suppmat1782.doc 808KB Jun 05 2011 09:30:53 PM

Derivation of Class II Force Fields. VIII. Derivation of a General

Quantum Mechanical Force Field For Organic Compounds

Carl S. Ewig, Rajiv Berry, Uri Dinur, Jörg-Rüdiger Hill, Ming-Jing Hwang, Haiying Li, Chris

Liang, Jon Maple, Zhengwei Peng, Thomas P. Stockfisch, Thomas S. Thacher,

Lisa Yan, Xiangshan Ni, and Arnold T. Hagler

Table 1. Compounds Comprising the Training Set of Molecular Structures Used for Parameterizing the Force Field.

Acetals/hemiacetals

1,1,2-trihydroxyethane 1,1,3-trihydroxypropane

1,1,4-trihydroxyethylmethyl ether 1,1-dihydroxybutane

1,2,2-trihydroxypropane 1,3,5,7-tetraoxacane 1,3,5-trioxane

1,3-dihydroxymethylethyl ether 1,3-dioxane

1,3-dioxolane

2,2,3-trihydroxybutane

2-deoxy--D-glycerotetrafuranose 2-hydroxytetrahydrofuran

3-hydroxymethylethyl ether dihydroxymethyl ether dimethoxymethane hydroxyethoxymethane hydroxymethoxymethane Alcohols

Aldehydes/ketones acetaldehyde acetone butanone cyclobutanone cyclopropanone

cyclopropylcarboxaldehyde formaldehyde

isobutyraldehyde n-butyraldehyde propionaldehyde succinaldehyde Alkanes

1,1-dimethylcyclopropane 1,2-dimethylcyclopropane cyclobutane

cyclohexane cyclopentane cyclopropane ethane isobutane isopentane methane

methylcyclobutane methylcyclopropane n-butane

neopentane n-pentane propane Alkenes

1,3,6,8-nonatetraene 1,3-hexadiene 1,3-pentadiene 1,4-pentadiene 1,5-hexadiene 1-butene

1-methyl-cyclopentadiene 1-pentene

1-vinyl-cyclohexene 2-(isopropenyl)butadiene 2,3-dimethylbutadiene 2-butadienyl-butadiene 2-butene

2-ethyl-butadiene 2-methyl-1-butene 2-methyl-2,4-pentadiene 2-methyl-2-butene 2-pentene

2-vinyl butadiene

3-methyl-penta-1,3-diene 4,5-dimethyloctatetraene 4-methyl-1,3,5,8-octatetraene 4-methyl-1,3,6,8-nonatetraene 4-methyl-1,3-hexadiene butadiene

dodecahexaene ethylene

hexadecaoctaene hexatriene isobutene octatetraene propene

tetradecaheptaene Alkynes

1-butyne acetylene phenyl ethyne propyne Amides

acetamide

azacyclopropanone butyrolactam formamide

methylazacyclopropanone N,N-dimethylacetamide N,N-dimethylformamide N-ethylformamide N-formylalanineamide N-formylazeridine N-formylformamide N-formylglycineamide

N-formyl-N-methylglycineamide N-methylacetamide

Amines

azetidine aziridine

cyclobutylamine cyclopropylamine dimethylamine ethylamine ethylenediamine isopropylamine methylamine Ammonium cations

ammonium choline

dimethylammonium ethanolammonium methylethylammonium methylphosphorylcholine n-propylammonium t-butylammonium tetramethylammonium trimethylammonium Aromatic acids/esters

2-imidazolecarboxylic acid 2-methylbenzoic acid 2-pyridinecarboxylic acid 2-pyrimidinecarboxylic acid 4-imidazolecarboxylic acid

4-methyl-5-imidazolecarboxylic acid 5-imidazolecarboxylic acid

benzoic acid methyl benzoate

methyl-4-imidazolecarboxylate Aromatic alcohols/ethers

2-ethoxy pyridine 2-hydroxy imidazole 2-hydroxy pyridine 2-hydroxy pyrimidine 2-methoxy imidazole 2-methoxy pyridine 2-methoxy pyrimidine 4-ethoxy imidazole 4-hydroxy imidazole 4-methoxy imidazole 5-hydroxy imidazole anisole

ethoxy benzene Aromatic amides

1,2,3,5-tetrazole-4-carboxamide 2-furamide

2-methylbenzamide 2-pyridinecarboxamide 2-pyrimidinecarboxamide

3-methyl-pyridine-2-carboxamide 4-methyl-imidazole-5-carboxamide 5-methyl-imidazole-4-carboxamide benzamide

N-methyl-2-pyridinecarboxamide N-methyl-2-pyrimidinecarboxamide N-methyl-benzamide

N-methyl-imidazole-2-carboxamide N-methyl-imidazole-5-carboxamide N-methyl-oxazole-2-carboxamide N-methyl-pyrrole-3-carboxamide N-methyl-thiazole-2-carboxamide N-phenyl-acetamide

N-phenyl-benzamide N-phenyl-urea

oxazole-2-carboxamide pyrimidine-5-carboxamide pyrrole-3-carboxamide thiazole-2-carboxamide thiophene-2-carboxamide Aromatic amines

(2-methyl-phenyl) methyl amine (5-methyl-4-imidazolyl) methyl amine 2-imidazolyl ethyl amine

2-pyridyl methyl amine 2-pyrimidinyl methyl amine 4-imidazolyl ethyl methyl amine 4-imidazolyl methyl amine 5-imidazolyl methyl amine phenyl ethyl amine

phenyl ethyl methyl amine phenyl methyl amine Aromatic ammonium

2-methyl-anilinium 2-pyridyl ammonium 2-pyrimidinyl ammonium 3-pyridyl ammonium 4-imidazolyl ammonium

4-methyl-imidazolyl-5-ammonium 5-imidazolyl ammonium

5-methyl-imidazolyl-4-ammonium anilinium

N-methyl-imidazolyl-2-ammonium N-methyl-imidazolyl-4-ammonium N-methyl-imidazolyl-5-ammonium N-methyl-pyridyl-2-ammonium N-methyl-pyrimidinyl-2-ammonium Aromatic amidine cation

2-imidazole-carboxamidine 2-pyridine-carboxamidine 2-pyrimidine-carboxamidine

3-methyl-pyridine-2-carboxamidine 3-pyrrole-carboxamidine

4-imidazole-carboxamidine

4-methyl-imidazole-5-carboxamidine 5-imidazole-carboxamidine

5-pyrimidine-carboxamidine benzamidine

Aromatic rings

1,3,4-oxadiazole 1,3,4-thiadiazole benzene

imidazolium cation oxazole

pyrazine pyrazole pyridazine pyridine pyrimidine pyrrole thiazole thiophene triazine

Aromatic sulfones/sulfonamides 2-(methylsulfonyl)imidazole 2-(methylsulfonyl)pyridine 2-(methylsulfonyl)pyrimidine 3-(methylsulfonyl)pyrrole 4-(ethylsulfonyl)imidazole 4-(methylsulfonyl)imidazole 5-(methylsulfonyl)imidazole benzenesulfonamide

imidazole-2-sulfonamide imidazole-4-sulfonamide imidazole-5-sulfonamide methylphenyl-sulfone

N-ethyl-methanesulfonamide

N-ethyl-N-methyl-methanesulfonamide N-methyl-benzenesulfonamide

N-phenyl-methanesulfonamide pyridine-2-sulfonamide

Azoalkanes

1-pyrazoline azomethane

di(diazenyl)-ethane di(diazenyl)-methane diazene

ethyldiazene isopropyldiazene methyldiazine propyldiazene t-butyldiazene

tetrahydro-pyradazine Bridged aromatics

2-(fur-2-yl)pyrrole 2-(pyridin-2-yl)furan 2-(pyridin-2-yl)pyrrole 2-(pyridin-2-yl)thiophene 2-(thien-2-yl)furan 2-(thien-2-yl)pyrrole 2,2’-bifuran

5-(thien-2-yl)pyrazole 5,5’-bithiazole

biphenyl Carbamates

carbamic acid methylcarbamate

N,N-dimethyl carbamate N,N-dimethyl methylcarbamate N-methyl carbamate

N-methyl methylcarbamate Carbonates

carbonic acid dimethyl carbonate methyl carbonate Carboxylate anions

acetate formate propionate Carboxylic acids

acetic acid formic acid glycolic acid glyoxylic acid propionic acid pyruvic acid Clorinated hydrocarbons

1,2-dichlorobenzene

1,5-dimethyl-4-chloro-imidazole 2-chloroimidazole

4-chloroimidazole

4-methyl-5-chloro imidazole 4-trichloromethyl imidazole

4-trichloromethyl-5-methyl imidazole 5-chloroimidazole

5-trichloromethyl imidazole benzotrichloride

chlorobenzene chlorotoluene Dialkyl phosphate anions

dimethyl phosphate methylethyl phosphate Disulfides/sulfides/thiols

1,2-dithiethane 1,2-dithiirane

1,3-dithiacyclopentane

1-methyl-1-methylthiocyclopropane 1-methylcyclopropane-1-thiol 2-methylthioethane-1-thiol 2-methylthiopropane cyclobutanethiol cyclopentanethiol cyclopropanethiol dimethyl disulfide ethane-1,2-dithiol ethanethiol

ethyl hydrogen disulfide hydrogen disulfide methanethiol

methylthiocyclopentane methylthiocyclopropane methylthioethane methylthiomethane methylthiomethanethiol methylthiopropane propane-2-thiol propanethiol

tetrahydrothiophene thietane

thiirane Esters

3-formyl propanal acetolactone ethyl formate methyl acetate methyl formate methyl glycolate propiolactone Ethers

dimethyl ether methyl ethyl ether methyl isopropyl ether oxetane

oxirane

Fluorinated hydrocarbons 1,2-difluorobenzene

1-trifluoromethyl-2-methylbenzene 2-fluoro imidazole

4-fluoroimidazole

4-methyl-5-fluoroimidazole 4-trifluoromethyl imidazole

4-trifluoromethyl-5-methyl imidazole 5-fluoroimidazole

5-trifluoromethyl imidazole benzotrifluoride

fluorobenzene fluorotoluene Hydrazines

hydrazine

methyl hydrazine

N,N,N’-trimethyl hydrazine N,N’-dimethyl hydrazine N,N’-methylethyl hydrazine N,N-dimethyl hydrazine N,N-methylethyl hydrazine N-isopropyl hydrazine pyrazolidine

tetramethyl hydrazine Nitroaromatics

1-nitro-2-methyl benzene 2-nitroimidazole

4-methyl-5-nitroimidazole 4-nitro-5-methylimidazole 4-nitroimidazole

5-nitroimidazole nitrobenzene Phosphate dianions

Table 2. Quantum Mechanical Force Field Parameters.

_____________________________________________________________ Bond length terms, E2K b b_b   K b b_b   K b b_b 

0 2 3 0 3 4 0 4

( ) ( ) ( )

_____________________________________________________________

b0 2_Kb 3_Kb 4_Kb

_____________________________________________________________

np sp 1.6485995 364.49484 -545.62257 650.00000 o p 1.6354935 398.67842 -622.23348 826.84440 o p- 1.6997274 341.95620 -557.68919 360.06776 o' s" 1.435736 814.84012 -1897.2198 2627.9370 o- p 1.4703638 785.88168 -1629.6214 2056.5623 o- p- 1.4968054 683.42789 -1401.9900 1933.7442 s s 2.0818932 212.21289 -102.04010 820.35444

_____________________________________________________________

Bond angle terms, E2K   K   K 

0 2 3 0 3 4 0 4

(  ) (  ) (  )

__________________________________________________________________

0 2K 3K 4K

__________________________________________________________________

cp sp cpb 91.936733 105.41543 -1.9188303 -21.287306 cp sp np 92.774216 86.161137 -2.2523088 -17.443818 np sp np 92.134128 214.89015 -4.3097771 -43.389864

_____________________________________________________

Torsion terms, E 1K ₁  2K ₁ ₂  3K ₁ ₃ ( cos ) ( cos ) ( cos )

_____________________________________________________

 1_K

 2K 3K

_____________________________________________________

cp np np cp 0.0000000 6.5781347 0.0000000 cp np np nh 0.0000000 12.174899 0.0000000 cp np np np 0.0000000 11.553981 0.0000000 cp np np op 0.0000000 7.9409292 0.0000000 cp np np sp 0.0000000 9.4254789 0.0000000 cpb np np nh 0.0000000 12.174899 0.0000000 nh np np np 0.0000000 12.189261 0.0000000 np np np np 0.0000000 17.747821 0.0000000 cp np op cp 0.0000000 4.7482919 0.0000000 cp np op np 0.0000000 7.5296380 0.0000000 cpb np op cp 0.0000000 4.7482919 0.0000000 np np op cp 0.0000000 8.7762279 0.0000000 cp np sp cp 0.0000000 1.7246826 0.0000000 cp np sp np 0.0000000 7.8825686 0.0000000 np np sp cp 0.0000000 12.302920 0.0000000 c o p o 0.2930291 0.6805675 -0.1131336 c o p o- 0.2227508 1.0763550 -0.0456873 c o p- o- 2.1848923 1.9290247 0.0135564 cp o p- o- 0.2972287 0.4133854 0.6403414 h o p- o- -0.1640977 0.0892636 -0.0876827 c s s c -0.9068356 -3.3890910 -0.6594178 c s s h -0.3561881 -3.3826880 -0.3484207 h s s h -0.1274146 -3.5179435 -0.2618985

__________________________________________________________________ Electrostatic energy, for atom pair i and j, E 332q q ri j/ ij

where qi ik k





 _{and }

ik >0 for charge donated from atom i to atom k

_______________________________________________________________________________

i k _ik

__________________________

br br 0.0000 br c -0.1920 br c" -0.0800 br c' -0.0800 br c- -0.0281 br c -0.0800 br cl 0.0507 br cp -0.0800 br ct 0.0173 br f 0.2099 br h -0.1978 br i -0.0110 br n 0.1422 br n+ 0.2452 br n 0.1422 br na 0.0601 br nh -0.0438 br nn 0.1422 br np 0.1422 br o 0.0818 br o' 0.3140 br op 0.3140 br p -0.2156 br s -0.0437 br s" 0.0034 br sp 0.0034 c c 0.0000 c c" 0.0000000 c c' 0.0000000 c c- -0.2300 c c 0.0000000 c cl -0.011 c co 0.0000000 c cp 0.0000000 c ct 0.0441500 c f 0.13275 c h -0.0530 c hc+ -0.2270000 c i 0.1120 c n 0.0000000 c n+ 0.205

c op 0.3957 c p 0.0110 c s 0.065000 c s" -0.0500000 c sp 0.1180 c" c" 0.0000 c" c' 0.0000 c" c- -0.1368 c" c 0.0000 c" cl 0.1020 c" co 0.0000000 c" cp 0.065 c" ct -0.0927 c" f 0.1300 c" h -0.0456000 c" i -0.1291 c" n 0.0000000 c" n+ 0.1331

c+ cp 0.0 c+ h -0.1268 c+ n -0.0680 c+ nr -0.0680 c- c- 0.0000 c- cl 0.0747 c- cp -0.0424 c- ct 0.0432 c- f 0.2241 c- h -0.0530 c- i 0.0185 c- n 0.1607 c- n+ 0.2597

c- n 0.1607 c- nh -0.0176 c- nn 0.1607 c- np 0.1607 c- o 0.1012 c- o' 0.3241 c- o- 0.0337 c- op 0.3241 c- p -0.0857 c- s -0.0087 c- s- -0.1223

c c 0.0000000 c cl 0.1020 c cp 0.03 c ct 0.0852 c f 0.1300 c h -0.1268000 c i 0.0642 c n 0.07

c n+ 0.2989 c n 0.36

cl i -0.0623 cl n 0.0897 cl n+ 0.1858 cl n 0.0897 cl na 0.0117 cl nh -0.0854 cl nn 0.0897 cl np 0.0897 cl o 0.0367 cl o' 0.2585 cl op 0.2585 cl p -0.2544 cl s -0.0898 cl s" -0.0457 cl sp -0.0457 co h -0.0530000 co na 0.0827000 co nh 0.2108000 co o 0.1133000 cp cp 0.0000000 cp cpb 0.0000000 cp ct 0.0790400 cp f 0.116 cp h -0.14 cp i 0.0642 cp n 0.1746000 cp n' 0.2 cp n+ 0.2989

cp n 0.1380000 cp nh 0.16 cp nn 0.04 cp np 0.18 cp o 0.1500000 cp o' 0.3964 cp op 0.11 cp s -0.0120 cp s" 0.0732 cp sp -0.02 cpb cpb 0.0000000 cpb nh 0.16 cpb nn 0.0000000 cpb np 0.18 cpb op 0.11 cpb sp -0.02 cr n 0.1380 cr nr .1746

ct f 0.1873 ct h -0.2183300 ct i -0.0281 ct n 0.1204 ct n+ 0.0992

ct n 0.1204 ct na -0.0636 ct nh -0.0568 ct nn 0.0920 ct np 0.1204 ct o 0.0675 ct o' 0.2874 ct op 0.2874 ct p -0.1335 ct s -0.0581 ct s" -0.0135 ct sp -0.0135 f f 0.0000 f h -0.3823 f i -0.2234 f n -0.0731 f n+ 0.0062 f n -0.0731 f na -0.1415 f nh -0.2220 f nn -0.0731 f np -0.0731 f o -0.1077 f o' 0.0888 f op 0.0888 f p -0.3869 f s -0.2380 f s" -0.2011 f sp -0.2011 h h 0.0000 h i 0.1923 h o' 0.4943 h op 0.4943 h p -0.0356 h s 0.1900000 h s" 0.0500000 h sp 0.1932 h n 0.0000000 h n' 0.4

h nn 0.37 h np 0.3278 h nr 0.4068 h o 0.4241000 h o 0.3991 h+ n+ .2800000 h+ nh 0.3700 i i 0.0000 i n 0.1554 i n+ 0.2615 i n 0.1554 i na 0.0714 i nh -0.0356 i nn 0.1554 i np 0.1554 i o 0.0924 i o' 0.3297 i op 0.3297 i p -0.2110 i s -0.0345 i s" 0.0140 i sp 0.0140 n n 0.0000 n n+ 0.0883

n+ s -0.2755 n+ s" -0.2341 n+ sp -0.2341

__________________________________________________________________ Van der Waals interaction energy, for atom pair i and j,

E[ ( */ )2 r r 9  3( */ ) ]r r 6 _where

r* [( *_ r_i 6_r_j* )/ ] ,6 ₂1 6/ _{ }(_{ i j})1 2/ ₂( * *) /( *r r_{i j} 3 r_i 6 _r_j* )6

__________________________________________________________________

atom r* 

______________________________________

nn 3.7000000 0.1700000 np 3.7000000 0.1700000 o 3.5350000 0.2400000 o' 3.5350000 0.2670000 o 3.6080 0.2740 o- 3.5960000 0.1670000 op 3.5350000 0.1090000 p 4.2000000 0.2000000 p- 4.2000000 0.2000000 s 4.0270000 0.0710000 s" 4.0270000 0.0710000 sp 4.0270000 0.0710000 ______________________________________ Bond-bond interaction terms,

EK_bb´(b b b b ₀)( ´ ₀´)

______________________________________

K_bb´

______________________________________

_______________________________

Bond-angle interaction terms, _______________________________

EK b b_b(  ₀)(  ₀)

The first two atoms listed form the bond. _______________________________

K_b

_______________________________

h s" o' -2.6941575 n s" c 34.057933 n s" cp 26.702044 n s" h 64.795392 n s" o' 82.126216 o' s" c 64.225460 o' s" cp 70.282762 o' s" h 70.658014 o' s" n 71.605035 o' s" o' 62.917644 cp sp cp 0.0000000 cp sp cpb 0.0000000 cp sp np 4.9599678 cpb sp cp 0.0000000 np sp cp 86.829685 np sp np 145.75877

_______________________________

Angle-angle interaction terms, EK_´(  ₀)(´₀´)

The first 3 atoms define the first angle. the second 3 atoms the second angle.

_______________________________ K_´

_______________________________ c c c c -0.2010870

c s" cp o' 12.472972 n s" cp o' 31.425422 o' s" cp o' 31.313629 c s" h o' 73.817972 h s" h o' 81.357701 n s" h o' 70.343039 o' s" h o' 94.781438 c s" n o' 43.084921 cp s" n o' 23.902135 h s" n o' 39.712683 o' s" n o' 66.606522 c s" o' c 22.014898 c s" o' cp 12.786932 c s" o' h 26.903645 c s" o' n 21.950105 c s" o' o' 14.372797 cp s" o' n 11.397878 cp s" o' o' 10.909840 h s" o' h 33.246240 h s" o' n 32.458035 h s" o' o' 16.745065 n s" o' o' 10.946279

___________________________________________________

Angle-torsion interaction terms,

E(  ₀)(1K_(₁ cos )  2K_(₁ cos )₂  3K_(₁ cos )₃

The first 3 atoms denote the angle.

___________________________________________________

1_K



2_K



3_K



___________________________________________________

cp np sp cp 0.0000000 -18.067722 0.0000000 cp sp np cp 0.0000000 0.0000000 0.0000000 cp np sp np 0.0000000 -32.154493 0.0000000 np sp np cp 0.0000000 -92.681625 0.0000000 cp sp np np 0.0000000 9.6677462 0.0000000 np np sp cp 0.0000000 0.0000000 0.0000000 c o p o -.7624835 .0000000 -.1917483 o p o c .7025883 1.4881885 .6918788 c o p o- .2447691 .0000000 -.1490948 o- p o c 1.0224856 2.7888059 -.0455052 c o p- o- 3.9202442 2.6763874 -0.7781807 o- p- o c 4.9698831 5.8822768 -1.3708918 cp o p- o- -1.1851771 -0.1080315 -0.7516878 o- p- o cp 0.1471697 2.1542669 -2.5203351 h o p- o- 0.5457203 -0.3040876 -0.2517514 o- p- o h -3.2598969 1.4348270 -0.3098329 c s s c -0.8891659 4.4206322 -0.8057607 c s s h -2.8675863 4.7025788 -0.2965184 h s s c -1.4888296 3.6333475 0.3498797 h s s h -2.2328295 2.7400153 -0.6892629

___________________________________________________

Central bond-torsion interaction terms, E (b b₀)(1K_b(₁ cos ) 2K_b(₁ cos )₂  3K_b(₁ cos )₃

     

___________________________________________________

1_K

b 

2_K

b 

3_K

b 

___________________________________________________

________________________________________________________ Terminal bond-torsion interaction terms,

E(b b´ ₀´)(1K_b´(₁ cos ) 2K_b´(₁ cos )₂  3K_b´(₁ cos )₃

     

The first 2 atoms denote the bond.

________________________________________________________

1_K

b

´ 2Kb´ 3Kb´

cp np np sp 0.0000000 -15.718746 0.0000000 sp np np cp 0.0000000 0.0000000 0.0000000 cpb np np nh 0.0000000 -48.580227 0.0000000 nh np np cpb 0.0000000 0.5298679 0.0000000 nh np np np 0.0000000 -17.936306 0.0000000 np np np nh 0.0000000 -35.999054 0.0000000 np np np np 0.0000000 -20.659569 0.0000000 cp np op cp 0.0000000 -16.867844 0.0000000 cp op np cp 0.0000000 0.0000000 0.0000000 cp np op np 0.0000000 -2.1974554 0.0000000 np op np cp 0.0000000 11.286416 0.0000000 cp op np cpb 0.0000000 0.0000000 0.0000000 cpb np op cp 0.0000000 -16.867844 0.0000000 cp op np np 0.0000000 0.0000000 0.0000000 np np op cp 0.0000000 0.0000000 0.0000000 cp np sp cp 0.0000000 12.554767 0.0000000 cp sp np cp 0.0000000 17.193865 0.0000000 cp np sp np 0.0000000 26.007690 0.0000000 np sp np cp 0.0000000 0.0000000 0.0000000 cp sp np np 0.0000000 0.0000000 0.0000000 np np sp cp 0.0000000 66.445782 0.0000000 c o p o -.4593485 -1.1087339 -.2708256 o p o c -6.0582395 1.6908438 .3746805 c o p o- -3.0190987 -1.3092218 -.0912146 o- p o c -5.7884348 .5788403 -.2691330 c o p- o- -5.3689239 -2.1094737 -3.3588287 o- p- o c -5.2084509 0.3831170 -0.7032106 cp o p- o- -2.8580464 -2.5796830 -3.4045218 o- p- o cp -4.8894572 0.2397342 -3.2676444 h o p- o- 1.2818410 1.3346600 1.4934464 o- p- o h -3.6855186 -0.2406764 -0.0277279 c s s c 0.1928236 0.0700661 -0.1260744 c s s h 0.3317717 1.1555102 -0.0994404 h s s c -0.0733998 0.4863718 0.2161753 h s s h 0.1527093 0.1702318 0.1954227 ______________________________________________

Angle-angle-torsion interaction terms EK_{ ´} (  ₀)( ´ ₀´)cos

______________________________________________ K_{ ´}

______________________________________________

c o p o- -12.755978 c o p- o- -28.006783 cp o p- o- -7.8604649 h o p- o- -2.7406064 c s s c -53.231114 c s s h -37.742894 h s s h -23.899085