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Table 7

Fitted parameters of the RAM Hamiltonian for 13C acetaldehyde isotopologs.

Operator1 n tr 2 Parameter 12CH312CHO 13CH312CHO 12CH313CHO
value3 (cm-1) value3 (cm-1) value3 (cm-1)
pα2 220 F 7.56708175(35) 7.5652791(19) 7.5068034(55)
(1/2) (1−cos(3α)) 220 V 3 407.59768(20) 407.91172(18) 407.58042(26)
pαPa 211 ρ 0.328632673(20) 0.328300981(35) 0.322541154(65)
Pa2 202 A RAM 1.884881309(61) 1.88013742(13) 1.848976496(63)
Pb2 202 B RAM 0.348706489(16) 0.337726802(58) 0.348720666(61)
Pc2 202 C RAM 0.3031840478(44) 0.294817724(11) 0.302179145(18)
{Pa,Pb} 202 D ab −0.122669037(94) −0.11817144(38) −0.12175149(39)
(1/2) (1−cos(6α)) 440 V 6 −11.63964(50) -11.67172(20) -11.71571(65)
pα4 440 F m −0.424416(38) × 10-3 −0.427829(96) × 10-3 −0.42834(12) × 10-3
pα3Pa 431 ρ m −0.832163(52) × 10-3 −0.82821(11) × 10-3 −0.81162(16) × 10-3
P2(1−cos(3α)) 422 V 3 J 0.5577894(41) × 10-3 0.524776(16) × 10-3 0.529795(22) × 10-3
Pa2(1−cos(3α)) 422 V 3 K −0.19313906(98) × 10-1 −0.1898291(17) × 10-1 −0.1953217(29) × 10-1
(Pb2− Pc2)(1−cos(3α)) 422 V 3 bc 0.2114597(52) × 10-3 0.192215(19) × 10-3 0.213208(15) × 10-3
(1/2){Pa,Pb}(1−cos(3α)) 422 V 3 ab 0.4235363(70) × 10-2 0.405222(33) × 10-2 0.443922(36) × 10-2
pα2P2 422 F J −0.291704(53) × 10-5 −0.26533(11) × 10-5 −0.31536(19) × 10-5
pα2Pa2 422 F K −0.982545(30) × 10-3 −0.972792(46) × 10-3 −0.944979(89) × 10-3
pα2(Pb2− Pc2) 422 F bc 0.14720(12) × 10-5 0.13745(31) × 10-5 0.16632(37) × 10-5
(1/2){Pa,Pc}sin(3α) 422 D 3 ac −0.1405503(49) × 10-1 −0.135881(17) × 10-1 −0.142957(18) × 10-1
(1/2){Pb,Pc}sin(3α) 422 D 3 bc 0.59173(13) × 10-3 0.55489(43) × 10-3 0.63859(48) × 10-3
pαPaP2 413 ρ J 0.958253(42) × 10-5 0.91984(11) × 10-5 0.91075(15) × 10-5
pαPa3 413 ρ K −0.571407(12) × 10-3 −0.564681(12) × 10-3 −0.549548(31) × 10-3
(1/2){Pa,(Pb2− Pc2)}pα 413 ρ bc −0.5536(12) × 10-6 −0.5763(36) × 10-6 −0.2887(35) × 10-6
− P4 404 Δ J 0.3244101(54) × 10-6 0.306742(20) × 10-6 0.325255(24) × 10-6
− P2Pa2 404 Δ JK −0.581770(50) × 10-5 −0.556079(86) × 10-5 −0.571647(79) × 10-5
− Pa4 404 Δ K 0.1308173(26) × 10-3 0.1292582(29) × 10-3 0.1263888(56) × 10-3
−2P2(Pb2− Pc2) 404 δ J 0.778472(23) × 10-7 0.709225(70) × 10-7 0.786517(69) × 10-7
−{Pa2,(Pb2− Pc2)} 404 δ K 0.94268 (17) × 10-6 0.105094(90) × 10−5 0.10255(12) × 10−5
{Pa,Pb}P2 404 D abJ 0.867443(32) × 10-6 0.81273(20) × 10-6 0.85179(14) × 10-6
{Pa3,Pb} 404 D abK 0.30498(25) × 10-5 0.29239(46) × 10-5 0.27116(33) × 10-5
(1/2) (1−cos(9α)) 660 V 9 −0.20318(11) −0.14153(11) −0.2413(14)
pα6 660 F mm −0.5066(16) × 10-6 −0.8986(20) × 10-6 −0.7480(55) × 10-6
pα5Pa 651 ρ mm −0.7147(34) × 10-6 −0.14955(33) × 10−5 −0.1138(11) × 10−5
P2(1−cos(6α)) 642 V 6 J 0.557997(73) × 10-4 0.54132(35) × 10-4 0.58065(59) × 10-4
Pa2(1−cos(6α)) 642 V 6 K −0.37191(19) × 10-3 −0.37805(54) × 10-3 −0.33899(20) × 10-3
(1/2){Pa,Pb}(1−cos(6α)) 642 V 6 ab −0.13638(10) × 10-3 −0.12574(40) × 10-3 −0.12853(37) × 10-3
(Pb2− Pc2)(1−cos(6α)) 642 V 6 bc − 0.1222(48) × 10−5 −0.829(29) × 10-6
pα4P2 642 F mJ −0.1259(18) × 10-9 − −0.4837(88) × 10−8
pα4Pa2 642 F mK −0.1712(36) × 10-6 −0.8131(17) × 10-6 −0.5599(92) × 10-6
pα4(Pb2− Pc2) 642 F mbc 0.1197(22) × 10-8 − −
(1/2){Pb,Pc}sin(6α) 642 D 6 bc −0.72949(57) × 10-4 −0.6730(24) × 10-4 −0.7137(16) × 10-4
(1/2){Pa,Pc}sin(6α) 642 D 6 ac 0.46097(67) × 10-3 0.4258(12) × 10-3 0.5301(10) × 10-3
(1/2){Pa,Pc,pα2,sin(3α)} 642 D 3 acm −0.3178(66) × 10-5 − −
pα3Pa3 633 ρ mK 0.2946(23) × 10-6 − 0.754(45) × 10−7
pα3PaP2 633 ρ mJ −0.5306(26) × 10-8 −0.3235(44) × 10-8 −0.968(12) × 10-8
(1/2){Pa,Pb,Pc,pα,sin(3α)} 633 ρ 3 bc 0.47689(54) × 10-5 − −
(1/2){Pa,(Pb2− Pc2)}pα3 633 ρ mbc 0.4347(38) × 10-8 −0.995(35) × 10-9 −
P4(1 − cos(3α)) 624 V 3 JJ −0.77602(42) × 10-8 −0.4499(21) × 10-8 −0.5065(25) × 10-8
P2Pa2(1−cos(3α)) 624 V 3 J K − −0.23620(40) × 10-6 −
Pa4(1−cos(3α)) 624 V 3 KK 0.115646(94) × 10-5 0.11145(17) × 10-5 0.12880(30) × 10-5
pα2P4 624 F J J − 0.2343(39) × 10-10 0.1768(58) × 10-10
pα2P2Pa2 624 F JK −0.6904(18) × 10-8 −0.5620(48) × 10-8 −0.9637(65) × 10-8
pα2Pa4 624 F KK 0.30324(86) × 10-6 0.22248(28) × 10-6 0.2220(15) × 10-6
P2(Pb2− Pc2)(1−cos(3α)) 624 V 3 bcJ −0.17920(30) × 10-8 − −0.2040(11) × 10-8
(1/2){Pa3,Pc}sin(3α) 624 D 3 acK 0.1246(10) × 10-5 0.1141(18) × 10-5 0.171(10) × 10−6
(1/2){Pa,Pc}P2sin(3α) 624 D 3 acJ 0.10821(34) × 10-7 0.7636(98) × 10-7 0.372(14) × 10-7
(1/2){Pb,Pc}P2sin(3α) 624 D 3 bcJ −0.7436(17) × 10-8 −0.6126(43) × 10-8 −0.5837(99) × 10-8
(1/2){Pa2,Pb,Pc}sin(3α) 624 D 3 bcK 0.14846(12) × 10-5 −0.2309(55) × 10-6 −0.713(55) × 10-7
(1/2)({Pa,Pb3} – {Pa,Pb,Pc2})cos(3α) 624 V 3 ab 3 0.38167(73) × 10-7 0.5101(22) × 10-7 0.5078(27) × 10-7
P2Pa2(1−cos(3α)) 624 V 3 JK −0.23880(36) × 10-6 − −0.28455(63) × 10-6
(1/2){Pa,Pb}P2(1−cos(3α)) 624 V 3ab J − −0.566(26) × 10−8 −
(1/2){Pa3,Pb}(1−cos(3α)) 624 V 3abK 0.1562(36) × 10-6 0.2057(47) × 10-6 0.4345(43) × 10-6
P2(Pb2− Pc2)(1−cos(3α)) 624 V 3 bcJ − −0.1586(11) × 10−8 −
(1/2){Pa2,(Pb2− Pc2)}(1−cos(3α)) 624 V 3 bc K − 0.28389(89) × 10-6 0.2838(15) × 10-6
(1/2){Pa,Pc3}sin(3α) 624 D 3ac3 −0.3543(24) × 10-7 −0.3397(69) × 10-7 −
(1/2)({Pb3,Pc} − {Pb,Pc3})sin(3α) 624 D 3 bc 3 − −0.2447(29) × 10-8 −0.2554(33) × 10-8
pα2(Pb2− Pc2)P2 624 F bc J − − 0.1505(72) × 10-10
(1/2){Pa2,(Pb2− Pc2)}pα2 624 F bcK 0.7853(30) × 10-8 − −
pα2(Pc4+Pb4) 624 F b 2 c 2 0.49528(94) × 10-10 − −
(Pb4+Pc4)cos3a 624 V 3 b 2 c 2 −0.32294(69) × 10-8 0.770(22) × 10−9 0.684(31) × 10−9
pαPaP4 615 ρ J J − − −0.2008(54) × 10-10
pαPa3P2 615 ρ JK −0.40507(54) × 10-8 −0.3627(21) × 10-8 −0.4479(25) × 10-8
pαPa5 615 ρ KK 0.12126(20) × 10-6 0.10832(11) × 10-6 0.10436(34) × 10-6
(1/2){Pa,(Pb2− Pc2)}P2pα 615 ρ bcJ −0.15285(93) × 10-10 − −
(1/2){Pa3,(Pb2− Pc2)}pα 615 ρ bcK 0.6381(14) × 10-8 0.1989(33) × 10-8 0.1085(16) × 10-8
{Pa,Pb}P4 606 D abJJ −0.16905(91) × 10-11 −0.1419(35) × 10-11 −
{Pa3,Pb}P2 606 D abJK 0.8607(65) × 10-10 0.478(19) × 10-10 0.3802(27) × 10−9
{Pa5,Pb} 606 D abKK 0.11154(82) × 10-8 0.888(24) × 10−9 0.4607(23) × 10-8
P6 606 Φ J 0.80483(59) × 10-12 0.6874(26) × 10-12 0.7043(10) × 10-12
P4Pa2 606 Φ JK −0.2732(10) × 10-10 −0.1630(29) × 10-10 −0.4677(31) × 10-10
P2Pa4 606 Φ K J −0.14266(25) × 10-8 −0.12937(55) × 10-8 −0.17957(56) × 10-8
Pa6 606 Φ K 0.19759(23) × 10-7 0.18596(18) × 10-7 0.18908(47) × 10-7
2P4(Pb2− Pc2) 606 φ J 0.38476(25) × 10-12 0.3246(12) × 10-12 0.34399(69) × 10-12
P2{Pa2,(Pb2− Pc2)} 606 φ JK −0.9470(55) × 10-11 −0.653(13) × 10-11 −0.1633(14) × 10-10
{Pa4,(Pb2− Pc2)} 606 φ K 0.54529(21) × 10-9 0.3514(59) × 10-9 −
(1/2) (1−cos(12α)) 880 V 12 0.10123(15) − 0.1718(18)
pα8 880 F mmm −0.8628(32) × 10-8 − −
pα7Pa 871 ρ mmm −0.20698(86) × 10-7 − −
P2(1−cos(9α)) 862 V 9 J −0.15703(56) × 10-5 −0.1211(24) × 10-5 −0.3590(49) × 10-5
Pa2(1 – cos(9α)) 862 V 9 K 0.3044(14) × 10-4 0.4013(45) × 10-4 −
(Pb2− Pc2)(1−cos(9α)) 862 V 9 bc −0.11444(53) × 10-5 −0.1967(52) × 10-5 −
(1/2){Pa,Pb}(1−cos(9α)) 862 V 9 ab − −0.608(35) × 10-5 −0.599(34) × 10-5
(1/2){Pa,Pc}sin(9α) 862 D 9 ac − 0.1409(39) × 10−4 −
(1/2){Pb,Pc}sin(9α) 862 D 9 bc 0.884(13) × 10-6 − −0.343(10) × 10-5
Pa2pα6 862 F mmK −0.2202(11) × 10-7 −0.1406(33) × 10−9 −
(1/2){Pa,Pb}pα6 862 F mmab 0.2906(29) × 10-10 − −
Pa3pα5 853 ρ mmK −0.13762(98) × 10-7 − −
pα4Pa4 844 F mKK −0.5734(57) × 10-8 − −
(1/2)P2{Pa,Pc}sin(6α) 844 D 6 acJ −0.6552(93) × 10-8 − −
(1/2){Pa3,Pc}sin(6α) 844 D 6 acK − − 0.425(37) × 10-7
(1/2){Pb,Pc}P2sin(6α) 844 D 6 bcJ 0.6210(71) × 10-9 0.748(21) × 10-9 −
(1/2)({Pb3,Pc} − {Pb,Pc3})sin(6α) 844 D 6bc 3 − 0.5148(97) × 10-9 0.996(12) × 10-9
P4(1−cos(6α)) 844 V 6 JJ −0.799(16) × 10-10 − −
Pa4(1−cos(6α)) 844 V 6 KK 0.15152(47) × 10-6 0.1380(11) × 10-6 0.1679(26) × 10-6
(1/2){Pa,Pb}P2(1 − cos(6α)) 844 V 6 ab J − −0.554(14) × 10-8 −
(1/2){Pa3,Pb}(1 − cos(6α)) 844 V 6 abK −0.6966(58) × 10-7 −0.542(21) × 10-7 −
P2(Pb2− Pc2)(1−cos(6α)) 844 V 6 bcJ − − −0.2303(65) × 10-9
(1/2)({Pa,Pb3} – {Pa,Pb,Pc2})cos(6α) 844 V 6 ab 3 − −0.430(16) × 10-8 −
pα3PaP4 835 ρ m JJ − −0.873(21) × 10-13 −
pα3Pa5 835 ρ mKK −0.1656(23) × 10-8 − −
(1/2){Pa,Pb3,Pc,pα,sin(3α)} − (1/2){Pa,Pb,Pc3,pα,sin(3α)} 835 ρ 3 bc 3 −0.2511(23) × 10-10 − −
P6(1 − cos(3α)) 826 V 3 JJJ 0.1344(25) × 10-13 − −
P4Pa2(1−cos(3α)) 826 V 3 JJK 0.3715(39) × 10-11 − −
P2Pa4(1−cos(3α)) 826 V 3 JKK 0.472(13) × 10-10 0.303(11) × 10-10 −
(1/2)P2{Pa3,Pb}(1−cos(3α)) 826 V 3 ab J K − 0.3426(55) × 10-10 −
(1/2){Pa5,Pb}(1−cos(3α)) 826 V 3 abKK −0.2031(61) × 10-9 − −
(1/2)P2{Pa3,Pc}sin(3α) 826 D 3 acJK −0.5470(82) × 10-10 − −
(1/2){Pa5,Pc}sin(3α) 826 D 3 ac K K − 0.450(27) × 10-10 −
(1/2)P2{Pa2,Pb,Pc}sin(3α) 826 D 3 bcJK 0.854(18) × 10-11 − −
(1/2){Pa4,Pb,Pc}sin(3α) 826 D 3 bc K K − − −0.1697(63) × 10-9
(1/2){Pa3,Pc3}sin(3α) 826 D 3 ac3K 0.6486(74) × 10-10 − −
pα2Pa6 826 F KKK −0.3325(57) × 10-9 − −
pαPa3P4 817 ρ J J K − − 0.842(43) × 10-13
pαPa7 817 ρ KKK −0.4300(86) × 10-10 − −0.353(15) × 10-11
P8 808 L J −0.3808(46) × 10-17 −0.358(24) × 10-17 −
P4Pa4 808 L JK 0.5000(60) × 10-14 − 0.372(14) × 10-13
Pa8 808 L K −0.2743(71) × 10-11 − −0.1256(48) × 10-11
2P6(Pb2− Pc2) 808 l J −0.1626(22) × 10-17 −0.1332(63) × 10-17 −0.1426(83) × 10-17
pα8Pa2 1082 F mmm K − 0.281(16) × 10-12 −
(1/2)P2{Pa,Pb}(1−cos(9α)) 1064 V 9 abJ −0.1960(28) × 10-8 − −
P4Pa2(1−cos(6α)) 1046 V 6 JJK − − 0.3164(60) × 10-11
P2Pa4(1−cos(6α)) 1046 V 6 JKK − − −0.222(11) × 10-10
(1/2){Pa5,Pb}(1−cos(6α)) 1046 V 6 abKK −0.585(13) × 10-10 − −
pα2Pa8 1028 F KKKK 0.1427(47) × 10-14 − −

Notes.

(1)

{ A,B } = AB + BA; { A,B,C } = ABC + CBA; { A,B,C,D } = ABCD + DCBA. The product of the operator in the first column of a given row and the parameter in the third column of that row gives the term actually used in the torsion-rotation Hamiltonian of the program, except for F, ρ, and ARAM, which occur in the Hamiltonian in the form .

(2)

n = t + r, where n is the total order of the operator, t is the order of the torsional part, and r is the order of the rotational part, respectively.

(3)

All values are in cm-1, except ρ which is unitless. Statistical uncertainties are shown as one standard uncertainty in the units of the last two digits.

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