4. Resonance in C6H6
4.1. Introduction
Benzene (C6H6) is a classical aromatic molecule in which electrons in \(\pi\) space are delocalized. The covalent valence bond structures are shown in the following table.
Covalent Structures of C6H6
The 5 covalent structures can be classified as 2 types: the first 2 structures are Kekulé structures and the others are regarded as Dewar structures. In this exercise, the resonance energy in the benzene molecule will be computed to show how resonance stabilize the aromatic molecule.
4.2. Computations with 5 Covalent Structures
First, we will compute the energy with all 5 covalent structures.
4.2.1. Input File
The input file is shown below.
C6H6
$ctrl
str=cov nao=6 nae=6 iprint=3
orbtyp=hao frgtyp=sao
basis=cc-pvdz
$end
$frag
12 2*6
spxpydxxdyydzzdxy 1-12
pzdxzdyz 1 2
pzdxzdyz 3 4
pzdxzdyz 5 6
pzdxzdyz 7 8
pzdxzdyz 9 10
pzdxzdyz 11 12
$end
$orb
1*18 1*6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
$end
$geo
C 0.6995000584 1.2115696411 0.0000000000
H 1.2460106991 2.1581538376 0.0000000000
C -0.6995000584 1.2115696411 0.0000000000
H -1.2460106991 2.1581538376 0.0000000000
C -1.3990001169 0.0000000000 0.0000000000
H -2.4920213982 0.0000000000 0.0000000000
C -0.6995000584 -1.2115696411 0.0000000000
H -1.2460106991 -2.1581538376 0.0000000000
C 0.6995000584 -1.2115696411 0.0000000000
H 1.2460106991 -2.1581538376 0.0000000000
C 1.3990001169 0.0000000000 0.0000000000
H 2.4920213982 0.0000000000 0.0000000000
$end
In this input file, the molecule lays in the XY plane. The fragments specified in $FRAG section are defined according to the symmetries. Since we concern on the resonance in \(\pi\)space, electrons locating in \(\sigma\)space is not in the interest. Thus eletrons in the \(\sigma\)orbitals can be treated as inactive ones and make them delocalized in all structures. To this reason, the first fragment is defined to include all basis functions in \(\sigma\) space in the molecule. The remained fragments denote \(\pi\) orbitals locates on each CH group.
4.2.2. Computational Results
5 VB structures will be generated in the computation and can be found in the output:
Number of Structures: 5
The following structures are used in calculation:
1 ***** 1:18 21 22 20 23 19 24
2 ***** 1:18 20 21 22 23 19 24
3 ***** 1:18 20 21 19 22 23 24
4 ***** 1:18 19 20 22 23 21 24
5 ***** 1:18 19 20 21 22 23 24
Final energy can be found after the VBSCF iteration as -230.63450 hartree.
Total Energy: -230.63449533
Both coefficients and weights of VB structures show that structures are in 2 groups. Structure 2 and 5 correspond to the Kekulé structures, which are the dominant ones, and the others denote Dewar structures.
****** COEFFICIENTS OF STRUCTURES ****** 1 0.14537 ****** 1:18 21 22 20 23 19 24 2 -0.40098 ****** 1:18 20 21 22 23 19 24 3 0.14537 ****** 1:18 20 21 19 22 23 24 4 0.14537 ****** 1:18 19 20 22 23 21 24 5 -0.40098 ****** 1:18 19 20 21 22 23 24
****** WEIGHTS OF STRUCTURES ****** 1 0.11245 ****** 1:18 21 22 20 23 19 24 2 0.33132 ****** 1:18 20 21 22 23 19 24 3 0.11245 ****** 1:18 20 21 19 22 23 24 4 0.11245 ****** 1:18 19 20 22 23 21 24 5 0.33132 ****** 1:18 19 20 21 22 23 24
The atomic population analysis shows that the bond orders of all CC bondings are the same as 1.188, showing the effect of conjugation.
****** BOND ORDER ****** ATOM 1 ATOM 2 DIST BOND ORDER 1 C 2 H 1.093 0.979 1 C 3 C 1.399 1.188 3 C 4 H 1.093 0.979 3 C 5 C 1.399 1.188 5 C 6 H 1.093 0.979 5 C 7 C 1.399 1.188 7 C 8 H 1.093 0.979 7 C 9 C 1.399 1.188 9 C 10 H 1.093 0.979 1 C 11 C 1.399 1.188 9 C 11 C 1.399 1.188 11 C 12 H 1.093 0.979
4.3. Computations with 2 Kekulé Structures
4.3.1. Input File
The input file is shown below.
C6H6
$ctrl
nstr=2 nao=6 nae=6 iprint=3
orbtyp=hao frgtyp=sao
basis=cc-pvdz
guess=read
$end
$str
1:18 19-24
1:18 19 24 20 21 22 23
$end
$frag
12 2*6
spxpydxxdyydzzdxy 1-12
pzdxzdyz 1 2
pzdxzdyz 3 4
pzdxzdyz 5 6
pzdxzdyz 7 8
pzdxzdyz 9 10
pzdxzdyz 11 12
$end
$orb
1*18 1*6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
$end
$geo
C 0.6995000584 1.2115696411 0.0000000000
H 1.2460106991 2.1581538376 0.0000000000
C -0.6995000584 1.2115696411 0.0000000000
H -1.2460106991 2.1581538376 0.0000000000
C -1.3990001169 0.0000000000 0.0000000000
H -2.4920213982 0.0000000000 0.0000000000
C -0.6995000584 -1.2115696411 0.0000000000
H -1.2460106991 -2.1581538376 0.0000000000
C 0.6995000584 -1.2115696411 0.0000000000
H 1.2460106991 -2.1581538376 0.0000000000
C 1.3990001169 0.0000000000 0.0000000000
H 2.4920213982 0.0000000000 0.0000000000
$end
$gus
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 5 5 5 5 5 5
# ORBITAL 1 NAO = 90
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# ORBITAL 2 NAO = 90
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0.0024591332 102 -0.0005052562 103 -0.0001393968 104 0.0000000109 105
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# ORBITAL 3 NAO = 90
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0.0000000315 113 0.0000000132 115 -0.0000000150 116 -0.0000000085 117
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# ORBITAL 4 NAO = 90
-0.2891567875 1 -0.0011788380 2 0.0006696401 3 0.0001162136 4
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0.0005581117 70 -0.0000174532 71 0.0007001480 73 0.0007639674 75
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# ORBITAL 5 NAO = 90
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0.0000674501 11 0.0010736729 13 0.0013237329 15 -0.0001590050 16
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0.0000001203 113 0.0000000496 115 -0.0000000565 116 -0.0000000328 117
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# ORBITAL 6 NAO = 90
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0.0007516425 70 -0.0000774650 71 0.0006732027 73 0.0009931240 75
0.0000366093 76 0.0002099034 77 0.0000247197 78 0.0000427155 79
0.4091251465 81 0.0018241721 82 -0.0036592195 83 0.0000021021 84
0.0000012118 85 0.0007742169 87 -0.0013386734 88 -0.0007514874 90
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0.0007910987 113 0.0009920653 115 0.0000386489 116 0.0002101309 117
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# ORBITAL 7 NAO = 90
-0.0107819418 1 0.1790782666 2 0.0924284475 3 -0.0318865151 4
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0.0127397007 113 -0.0063317099 115 0.0456056422 116 0.0036342781 117
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# ORBITAL 8 NAO = 90
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0.0639689094 5 -0.0281707685 7 0.0270146479 8 0.0056247890 10
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0.0143938213 118 0.0000000001 119
# ORBITAL 9 NAO = 90
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-0.0000000003 113 0.0000000001 115 -0.0000000020 116 -0.0000000003 117
0.0000000003 118 -0.0029035175 119
# ORBITAL 10 NAO = 90
-0.0051440589 1 0.1005718952 2 0.0845638529 3 -0.1397337895 4
0.1141834782 5 -0.0457303867 7 0.0234939964 8 0.0117656669 10
0.0056709072 11 -0.0100493795 13 -0.0019026433 15 0.0770087430 16
0.0164532289 17 -0.0072726819 18 -0.0060307741 19 -0.0051440587 21
0.1005718915 22 0.0845638499 23 0.1397337914 24 0.1141834780 25
0.0457303869 27 0.0234939969 28 0.0117656673 30 -0.0056709071 31
-0.0100493799 33 -0.0019026433 35 0.0770087402 36 0.0164532285 37
0.0072726816 38 -0.0060307738 39 0.0102806964 41 -0.2012853772 42
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0.0038040175 55 -0.1542612985 56 -0.0329629875 57 -0.0177440549 58
-0.0000000001 59 -0.0051440589 61 0.1005718952 62 0.0845638530 63
0.1397337890 64 -0.1141834780 65 0.0457303863 67 -0.0234939964 68
0.0117656672 70 0.0056709069 71 -0.0100493798 73 -0.0019026434 75
0.0770087429 76 0.0164532290 77 0.0072726817 78 0.0060307742 79
-0.0051440587 81 0.1005718916 82 0.0845638498 83 -0.1397337917 84
-0.1141834783 85 -0.0457303871 87 -0.0234939968 88 0.0117656669 90
-0.0056709073 91 -0.0100493797 93 -0.0019026432 95 0.0770087402 96
0.0164532283 97 -0.0072726818 98 0.0060307738 99 0.0102806964 101
-0.2012853761 102 -0.1692333681 103 -0.0582639958 104 0.0000000022 105
0.0049551210 107 0.0000000006 108 0.0006914310 110 -0.0000000001 111
-0.0041279019 113 0.0038040170 115 -0.1542612996 116 -0.0329629888 117
0.0177440529 118 0.0000000001 119
# ORBITAL 11 NAO = 90
-0.0088998256 1 0.1742799331 2 0.1465089246 3 0.1142486920 4
-0.0078131644 5 0.0235313039 7 -0.0185720459 8 -0.0034227095 10
-0.0060473270 11 0.0064037031 13 -0.0032944140 15 0.1334935781 16
0.0285120656 17 -0.0060359574 18 -0.0142465448 19 0.0088998257 21
-0.1742799344 22 -0.1465089259 23 0.1142486881 24 0.0078131605 25
0.0235313028 27 0.0185720448 28 0.0034227096 30 -0.0060473293 31
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-0.0060359581 38 0.0142465441 39 -0.0000000001 41 0.0000000021 42
0.0000000018 43 -0.0000000007 44 -0.2057016988 45 0.0000000001 47
-0.0593097208 48 -0.0000000000 50 -0.0158714988 51 0.0000000000 53
-0.0000000001 55 0.0000000016 56 0.0000000003 57 0.0000000002 58
-0.0037832706 59 -0.0088998256 61 0.1742799323 62 0.1465089241 63
-0.1142486911 64 0.0078131629 65 -0.0235313036 67 0.0185720454 68
-0.0034227098 70 -0.0060473291 71 0.0064037028 73 -0.0032944147 75
0.1334935779 76 0.0285120654 77 0.0060359579 78 0.0142465442 79
0.0088998257 81 -0.1742799352 82 -0.1465089263 83 -0.1142486891 84
-0.0078131618 85 -0.0235313029 87 -0.0185720453 88 0.0034227093 90
-0.0060473274 91 -0.0064037028 93 0.0032944141 95 -0.1334935796 96
-0.0285120659 97 0.0060359576 98 -0.0142465449 99 -0.0000000001 101
0.0000000020 102 0.0000000018 103 0.0000000006 104 0.2057016950 105
-0.0000000001 107 0.0593097217 108 0.0000000001 110 -0.0158715011 111
-0.0000000001 113 -0.0000000000 115 0.0000000017 116 0.0000000004 117
-0.0000000002 118 0.0037832703 119
# ORBITAL 12 NAO = 90
-0.0037231920 1 0.0111150627 2 0.0324659518 3 0.0947657341 4
0.1639019892 5 0.0324226748 7 0.0560650241 8 0.0004508783 10
-0.0049970547 11 -0.0045275601 13 0.0010122333 15 0.1490787651 16
0.0353865179 17 -0.0067083649 18 -0.0116265648 19 -0.0037231920 21
0.0111150626 22 0.0324659517 23 -0.0947657340 24 0.1639019892 25
-0.0324226745 27 0.0560650241 28 0.0004508780 30 0.0049970544 31
-0.0045275597 33 0.0010122332 35 0.1490787652 36 0.0353865178 37
0.0067083650 38 -0.0116265647 39 -0.0037179973 41 0.0112563382 42
0.0325136599 43 -0.1892984480 44 -0.0000000001 45 -0.0647516749 47
-0.0000000000 48 -0.0070415813 50 0.0000000001 51 0.0029723742 53
0.0010156547 55 0.1491231283 56 0.0353714222 57 0.0134368381 58
-0.0000000001 59 -0.0037231920 61 0.0111150627 62 0.0324659517 63
-0.0947657339 64 -0.1639019894 65 -0.0324226745 67 -0.0560650241 68
0.0004508780 70 -0.0049970546 71 -0.0045275596 73 0.0010122332 75
0.1490787651 76 0.0353865177 77 0.0067083651 78 0.0116265647 79
-0.0037231920 81 0.0111150627 82 0.0324659518 83 0.0947657342 84
-0.1639019894 85 0.0324226748 87 -0.0560650241 88 0.0004508784 90
0.0049970549 91 -0.0045275602 93 0.0010122333 95 0.1490787649 96
0.0353865178 97 -0.0067083650 98 0.0116265649 99 -0.0037179973 101
0.0112563375 102 0.0325136602 103 0.1892984469 104 -0.0000000001 105
0.0647516758 107 -0.0000000000 108 -0.0070415822 110 -0.0000000002 111
0.0029723750 113 0.0010156554 115 0.1491231290 116 0.0353714232 117
-0.0134368367 118 -0.0000000001 119
# ORBITAL 13 NAO = 90
-0.0077795017 1 0.1114492677 2 0.1676977293 3 0.0636506129 4
0.1102407267 5 0.0070910121 7 0.0122634165 8 -0.0147424044 10
0.0206787210 11 0.0059331824 13 0.0000881382 15 0.1838685648 16
0.0693750950 17 -0.0071295317 18 -0.0123503755 19 0.0077795017 21
-0.1114492677 22 -0.1676977292 23 0.0636506129 24 -0.1102407270 25
0.0070910126 27 -0.0122634173 28 0.0147424042 30 0.0206787185 31
-0.0059331821 33 -0.0000881383 35 -0.1838685644 36 -0.0693750951 37
-0.0071295320 38 0.0123503750 39 -0.0077800331 41 0.1114379546 42
0.1677056671 43 -0.1272989969 44 0.0000000000 45 -0.0141745507 47
-0.0000000000 48 0.0162731269 50 0.0000000000 51 -0.0250837618 53
0.0000883276 55 0.1838681097 56 0.0693765065 57 0.0142627162 58
0.0000000000 59 0.0077795017 61 -0.1114492678 62 -0.1676977292 63
0.0636506129 64 0.1102407272 65 0.0070910126 67 0.0122634172 68
0.0147424042 70 -0.0206787184 71 -0.0059331821 73 -0.0000881382 75
-0.1838685642 76 -0.0693750950 77 -0.0071295322 78 -0.0123503751 79
-0.0077795017 81 0.1114492677 82 0.1676977293 83 0.0636506130 84
-0.1102407268 85 0.0070910120 87 -0.0122634164 88 -0.0147424044 90
-0.0206787211 91 0.0059331824 93 0.0000881382 95 0.1838685647 96
0.0693750949 97 -0.0071295317 98 0.0123503756 99 0.0077800331 101
-0.1114379543 102 -0.1677056673 103 -0.1272989948 104 0.0000000001 105
-0.0141745525 107 0.0000000000 108 -0.0162731262 110 0.0000000001 111
0.0250837603 113 -0.0000883285 115 -0.1838681106 116 -0.0693765078 117
0.0142627144 118 0.0000000001 119
# ORBITAL 14 NAO = 90
0.0000011888 1 -0.0000428122 2 -0.0000193406 3 -0.2396559405 4
0.1383212588 5 -0.0833040860 7 0.0480780537 8 0.0081916407 10
0.0054632598 11 -0.0081940522 13 0.0000021430 15 -0.0000568495 16
-0.0000165189 17 -0.0045591993 18 0.0026388067 19 0.0000011888 21
-0.0000428110 22 -0.0000193401 23 0.2396559394 24 0.1383212571 25
0.0833040855 27 0.0480780535 28 0.0081916411 30 -0.0054632601 31
-0.0081940519 33 0.0000021439 35 -0.0000568498 36 -0.0000165188 37
0.0045591989 38 0.0026388061 39 -0.0000000000 41 -0.0000000000 42
-0.0000000000 43 -0.0000000000 44 -0.2767352102 45 -0.0000000000 47
-0.0961986294 48 -0.0000000001 50 -0.0109202542 51 0.0000000000 53
-0.0000000001 55 -0.0000000000 56 -0.0000000000 57 -0.0000000000 58
-0.0052745169 59 -0.0000011888 61 0.0000428110 62 0.0000193401 63
-0.2396559396 64 0.1383212572 65 -0.0833040854 67 0.0480780534 68
-0.0081916410 70 -0.0054632601 71 0.0081940518 73 -0.0000021439 75
0.0000568498 76 0.0000165188 77 -0.0045591989 78 0.0026388060 79
-0.0000011888 81 0.0000428123 82 0.0000193406 83 0.2396559404 84
0.1383212588 85 0.0833040860 87 0.0480780537 88 -0.0081916407 90
0.0054632598 91 0.0081940523 93 -0.0000021429 95 0.0000568495 96
0.0000165189 97 0.0045591993 98 0.0026388067 99 -0.0000000000 101
0.0000000001 102 -0.0000000000 103 -0.0000000000 104 -0.2767352058 105
-0.0000000000 107 -0.0961986304 108 -0.0000000000 110 0.0109202573 111
0.0000000000 113 -0.0000000001 115 -0.0000000000 116 -0.0000000000 117
0.0000000000 118 -0.0052745165 119
# ORBITAL 15 NAO = 90
-0.0006266032 1 -0.0382787571 2 -0.0483997291 3 0.1633248653 4
0.1961580581 5 0.0509526263 7 0.0877847417 8 -0.0121775015 10
0.0001460825 11 0.0084420611 13 0.0009659969 15 0.1928592179 16
0.0682975466 17 -0.0044975288 18 -0.0126684488 19 -0.0006266032 21
-0.0382787568 22 -0.0483997291 23 -0.1633248656 24 0.1961580581 25
-0.0509526265 27 0.0877847418 28 -0.0121775014 30 -0.0001460827 31
0.0084420612 33 0.0009659972 35 0.1928592182 36 0.0682975468 37
0.0044975286 38 -0.0126684491 39 -0.0000000000 41 -0.0000000001 42
-0.0000000001 43 -0.0000000005 44 -0.0865394535 45 -0.0000000003 47
-0.0003977750 48 -0.0000000001 50 -0.0204636857 51 0.0000000000 53
-0.0000000000 55 0.0000000003 56 0.0000000001 57 0.0000000000 58
-0.0048836154 59 0.0006266032 61 0.0382787567 62 0.0483997289 63
0.1633248656 64 0.1961580577 65 0.0509526263 67 0.0877847414 68
0.0121775015 70 -0.0001460828 71 -0.0084420614 73 -0.0009659972 75
-0.1928592177 76 -0.0682975466 77 -0.0044975287 78 -0.0126684492 79
0.0006266032 81 0.0382787572 82 0.0483997292 83 -0.1633248655 84
0.1961580588 85 -0.0509526262 87 0.0877847417 88 0.0121775015 90
0.0001460827 91 -0.0084420611 93 -0.0009659969 95 -0.1928592181 96
-0.0682975467 97 0.0044975289 98 -0.0126684490 99 0.0000000000 101
0.0000000001 102 0.0000000001 103 -0.0000000005 104 -0.0865394521 105
-0.0000000001 107 -0.0003977754 108 -0.0000000001 110 0.0204636868 111
0.0000000001 113 -0.0000000000 115 -0.0000000004 116 -0.0000000001 117
0.0000000000 118 -0.0048836154 119
# ORBITAL 16 NAO = 90
-0.0003586348 1 -0.0220988824 2 -0.0279910890 3 0.0076641824 4
0.1632676821 5 0.0290045303 7 0.0509421160 8 0.0106923614 10
0.0119090275 11 -0.0128503064 13 0.0005575567 15 0.1113324509 16
0.0394219179 17 -0.0074820090 18 -0.0044959757 19 0.0003586348 21
0.0220988823 22 0.0279910887 23 0.0076641819 24 -0.1632676811 25
0.0290045293 27 -0.0509421154 28 -0.0106923611 30 0.0119090285 31
0.0128503057 33 -0.0005575565 35 -0.1113324506 36 -0.0394219177 37
-0.0074820091 38 0.0044959755 39 0.0007213416 41 0.0441838214 42
0.0558579175 43 0.2904548744 44 -0.0000000001 45 0.1171902403 47
0.0000000000 48 -0.0039203318 50 -0.0000000002 51 0.0082203577 53
-0.0011142396 55 -0.2227104062 56 -0.0788656304 57 -0.0152767920 58
0.0000000001 59 0.0003586348 61 0.0220988824 62 0.0279910889 63
0.0076641824 64 0.1632676819 65 0.0290045295 67 0.0509421156 68
-0.0106923610 70 -0.0119090285 71 0.0128503057 73 -0.0005575565 75
-0.1113324512 76 -0.0394219179 77 -0.0074820092 78 -0.0044959756 79
-0.0003586348 81 -0.0220988822 82 -0.0279910888 83 0.0076641819 84
-0.1632676816 85 0.0290045301 87 -0.0509421157 88 0.0106923615 90
-0.0119090275 91 -0.0128503065 93 0.0005575567 95 0.1113324502 96
0.0394219176 97 -0.0074820090 98 0.0044959758 99 -0.0007213416 101
-0.0441838220 102 -0.0558579171 103 0.2904548721 104 -0.0000000001 105
0.1171902421 107 -0.0000000000 108 0.0039203305 110 -0.0000000001 111
-0.0082203568 113 0.0011142407 115 0.2227104073 116 0.0788656318 117
-0.0152767901 118 -0.0000000001 119
# ORBITAL 17 NAO = 90
-0.0012911822 1 -0.0067578701 2 0.0010641979 3 0.2465963959 4
-0.0149150249 5 0.1010029777 7 -0.0041486050 8 -0.0213705643 10
0.0047272710 11 0.0206865037 13 0.0003120959 15 0.1092296435 16
0.0566451247 17 0.0032481770 18 -0.0081643699 19 -0.0012911822 21
-0.0067578703 22 0.0010641980 23 -0.2465963959 24 -0.0149150244 25
-0.1010029782 27 -0.0041486050 28 -0.0213705640 30 -0.0047272709 31
0.0206865033 33 0.0003120959 35 0.1092296434 36 0.0566451249 37
-0.0032481772 38 -0.0081643701 39 0.0025873290 41 0.0136123011 42
-0.0020704704 43 0.2206386849 44 0.0000000000 45 0.0937752162 47
0.0000000000 48 -0.0274275908 50 -0.0000000002 51 0.0288047866 53
-0.0006225099 55 -0.2183274679 56 -0.1132505770 57 -0.0108749007 58
0.0000000001 59 -0.0012911822 61 -0.0067578702 62 0.0010641980 63
-0.2465963960 64 0.0149150247 65 -0.1010029780 67 0.0041486050 68
-0.0213705641 70 0.0047272710 71 0.0206865034 73 0.0003120959 75
0.1092296434 76 0.0566451248 77 -0.0032481771 78 0.0081643701 79
-0.0012911822 81 -0.0067578700 82 0.0010641980 83 0.2465963959 84
0.0149150247 85 0.1010029776 87 0.0041486049 88 -0.0213705644 90
-0.0047272711 91 0.0206865038 93 0.0003120960 95 0.1092296435 96
0.0566451247 97 0.0032481769 98 0.0081643700 99 0.0025873290 101
0.0136123017 102 -0.0020704709 103 -0.2206386828 104 0.0000000001 105
-0.0937752175 107 0.0000000001 108 -0.0274275902 110 0.0000000002 111
0.0288047861 113 -0.0006225107 115 -0.2183274690 116 -0.1132505784 117
0.0108748991 118 0.0000000001 119
# ORBITAL 18 NAO = 90
-0.0022422568 1 -0.0117861316 2 0.0017961586 3 -0.0149685697 4
0.2293271422 5 -0.0041574297 7 0.0961923012 8 -0.0046743391 10
0.0297488877 11 0.0035011315 13 0.0005353083 15 0.1890734811 16
0.0980836424 17 -0.0081565837 18 -0.0061626988 19 0.0022422568 21
0.0117861306 22 -0.0017961592 23 -0.0149685686 24 -0.2293271406 25
-0.0041574294 27 -0.0961923006 28 0.0046743386 30 0.0297488899 31
-0.0035011317 33 -0.0005353092 35 -0.1890734813 36 -0.0980836425 37
-0.0081565829 38 0.0061626996 39 0.0000000000 41 -0.0000000000 42
0.0000000000 43 0.0000000001 44 0.2552406577 45 0.0000000001 47
0.1034033602 48 0.0000000001 50 0.0215474843 51 -0.0000000000 53
0.0000000001 55 -0.0000000000 56 -0.0000000000 57 0.0000000000 58
0.0079649290 59 -0.0022422568 61 -0.0117861306 62 0.0017961592 63
0.0149685685 64 -0.2293271407 65 0.0041574294 67 -0.0961923005 68
-0.0046743388 70 0.0297488900 71 0.0035011319 73 0.0005353091 75
0.1890734813 76 0.0980836425 77 0.0081565830 78 0.0061626997 79
0.0022422568 81 0.0117861315 82 -0.0017961586 83 0.0149685698 84
0.2293271424 85 0.0041574298 87 0.0961923012 88 0.0046743391 90
0.0297488876 91 -0.0035011315 93 -0.0005353084 95 -0.1890734810 96
-0.0980836423 97 0.0081565838 98 -0.0061626989 99 0.0000000000 101
0.0000000000 102 -0.0000000000 103 -0.0000000000 104 -0.2552406531 105
-0.0000000000 107 -0.1034033615 108 -0.0000000001 110 0.0215474869 111
0.0000000001 113 -0.0000000001 115 -0.0000000001 116 -0.0000000000 117
-0.0000000000 118 -0.0079649287 119
# ORBITAL 19 NAO = 5
-0.6069137242 6 -0.5288219485 9 0.0160778803 12 0.0278520511 14
-0.0182054019 20
# ORBITAL 20 NAO = 5
-0.6069137241 26 -0.5288219486 29 -0.0160778803 32 0.0278520509 34
-0.0182054020 40
# ORBITAL 21 NAO = 5
-0.6069128657 46 -0.5288236221 49 -0.0321586274 52 0.0000000002 54
-0.0182031846 60
# ORBITAL 22 NAO = 5
-0.6069137243 66 -0.5288219484 69 -0.0160778802 72 -0.0278520508 74
-0.0182054020 80
# ORBITAL 23 NAO = 5
-0.6069137243 86 -0.5288219484 89 0.0160778802 92 -0.0278520511 94
-0.0182054019 100
# ORBITAL 24 NAO = 5
-0.6069128633 106 -0.5288236244 109 0.0321586294 112 0.0000000002 114
-0.0182031848 120
$end
In the input file, VB structures are specified by the users rather than automatically generated. The keyword NSTR=2 in $CTRL section specifies 2 VB structures in the computation, and the 2 Kekulé structures are described in section $STR. The keyword GUESS=READ means that the initial guess is obtained by the user in $GUS section. The initial guess comes from the optmized orbitals of previous computation, stored in corresponding ORB file.
4.3.2. Computational Results
The final energy of the computation can be found after VBSCF iterations as -230.62995 hartree. Compared with the energy including 5 covalent structures, the corresponding resonance energy from Dewar structures can be obtained as 2.8 kcal/mol.
****** BOND ORDER ****** ATOM 1 ATOM 2 DIST BOND ORDER 1 C 2 H 1.093 0.979 1 C 3 C 1.399 1.186 3 C 4 H 1.093 0.979 3 C 5 C 1.399 1.186 5 C 6 H 1.093 0.979 5 C 7 C 1.399 1.186 7 C 8 H 1.093 0.979 7 C 9 C 1.399 1.186 9 C 10 H 1.093 0.979 1 C 11 C 1.399 1.186 9 C 11 C 1.399 1.186 11 C 12 H 1.093 0.979
The atomic population analysis also shows that the bonder orders of CC bondings are the same as 1.186. Compared with the bond order from 5 covalent structures and the resonance energy from Dewar structures, the conjugation and stability mainly come from the resonance of Kekulé structures.
4.4. Computations with 1 Kekulé Structure
Finally, the computation including only 1 Kekulé structure, which represents a cycloalkene, will be proceeded to check the resonance from Kekulé structures.
4.4.1. Input File
The input file of this computation is shown below. The initial guess comes from the optimized orbitals of the computation including 2 Kekulé structures.
C6H6
$ctrl
nstr=1 nao=6 nae=6 iprint=3
orbtyp=hao frgtyp=sao
basis=cc-pvdz
guess=read
$end
$str
1:18 19-24
$end
$frag
12 2*6
spxpydxxdyydzzdxy 1-12
pzdxzdyz 1 2
pzdxzdyz 3 4
pzdxzdyz 5 6
pzdxzdyz 7 8
pzdxzdyz 9 10
pzdxzdyz 11 12
$end
$orb
1*18 1*6
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
3
4
5
6
7
$end
$geo
C 0.6995000584 1.2115696411 0.0000000000
H 1.2460106991 2.1581538376 0.0000000000
C -0.6995000584 1.2115696411 0.0000000000
H -1.2460106991 2.1581538376 0.0000000000
C -1.3990001169 0.0000000000 0.0000000000
H -2.4920213982 0.0000000000 0.0000000000
C -0.6995000584 -1.2115696411 0.0000000000
H -1.2460106991 -2.1581538376 0.0000000000
C 0.6995000584 -1.2115696411 0.0000000000
H 1.2460106991 -2.1581538376 0.0000000000
C 1.3990001169 0.0000000000 0.0000000000
H 2.4920213982 0.0000000000 0.0000000000
$end
$gus
90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 90 5 5 5 5 5 5
# ORBITAL 1 NAO = 90
-0.4085796783 1 -0.0018719283 2 0.0001665818 3 0.0001010685 4
0.0001981258 5 0.0000450438 7 0.0000863179 8 0.0009553396 10
0.0000836069 11 0.0010374634 13 0.0011387900 15 -0.0001957720 16
-0.0001340753 17 0.0000469703 18 0.0000827563 19 -0.4085796727 21
-0.0018719292 22 0.0001665812 23 -0.0001010727 24 0.0001981248 25
-0.0000450425 27 0.0000863186 28 0.0009553426 30 -0.0000836019 31
0.0010374618 33 0.0011387887 35 -0.0001957726 36 -0.0001340767 37
-0.0000469715 38 0.0000827567 39 -0.4090835453 41 -0.0018962542 42
0.0001483676 43 -0.0002144425 44 0.0000004419 45 -0.0001064444 47
0.0000000620 48 0.0010795224 50 -0.0000005324 51 0.0009089949 53
0.0011398054 55 -0.0002140828 56 -0.0001424123 57 -0.0000967357 58
0.0000004933 59 -0.4085796762 61 -0.0018717962 62 0.0001665571 63
-0.0001012559 64 -0.0001976491 65 -0.0000451304 67 -0.0000860262 68
0.0009554043 70 0.0000853899 71 0.0010374754 73 0.0011388868 75
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# ORBITAL 2 NAO = 90
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# ORBITAL 3 NAO = 90
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# ORBITAL 4 NAO = 90
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# ORBITAL 5 NAO = 90
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# ORBITAL 6 NAO = 90
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# ORBITAL 7 NAO = 90
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# ORBITAL 8 NAO = 90
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# ORBITAL 9 NAO = 90
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# ORBITAL 10 NAO = 90
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# ORBITAL 11 NAO = 90
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# ORBITAL 12 NAO = 90
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# ORBITAL 13 NAO = 90
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0.0071157446 27 -0.0123065681 28 0.0147668120 30 0.0206625205 31
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0.1678699600 43 -0.1273340648 44 0.0000000002 45 -0.0142192259 47
0.0000000000 48 0.0162548947 50 -0.0000000003 51 -0.0250878035 53
0.0000679682 55 0.1838193396 56 0.0692809136 57 0.0142604356 58
0.0000000003 59 0.0075512263 61 -0.1114726350 62 -0.1678599130 63
0.0636737898 64 0.1102660184 65 0.0071157445 67 0.0123065680 68
0.0147668119 70 -0.0206625199 71 -0.0059162563 73 -0.0000670168 75
-0.1838227412 76 -0.0692803792 77 -0.0071326782 78 -0.0123576825 79
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0.0692803794 97 -0.0071326781 98 0.0123576822 99 0.0075377838 101
-0.1114776371 102 -0.1678699601 103 -0.1273340659 104 0.0000000001 105
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0.0250878041 113 -0.0000679680 115 -0.1838193399 116 -0.0692809135 117
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# ORBITAL 14 NAO = 90
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0.0054692760 11 -0.0082072207 13 -0.0000008338 15 -0.0000306106 16
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0.0832832749 27 0.0480626416 28 0.0082077197 30 -0.0054692758 31
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0.0045627709 38 0.0026348912 39 0.0000000316 41 0.0000000004 42
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-0.0082077198 70 -0.0054692758 71 0.0082072205 73 0.0000008342 75
0.0000306102 76 0.0000082652 77 -0.0045627709 78 0.0026348913 79
0.0000243167 81 -0.0000211335 82 -0.0000197398 83 0.2396956060 84
0.1383354953 85 0.0832832746 87 0.0480626416 88 -0.0082077199 90
0.0054692764 91 0.0082072205 93 0.0000008337 95 0.0000306105 96
0.0000082651 97 0.0045627706 98 0.0026348910 99 0.0000000566 101
0.0000000007 102 -0.0000000000 103 -0.0000000000 104 -0.2767109937 105
0.0000000000 107 -0.0961484638 108 -0.0000000005 110 0.0109288338 111
0.0000000001 113 -0.0000000005 115 0.0000000001 116 -0.0000000001 117
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# ORBITAL 15 NAO = 90
-0.0009983767 1 -0.0386214684 2 -0.0485455904 3 0.1633421841 4
0.1962473237 5 0.0509807373 7 0.0877733678 8 -0.0121832761 10
0.0001162009 11 0.0084147730 13 0.0010223542 15 0.1927411229 16
0.0682210653 17 -0.0045132574 18 -0.0126863282 19 -0.0009983769 21
-0.0386214683 22 -0.0485455906 23 -0.1633421842 24 0.1962473245 25
-0.0509807372 27 0.0877733679 28 -0.0121832762 30 -0.0001162008 31
0.0084147732 33 0.0010223542 35 0.1927411233 36 0.0682210653 37
0.0045132578 38 -0.0126863281 39 0.0000000963 41 0.0000000005 42
0.0000000001 43 -0.0000000002 44 -0.0868168628 45 -0.0000000004 47
-0.0005743185 48 -0.0000000005 50 -0.0204836282 51 0.0000000001 53
-0.0000000004 55 0.0000000002 56 0.0000000002 57 0.0000000001 58
-0.0048842722 59 0.0009985499 61 0.0386214692 62 0.0485455904 63
0.1633421850 64 0.1962473253 65 0.0509807366 67 0.0877733671 68
0.0121832762 70 -0.0001162013 71 -0.0084147740 73 -0.0010223547 75
-0.1927411227 76 -0.0682210647 77 -0.0045132580 78 -0.0126863286 79
0.0009985050 81 0.0386214693 82 0.0485455908 83 -0.1633421847 84
0.1962473254 85 -0.0509807370 87 0.0877733675 88 0.0121832760 90
0.0001162015 91 -0.0084147736 93 -0.0010223547 95 -0.1927411232 96
-0.0682210651 97 0.0045132576 98 -0.0126863286 99 0.0000001212 101
0.0000000009 102 -0.0000000000 103 -0.0000000009 104 -0.0868168632 105
0.0000000001 107 -0.0005743183 108 -0.0000000006 110 0.0204836287 111
0.0000000000 113 -0.0000000004 115 -0.0000000005 116 -0.0000000001 117
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# ORBITAL 16 NAO = 90
-0.0005894251 1 -0.0223088039 2 -0.0280654665 3 0.0075643177 4
0.1633898810 5 0.0288954659 7 0.0510111520 8 0.0106960596 10
0.0118868073 11 -0.0128757602 13 0.0005926770 15 0.1112764638 16
0.0393823208 17 -0.0074782289 18 -0.0045126110 19 0.0005894222 21
0.0223088033 22 0.0280654662 23 0.0075643166 24 -0.1633898807 25
0.0288954662 27 -0.0510111513 28 -0.0106960596 30 0.0118868066 31
0.0128757604 33 -0.0005926768 35 -0.1112764634 36 -0.0393823205 37
-0.0074782291 38 0.0045126111 39 0.0011830149 41 0.0446313651 42
0.0560606696 43 0.2905583224 44 0.0000000000 45 0.1172071367 47
0.0000000000 48 -0.0038785631 50 -0.0000000003 51 0.0082267364 53
-0.0011834653 55 -0.2225345572 56 -0.0787777804 57 -0.0152898732 58
0.0000000001 59 0.0005894244 61 0.0223088033 62 0.0280654664 63
0.0075643175 64 0.1633898819 65 0.0288954663 67 0.0510111514 68
-0.0106960595 70 -0.0118868065 71 0.0128757604 73 -0.0005926768 75
-0.1112764638 76 -0.0393823205 77 -0.0074782294 78 -0.0045126114 79
-0.0005894255 81 -0.0223088036 82 -0.0280654663 83 0.0075643174 84
-0.1633898808 85 0.0288954658 87 -0.0510111516 88 0.0106960598 90
-0.0118868073 91 -0.0128757604 93 0.0005926770 95 0.1112764630 96
0.0393823204 97 -0.0074782290 98 0.0045126113 99 -0.0011830119 101
-0.0446313639 102 -0.0560606692 103 0.2905583232 104 -0.0000000002 105
0.1172071362 107 -0.0000000001 108 0.0038785629 110 -0.0000000005 111
-0.0082267365 113 0.0011834651 115 0.2225345571 116 0.0787777798 117
-0.0152898742 118 -0.0000000002 119
# ORBITAL 17 NAO = 90
-0.0014020326 1 -0.0068956525 2 0.0009658087 3 0.2467463560 4
-0.0150840636 5 0.1010428983 7 -0.0041186505 8 -0.0213912436 10
0.0047152733 11 0.0207049577 13 0.0003249521 15 0.1090368977 16
0.0565120453 17 0.0032595528 18 -0.0081550919 19 -0.0014020678 21
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-0.1010428983 27 -0.0041186508 28 -0.0213912436 30 -0.0047152734 31
0.0207049580 33 0.0003249521 35 0.1090368981 36 0.0565120453 37
-0.0032595527 38 -0.0081550918 39 0.0028093586 41 0.0137482733 42
-0.0019741520 43 0.2207188322 44 -0.0000000000 45 0.0939446698 47
0.0000000001 48 -0.0274316281 50 -0.0000000006 51 0.0288061575 53
-0.0006463540 55 -0.2181730829 56 -0.1130340303 57 -0.0108702108 58
0.0000000003 59 -0.0014020302 61 -0.0068956520 62 0.0009658091 63
-0.2467463568 64 0.0150840636 65 -0.1010428979 67 0.0041186509 68
-0.0213912440 70 0.0047152734 71 0.0207049583 73 0.0003249522 75
0.1090368979 76 0.0565120450 77 -0.0032595523 78 0.0081550920 79
-0.0014020296 81 -0.0068956524 82 0.0009658087 83 0.2467463565 84
0.0150840633 85 0.1010428980 87 0.0041186504 88 -0.0213912437 90
-0.0047152733 91 0.0207049580 93 0.0003249522 95 0.1090368978 96
0.0565120452 97 0.0032595526 98 0.0081550920 99 0.0028092657 101
0.0137482731 102 -0.0019741520 103 -0.2207188329 104 0.0000000001 105
-0.0939446697 107 0.0000000002 108 -0.0274316280 110 0.0000000006 111
0.0288061582 113 -0.0006463534 115 -0.2181730832 116 -0.1130340301 117
0.0108702120 118 0.0000000003 119
# ORBITAL 18 NAO = 90
-0.0023749933 1 -0.0118884368 2 0.0017334005 3 -0.0150505701 4
0.2293750064 5 -0.0041065876 7 0.0962994757 8 -0.0046652132 10
0.0297409795 11 0.0034826782 13 0.0005612091 15 0.1889435721 16
0.0978899218 17 -0.0081612949 18 -0.0061659303 19 0.0023750060 21
0.0118884365 22 -0.0017334006 23 -0.0150505706 24 -0.2293750071 25
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-0.0081612954 38 0.0061659300 39 0.0000000032 41 0.0000000001 42
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0.1034266384 48 0.0000000003 50 0.0215760829 51 -0.0000000002 53
0.0000000002 55 -0.0000000000 56 -0.0000000001 57 -0.0000000001 58
0.0079574230 59 -0.0023750295 61 -0.0118884363 62 0.0017334008 63
0.0150505703 64 -0.2293750078 65 0.0041065877 67 -0.0962994753 68
-0.0046652137 70 0.0297409788 71 0.0034826785 73 0.0005612087 75
0.1889435718 76 0.0978899212 77 0.0081612957 78 0.0061659304 79
0.0023750118 81 0.0118884367 82 -0.0017334007 83 0.0150505701 84
0.2293750071 85 0.0041065877 87 0.0962994753 88 0.0046652134 90
0.0297409792 91 -0.0034826785 93 -0.0005612092 95 -0.1889435718 96
-0.0978899216 97 0.0081612954 98 -0.0061659305 99 0.0000000018 101
0.0000000003 102 0.0000000001 103 0.0000000001 104 -0.2554222907 105
0.0000000000 107 -0.1034266382 108 -0.0000000003 110 0.0215760822 111
0.0000000001 113 -0.0000000004 115 0.0000000000 116 -0.0000000001 117
0.0000000000 118 -0.0079574229 119
# ORBITAL 19 NAO = 5
-0.6051361517 6 -0.5306934722 9 0.0161429500 12 0.0279606935 14
-0.0181292011 20
# ORBITAL 20 NAO = 5
-0.6051361519 26 -0.5306934721 29 -0.0161429499 32 0.0279606936 34
-0.0181292011 40
# ORBITAL 21 NAO = 5
-0.6051369321 46 -0.5306928695 49 -0.0322859902 52 0.0000000004 54
-0.0181286009 60
# ORBITAL 22 NAO = 5
-0.6051361524 66 -0.5306934715 69 -0.0161429500 72 -0.0279606933 74
-0.0181292011 80
# ORBITAL 23 NAO = 5
-0.6051361522 86 -0.5306934717 89 0.0161429499 92 -0.0279606934 94
-0.0181292011 100
# ORBITAL 24 NAO = 5
-0.6051369320 106 -0.5306928697 109 0.0322859897 112 0.0000000007 114
-0.0181286008 120
$end
4.4.2. Computational Results
The final energy can be found as -230.59097 hartree, and the resonance Kekulé structures is 24.5 kcal/mol, which is the stablization energy to delocalize the \(\pi\) orbitals in a cycloalkene.
****** BOND ORDER ****** ATOM 1 ATOM 2 DIST BOND ORDER 1 C 2 H 1.093 0.980 1 C 3 C 1.399 1.425 3 C 4 H 1.093 0.980 3 C 5 C 1.399 1.011 5 C 6 H 1.093 0.980 5 C 7 C 1.399 1.425 7 C 8 H 1.093 0.980 7 C 9 C 1.399 1.012 9 C 10 H 1.093 0.980 1 C 11 C 1.399 1.011 9 C 11 C 1.399 1.425 11 C 12 H 1.093 0.980
The bond orders show that the CC bondings are not equivalent now, there are 3 CC bondings with bond order 1.425, close to double bonding, and 3 bondings with 1.011, close to single bonding. The computational results suggest that the resonance from Kekulé structrues mainly stabilizes the molecule and makes all CC bondings equivalent. Both resonance from Kekulé structures and equalization of CC bonds thus present the aromaticity in C6H6.