Synthesis, structures, and photochromic properties of 2-methylthieno[3,2-b][1]benzothiophen-3-ylfulgide

The novel heterocyclic fulgide, 3-(1-methylethylidene)-4-[1-(2-methylthieno[3,2-b]-[1]benzothiophen-3-yl)ethylidene]dihydrofuran-2,5-dione, was obtained and isolated as a Z-isomer. Its structure was confirmed by electronic absorption spectroscopy, ¹H NMR spectroscopy, and X-ray diffraction analysis. The novel thienothiophene fulgide is photochromic in both solutions and the crystalline state. Its colored cyclic form resists photodegradation and is thermally stable. When benzothiophene is annulated with the thienyl fragment, the long-wavelength absorption peak of the cyclic isomer of the novel fulgide experiences a bathochromic shift compared to the earlier described 3-thienylfulgide. The TD B3LYP/6-311G**-calculated spectroscopic characteristics of the fulgide isomers suggest the formation of their photostationary mixture under irradiation with λ = 365 nm. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 12, pp. 2318–2324, December, 2007.     

Structures and photochromic properties of fulgides based on naphtho[1,2-<Emphasis Type="Italic">b</Emphasis>]furan and benzo[<Emphasis Type="Italic">g</Emphasis>]indole

The novel heterocyclic fulgides, i.e. 3-isopropylidene-4-{1-[5-methoxy-1-(4-methoxy-phenyl)-2-methyl-1H-benzo[g]indol-3-yl]ethylidene}dihydrofuran-2,5-dione and 3-isopropylidene-4-[1-(1-benzyl-5-methoxy-2-methyl-1H-benzo[g]indol-3-yl)ethylidene]dihydrofuran-2,5-dione, were prepared and isolated as E-isomers. Photochromism, E-configuration, and high resistance to photocoloration—photobleaching of solutions of these fulgides as well as 3-isopropylidene-4-[1-(5-methoxy-2-methylnaphtho[1,2-b]furan-3-yl)ethylidene]dihydro-furan-2,5-dione and 3-isopropylidene-4-[1-(5-methoxy-2-methyl-1-phenyl-1H-benzo[g]indol-3-yl)ethylidene]dihydrofuran-2,5-dione synthesized previously were shown by X-ray diffraction analysis, ¹H NMR spectroscopy and electronic absorption spectroscopy. The novel fulgides show fluorescence and high thermal stability of photogenerated cyclic form. Repeated photo-coloration—photobleaching is accompanied by reversible photoinduced E—Z isomerization. Benzo[g]indolyl fulgides are characterized by the longer wavelength absorption of both original (E) and photoisomeric cyclic (C) forms as compared to naphthofuran fulgide.     

Conformational Properties of (Z,Z)-, (E,Z)-, and (E,E)-Cycloocta-1,4-dienes

Summary.  Ab initio calculations at the HF/6-31G^* level of theory for geometry optimization and the MP2/6-31G^*//HF/6-31G^* level for a single point total energy calculation are reported for (Z,Z)-, (E,Z)-, and (E,E)-cycloocta-1,4-dienes. The C ₂-symmetric twist-boat conformation of (Z,Z)-cycloocta-1,4-diene was calculated to be by 3.6 kJ·mol⁻¹ more stable than the C _S-symmetric boat-chair form; the calculated energy barrier for ring inversion of the twist-boat conformation via the C _S-symmetric boat-boat geometry is 19.1 kJ·mol⁻¹. Interconversion between twist-boat and boat-chair conformations takes place via a half-chair (C ₁) transition state which is 43.5 kJ·mol⁻¹ above the twist-boat form. The unsymmetrical twist-boat-chair conformation of (E,Z)-cycloocta-1,4-diene was calculated to be by 18.7 kJ·mol⁻¹ more stable than the unsymmetrical boat-chair form. The calculated energy barrier for the interconversion of twist-boat-chair and boat-chair is 69.5 kJ·mol⁻¹, whereas the barrier for swiveling of the trans-double bond through the bridge is 172.6 kJ·mol⁻¹. The C _S symmetric crown conformation of the parallel family of (E,E)-cycloocta-1,4-diene was calculated to be by 16.5 kJ·mol⁻¹ more stable than the C _S-symmetric boat-chair form. Interconversion of crown and boat-chair takes place via a chair (C _S) transition state which is 37.2 kJ·mol⁻¹ above the crown conformation. The axial- symmetrical twist geometry of the crossed family of (E,E)-cycloocta-1,4-diene is 5.9 kJ·mol⁻¹ less stable than the crown conformation. Corresponding author. E-mail: isayavar@yahoo.com Received March 25, 2002; accepted April 3, 2002     

Photochromism of fulgides and stereoelectronic factors: Synthesis of (E)-4(3)-adamantyhdene-(5)-dicyanomethylene-3(4)-[1-(2,5-dimethylfuryl)ethylidene]tetrahydrofuran-2-one 4 and 4a and (E)-4-adamantylidene-3-[2,6-dimethyl-3,5-bis(p-diethyl-aminostyryl)benzylidene]tetrahydrofuran-2,5-dione 10

In the search for fulgides with potential semiconductor laser compatibility, 4-adamantylidene-5-dicyanomemylene-3-[1-(2,5-dimethyl-3-furyl)ethylidene]tetrahydrofuran-2-one ( 4 ), along with its regioisomer 4a , have been synthesized from the corresponding fulgide 6 containing a succinic anhydride ring by reaction with malononitrile in the presence of diethylamine. Upon irradiation with a uv light at λ_max 350 run, a mixture of 4 and 4a revealed a considerably enhanced bathochromic shift to the visible region, λ_max 605 nm as compared with the starting fulgide 6 which, upon analogous uv irradiation, absorbed at λ_max 515 nm. In the search for semiconductor-laser-compatible fulgides with increased efficiency for the reverse bleaching reaction, another fulgide (E)–adamantylidene-3-[2,6-dimethyl-3,5-bis(p-diethylaminostyryl)-benzylidene]tetrahydrofuran-2,5-dione ( 10 ) was synthesized in seven steps starting from 2-bromo-m-xylene. However, 10 failed to undergo electrocyclic ring-closure upon irradiation with a uv light at λ_max 350 nm. The analogous fulgide 8 , which contains an isopropylidene functionality in place of the adamantyl group of 10 , was resynthesized for comparison, and showed two absorption maxima, one at 545 nm and the other at 620 nm. The missing physico-chemical data for 8 have also been provided.     

Photochemische Reaktionen. 118. Mitteilung [1] Zur vinylogen β-Spaltung von Enonen. UV.-Bestrahlung von 4-(3',7',7'-Trimethyl-2'-oxabicyclo [3.2.0]hept-3'-en-1'-yl)but-3-en-2-on

Vinylogous β-Cleavage of Enones: UV.-irradiation of 4-(3′,7′,7′-trimethyl-2′-oxabicyclo[3.2.0]hept-3′-ene-1′-yl)but-3-ene-2-on On ¹π,π*-excitation (λ = 254 nm) in acetonitrile (E/Z)- 2 is converted into the isomers 4–9 and undergoes fragmentation yielding 10 ; in methanol (E/Z)- 2 gives 7–10 and is transformed into 11 by incorporation of the solvent. On ¹π,π*-excitation (λ λ?347 nm; benzene-d₆) (E)- 2 is isomerized into (Z)- 2 , which is converted into the isomers 3 and 4 by further irradiation. ¹π,π*-Excitation (λ = 254 nm; acetonitrile) of 4 gives 6 and (E)- 9 , whereas UV.-irradiation (λ = 254 nm; acetonitrile-d₃) of 5 yields (E)- 7 and 8 . On ¹π,π*-excitation (λ = 254 nm; acetonitrile) of (E/Z)- 12 the compounds (E)- 14 and (E)- 15 are obtained.     

Reduction of (<Emphasis Type="Italic">E</Emphasis>)-3-aryl-2-(thiazol-2-yl)acrylonitriles with lithium aluminum hydride

Reduction of (E)-3-aryl-2-(4-arylthiazol-2-yl)acrylonitriles with lithium aluminum hydride in dry ether afforded (Z)-1-amino-3-aryl-2-(thiazol-2-yl)prop-1-ene derivatives in 15 to 40% yields. The structure of (Z)-1-amino-3-(2-chlorophenyl)-2-[4-(4-methylphenyl)thiazol-2-yl]prop-1-ene was confirmed by X-ray diffraction analysis. Dedicated to Academician N. K. Kochetkov on the occasion of his 90th birthday. __________ Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 5, pp. 1301–1303, May, 2005.     

Structure of (Z)-(5R)-methyl-2-(4-phenylbenzylidene)cyclohexanone as chiral component of liquid-crystalline systems

The spatial structure of (Z)-(5R)-methyl-2-(4-phenylbenzylidene)cyclohexanone prepared by photochemical isomerization of the E-isomer was studied by analyzing the magnitudes and temperature dependence of the proton spin-spin coupling constants obtained by ¹H NMR spectroscopy and the results of molecular modeling using semiempirical quantum chemical AM1 and PM3 methods and the density functional theory (DFT). Comparison of the results obtained for the Z-and E-isomers shows that in both cases the conformational equilibrium for both isomers is characterized by significant preference of the chair conformer having an equatorial methyl group, namely, − ΔH (chair a ⇌ chair e) = 1.98–2.12 and 1.36–1.54 kcal mole⁻¹ for the Z-and E-isomers, respectively. Distinctions in the non-planarity of the enone fragment and cyclohexanone ring in the Z-and E-isomers under study following from the results of mathematical modeling were confirmed by the experimental values of the geminal spin-spin coupling constants of protons of the methylene groups in α,α ′-positions with respect to the enone group. Quantum chemical calculations of the Z-isomer revealed the existence of intramolecular hydrogen bond between the carbonyl oxygen and the nearest aromatic proton in ortho-position of the benzene ring. Possible reasons for different helical twisting power of (Z)-(5R)-methyl-2-(4-phenylbenzylidene)cyclohexanone and the E-and Z-arylidene derivatives of 1R, 4R-isomenthone in the mesophase are discussed based on the results of molecular structure studies for these compounds. In the text below the unsaturated ketones under study will be called “arylidene cyclohexanone derivatives” for convenience of comparing the characteristics of methylcyclohexanone and isomenthone derivatives. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 6, pp. 962–972, June, 2006.     

DFT studies on tautomeric preferences of 1-(pyridin-2-yl)-4-(quinolin-2-yl)butane-2,3-dione in the gas phase and in solution

Density functional theory (DFT) calculations show that in vacuum such α-diketone as 1-(pyridin-2-yl)-4-(quinolin-2-yl)butane-2,3-dione is much less stable than its enolimine–enaminone ((1Z,3Z)-3-hydroxy-4-(pyridin-2-yl)-1-(quinolin-2(1H)-ylidene)but-3-en-2-one) and dienaminone tautomers ((1Z,3Z)-1-(pyridin-2-yl)-4-(quinolin-2-yl)buta-1,3-diene-2,3-diol). Other its tautomers (multiple basic and acidic centers in their molecules enable multiple proton transfer to take place) are even more labile. Strength of the intramolecular hydrogen bonds and aromatic character of the (quasi)rings [proved by the Harmonic Oscillator Model of Aromaticity (HOMA) index] in their molecules were found to be responsible for the observed tautomeric preferences. Polar and basic solvent disfavors and favors the enolimine and enaminone tautomers, respectively.     

On the chemistry of pyrrole pigments,XCVII: Synthesis,stereochemistry, and solvatochromic effects of a 1-(dipyrrinon-9-yl)-3-(dipyrrinon-9-ylidene)-1-propene

Summary By condensing a dipyrrinon-9-yl-acrolein with a dipyrrinone unsubstituted in position 9, an unsymmetricb-vinylogous verdinoid pigment was prepared. The configuration of this molecule was elucidated by means of 2D¹H NMR experiments to be (4Z,10E,12Z,17Z) in solutions of chloroform and dimethylsulfoxide, and its conformation was derived to be 5syn,9syn,11anti,16syn. The pronounced solvatochromic effect of this molecule with a cyclic helical geometry could be explained by solvent induced conformational changes.

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Zur Chemie von Pyrrolpigmenten, 97. Mitt.: Synthese, Stereochemie und Solvatochromer Effekt eines 1-(Dipyrrinon-9-yl)-3-(dipyrrinon-9-yliden)-1-propens

Zusammenfassung Durch Kondensation eines Dipyrrinon-9-yl-acroleins mit einem in Position 9 unsubstituierten Dipyrrinon wurde ein unsymmetrischesb-vinyloges verdinoides Pigment dargestellt. Die Konfiguration dieses Moleküls wurde mit Hilfe von 2D-¹H-NMR-Experimenten in Lösungen von und Dimethylsulfoxid als (4Z,10E,12Z,17Z) abgeleitet; seine Konformation konnte zu 5syn,9syn,11anti,16syn festgelegt werden. Die ausgeprägte Solvatochromie dieses Moleküls mit cyclisch helikaler Geometrie konnte auf solvensinduzierte konformative Veränderungen zurückgeführt werden.

     

Photochemical reactions. 137th communication. Preparation and photolysis of (E/Z)-7-methyl-β-ionone

The title compounds (E/Z)- 7 were prepared in 66% overall yield by reaction of β-ionone ((E)-( 1 ) with lithium dimethylcuprate, trapping of the intermediate enolate with benzeneselenenyl bromide and oxidation with H₂O₂. Analogously, (E/Z)-7-methyl-α-inone ((E/Z)- 12 ) was obtained in 65% yield from α-ionone ((E)- 11 ). ¹n, π^*- Excitation (λ > 347 nm, pentane) of (E)-7 causes rapid (E/Z)-isomerization and subsequent reaction of (Z)- 7 to 15 (66%). The formation of 15 is explained by twisting of the dienone chromophore due to repulsive interaction of the 7-CH₃-group with the CH₃-groups of the cyclohexene ring. On the other hand, irradiation λ > 347 nm, Et₂O) of (E)- 7 in the presence of acid leads to (Z)- 7 (5%) and to the novel compound 16 (88%).     

Candidate Trail Attractants of Reticultermes lucifugus: Stereoselective Syntheses of (3Z, 6E,BE)-(3Z, 6E, 8Z)-and (3Z, 6Z, 8Z)-3,6,8 -Dodecatrien-1-OL1

Stereoselective syntheses on a gram scale of (3Z,6E,8E)-, (3Z,6E,8Z)-and (3Z,6Z,8Z)-3,6,8-dodecatrien-1-ol, 8, 9 and 10, respectively, are described. A key step of the synthesis of 8 consisted of a copper-mediated coupling reaction between 4-(2-tetrahydropyranyloxy)-1-butynylmagnesium bromide (15) and the mesyl ester of (2E,4E)-2,4-octadien-1-ol (14). A similar copper-mediated reaction between 15 and the mesyl ester of (E)-2-octen-4-yn-1-ol (19) was used to construct the C-12 carbon skeleton of 9. On the other hand, the synthesis of 10 was based on a palladium-promoted reaction between (Z)-1-bromo-1-pentene (23) and the organozinc bromide derived from 3,6-heptadiyn-1-yl acetate (27).     

The Photoisomerization Behavior of the (4-Hydroxycinnamoyl)-spermidines

The photoisomerization behavior of three mono[(E)-3-(4-hydroxyphenyl)prop-2-enoyl]spermidines, 1, 2 , and 3 , and three bis[(E)-3-(4-hydroxyphenyl)prop-2-enoyl]spermidines, 4, 5 , and 6 , are investigated. The synthetic product (E)- 1 could be almost quantitatively (> 96%) converted into its isomer (Z)- 1 under UV light irradiation. In the cases of (E)- 2 and (E)- 3 , a mixture of (E)/(Z) ca. 1:2 was obtained, when the same conditions were applied. The comparison of their UV spectra provides the possible explanation for these different behaviors. Furthermore, it was noticed that the (Z) → (E) isomerization of the C?C bond took place during the purification by reverse-phase high-performance liquid chromatography (RP-HPLC), and the (E)/(Z)-mixture is thus inseparable. The same feature could be observed during the isolation of the (Z,Z)-N,N′-bis[3-(4-hydroxyphenyl)prop-2-enoyl]-spermidines, (Z,Z)- 4 , (Z,Z)- 5 , and (Z,Z)- 6 . Nevertheless, the fractions of (Z,Z)- 5 and (Z,Z)- 6 were in almost pure state collected, and their ¹-NMR spectra are presented.     

Conformation and Crystal Structure of Dipyrrinones with Oxindole Components

 Pyrrole α-aldehydes (2-formyl-4,5-dimethyl-1H-pyrrole and 2-formyl-N-methylpyrrole) condense readily at C(3) of indolin-2-ones to give dipyrrinone analogs, such as (3Z)-[(4,5-dimethylpyrrol-2-yl)-methylidenyl]-indolin-2-one and (3E)-[(1-methylpyrrol-2-yl)-methylidenyl]-indolin-2-one. ¹H-NMR NOE analyses and X-ray crystallography confirm the syn-(Z) configuration for the former and the syn-(E) configuration for the latter. The former is stabilized by intramolecular hydrogen bonding. Molecular mechanics calculations of the latter indicate no energy difference between the syn and anti conformations.     

Properties and structures of porphyrexides

The molecular and crystal structures of porphyrexides, viz., 4-amino-2-imino-(1) and (Z)-2-amino-4-bromoimino-5,5-dimethyl-4,5-dihydro-1H-imidazole 1-oxyls (7), and their diamagnetic precursors were determined. Compound 1 is kinetically unstable because it is oxidized with atmospheric oxygen to form (E)-1,2-bis[1-amino-1-(cyanoimino)-2-methyl-propan-2-yl]diazene 1,2-dioxide. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 3, pp. 441–447, March, 2006.     

1-R-2-[(1E,3E)-4-Aminobuta-1,3-dien-1-yl]-1H-benzimidazoles. Synthesis and some transformations

7-Nitropyridobenzimidazolium salts are cleaved with secondary amines to form 2-[(E,E)-4-aminobuta-1,3-dienyl]-1H-benzimidazoles. The latter react with dimethyl acetylene-dicarboxylate to yield 4a-[(E,Z,E)-6-amino-4,5-dimethoxycarbonylhexa-1,3,5-trien-1-yl]-1,2,3,4-tetra(methoxycarbonyl)-4a,5-dihydropyrido[1,2-a]benzimidazoles.     

Isolation by HPLC. and Identification by NMR. Spectroscopy of 11 mono-, di- and tri-cis isomers of an aromatic analogue of retinoic acid,ethyl all-trans-9-(4-methoxy-2,3,6-trimethylphenyl)-3,7-dimethyl-nona-2,4,6, 8-tetraenoate

Photoisomerization of an aromatic analogue of retinoic acid, ethyl all-trans-9-(4-methoxy-2,3,6-trimethylphenyl)-3,7-dimethyl-nona-2,4,6, 8-tetraenoate 1 in dilute solutions of hexane, benzene, and ethanol yielded multi-component mixtures of cis isomers which were separated by HPLC. FT-¹H-NMR. at 270 MHz and, in some cases, homonuclear decoupling and Overhauser experiments as well as ¹³C-NMR. were applied to establish the structures of 4 mono-cis, 4 (of 6 possible) di-cis, and 3 (of 4 possible) tri-cis isomers. The structures of 3 isomeric esters, namely (2Z, 4E, 6E, 8E) 6 , (2Z, 4Z, 6E, 8E) 9 , and (2Z, 4Z, 6Z, 8E) 7 were independently confirmed by direct syntheses. The ¹H-NMR. data of all these compounds and the ¹³C-NMR. data of the all-trans and of 6 cis isomers available in sufficiently large quantities are discussed.     

Synthesis and Stereoisomerization of 2-(1-Alkoxyimino-2,2,2-trifluoroethyl)-5-trimethylsilylfurans

2-(1-Alkoxyimino-2,2,2-trifluoroethyl)-5-trimethylsilylfurans were synthesized by the condensation of 2-(trifluoroacetyl)-5-trimethylsilylfuran with alkoxyamines. According to ¹H and ¹⁹F NMR spectroscopic data, the alkoxyimino group in the E-isomers descreens the H-3 and H-4 protons of the furan ring more strongly than in the Z-isomers, shifting their signals downfield. The fluorine atoms of the α-trifluoromethyl group in the Z-isomer are characterized by a downfield shift in relation to the E-isomer. __________ Translated from Khimiya Geterotsiklicheskikh Soedinenii, No. 6, pp. 834–838, June, 2005.     

The Behavior of 3,3-Diphenylindan-1,2-dione Towards Phosphonium Ylides

Summary. The reaction of alkoxycarbonyl- and cyanomethylene(triphenyl)phosphoranes with 3,3-diphenylindan-1,2-dione in dry benzene at room temperature for about 5 h led to the formation of a mixture of (E)- and (Z)-diastereomers. On the other hand, treatment of the dione with acetylmethylene(triphenyl)phosphorane afforded a mixture of (E)-3,3-diphenyl-1-(2-oxo-2-methylethylidene)indan-2-one and unexpected product (E)-3-(3,3-diphenyl-2-oxoindan-1-ylidene)-4-(triphenyl-λ⁵-phosphanylidene)hexane-2,5-dione, whereas with benzoylmethylene(triphenyl)phosphorane gave a mixture of (E)-3,3-diphenyl-1-(2-oxo-2-phenylethylidene)indan-2-one, [(2R ^*,3S ^*)-3-benzoyl-8,8-diphenyl-3,8-dihydro-2H-indeno{2,1-b}furan-2-yl]phenylmethanone and 1,4-diphenyl-2-(3,3-diphenyl-2-hydroxy-3H-inden-1-yl)but-2-ene-1,4-dione. The reaction mechanisms are considered and structural assignments of the new compounds are based on spectroscopic evidence. The molecular structures of the two diastereomers and the unexpected product were elucidated by X-ray crystallography.     

The Behavior of 3,3-Diphenylindan-1,2-dione Towards Phosphonium Ylides

The reaction of alkoxycarbonyl- and cyanomethylene(triphenyl)phosphoranes with 3,3-diphenylindan-1,2-dione in dry benzene at room temperature for about 5 h led to the formation of a mixture of (E)- and (Z)-diastereomers. On the other hand, treatment of the dione with acetylmethylene(triphenyl)phosphorane afforded a mixture of (E)-3,3-diphenyl-1-(2-oxo-2-methylethylidene)indan-2-one and unexpected product (E)-3-(3,3-diphenyl-2-oxoindan-1-ylidene)-4-(triphenyl-λ⁵-phosphanylidene)hexane-2,5-dione, whereas with benzoylmethylene(triphenyl)phosphorane gave a mixture of (E)-3,3-diphenyl-1-(2-oxo-2-phenylethylidene)indan-2-one, [(2R ^*,3S ^*)-3-benzoyl-8,8-diphenyl-3,8-dihydro-2H-indeno{2,1-b}furan-2-yl]phenylmethanone and 1,4-diphenyl-2-(3,3-diphenyl-2-hydroxy-3H-inden-1-yl)but-2-ene-1,4-dione. The reaction mechanisms are considered and structural assignments of the new compounds are based on spectroscopic evidence. The molecular structures of the two diastereomers and the unexpected product were elucidated by X-ray crystallography.     

Photochemistry of dienones—X: Photochemistry of (e)-β-ionone oxime ethyl ether

(E)-β-ionone oximc ethyl ether [(E, E)-4] upon direct irradiation with λ either254or 313 nm yields the geometrical isomer (E, Z)-4 and (Z)-retro-γ-ionone oxime ethyl ether (Z,E)-5 as the sole primary products, illustrating (E)-(Z) isomerization (φ₃₁₃ =0.49) and a 1, 5-hydrogen shift (φ313 =0.15) respectively. From studies with triplet photosensitizers and with ethyl iodide (to enhance the singlet-triplet intersystem crossing) it is concluded that these two products in the direct irradiation result only from the singlet excited state, and that the inter-system crossing quantum yield is relatively low. Upon prolonged irradiation of (E,E)-4 with λ 313 nm the eventual products are (Z,E)-5 and (Z,Z)-5, whereas with λ 254 nm they are (E,E)-5 and [(Z,E)-5 and/or (E,Z)-5]. Upon triplet photosensitization (E,E)-4 undergoes only (E)-(Z) isomerization, leading to a mixture of all the four geometrical isomers of4. From the dependence of the geometrical isomer distribution in the photostationary state on the triplet energy of the sensitizer the triplet energies of (E,E)-4, (E, Z)-4, (Z, E)-4, and (Z, Z)-4 have been determined to be ca 55, < 55,57, and 57 $\text{kcal}\text{mol}$ respectively.