.742570 1.876093 1.685321
NH3 1.789398 1.948819 1.859953
NH2Me 1.982452 2.696721 2.322253
NH2EtOH 2.002485 2.868522 2.326860
NH2Me3OH 2.046896 2.853271 2.537361
NH2Pr(OH)2 2.051723 2.988734 2.184162
PF3 2.169660 2.609292 2.599425
PBr3 2.472132 3.156167 3.154727
PEt3 2.663106 4.122755 3.433989
PBz3 3.000215 5.088678 4.221555
P(C6F5)3 3.216643 4.673923 4.017541