The original statement given by Woodward and Hoffmann is somewhat more elaborate:
A ground-state pericyclic change is symmetry-allowed when the total number of (4q+2)s and (4r)a components is odd.
However, the statement given here is mathematically equivalent. For a proof, see: https://archive.org/details/whdz_red
Woodward, R. B.; Hoffmann, Roald (1965). "Stereochemistry of Electrocyclic Reactions". Journal of the American Chemical Society. 87 (2): 395. doi:10.1021/ja01080a054.
Zimmerman, H. E. (1966). "On Molecular Orbital Correlation Diagrams, the Occurrence of Möbius Systems in Cyclization Reactions, and Factors Controlling Ground- and Excited-State Reactions. I". Journal of the American Chemical Society. 88 (7): 1564–1565. doi:10.1021/ja00959a052.
Zimmerman, Howard E. (1971-08-01). "Moebius-Hueckel concept in organic chemistry. Application of organic molecules and reactions". Accounts of Chemical Research. 4 (8): 272–280. doi:10.1021/ar50044a002. ISSN0001-4842.
Dewar, M.J.S. (1966-01-01). "A molecular orbital theory of organic chemistry—VIII: romaticity and electrocyclic reactions". Tetrahedron. 22: 75–92. doi:10.1016/S0040-4020(01)82171-2. ISSN0040-4020.
Dewar, Michael J. S. (1971-11-01). "Aromaticity and Pericyclic Reactions". Angewandte Chemie International Edition in English. 10 (11): 761–776. doi:10.1002/anie.197107611. ISSN1521-3773.
Fukui, Kenichi (1965-01-01). "Stereoselectivity associated with noncycloaddition to unsaturated bonds". Tetrahedron Letters. 6 (28): 2427–2432. doi:10.1016/S0040-4039(00)90203-X. ISSN0040-4039.
Geerlings, Paul; Ayers, Paul W.; Toro-Labbé, Alejandro; Chattaraj, Pratim K.; De Proft, Frank (2012). "The Woodward–Hoffmann Rules Reinterpreted by Conceptual Density Functional Theory". Accounts of Chemical Research. 45 (5): 683–95. doi:10.1021/ar200192t. hdl:10533/131820. PMID22283422.
Criegee, Rudolf; Noll, Klaus (1959). "Umsetzungen in der Reihe des 1.2.3.4-Tetramethyl-cyclobutans". Justus Liebigs Annalen der Chemie. 627: 1–14. doi:10.1002/jlac.19596270102.
Srinivasan, R. (1968-07-01). "Mechanism of the photochemical valence tautomerization of 1,3-butadienes". Journal of the American Chemical Society. 90 (16): 4498–4499. doi:10.1021/ja01018a080. ISSN0002-7863.
Winter, Rudolph Ernst K. (1965-01-01). "The preparation and isomerization of cis- and trans-3,4-dimethylcyclobutene". Tetrahedron Letters. 6 (17): 1207–1212. doi:10.1016/S0040-4039(01)83997-6. ISSN0040-4039.
Woodward, R. B.; Hoffmann, Roald (1969). "The Conservation of Orbital Symmetry". Angew. Chem. Int. Ed. 8 (11): 781–853. doi:10.1002/anie.196907811.
Longuet-Higgins, H. C.; Abrahamson, E. W. (1965). "The Electronic Mechanism of Electrocyclic Reactions". Journal of the American Chemical Society. 87 (9): 2045. doi:10.1021/ja01087a033.
Hoffmann, Roald; Woodward, R. B. (1965). "Selection Rules for Concerted Cycloaddition Reactions". J. Am. Chem. Soc. 87 (9): 2046. doi:10.1021/ja01087a034.
Berson, Jerome A.; Nelson, George L. (1967-10-01). "Inversion of configuration in the migrating group of a thermal 1,3-sigmatropic rearrangement". Journal of the American Chemical Society. 89 (21): 5503–5504. doi:10.1021/ja00997a065. ISSN0002-7863.
Zimmerman, H. E. (1966). "On Molecular Orbital Correlation Diagrams, Möbius Systems, and Factors Controlling Ground and Excited State Reactions. II". J. Am. Chem. Soc. 88 (7): 1566–156. doi:10.1021/ja00959a053.
Geerlings, Paul; Ayers, Paul W.; Toro-Labbé, Alejandro; Chattaraj, Pratim K.; De Proft, Frank (2012). "The Woodward–Hoffmann Rules Reinterpreted by Conceptual Density Functional Theory". Accounts of Chemical Research. 45 (5): 683–95. doi:10.1021/ar200192t. hdl:10533/131820. PMID22283422.
Ayers, Paul W.; Morell, Christophe; De Proft, Frank; Geerlings, Paul (5 October 2007). "Understanding the Woodward–Hoffmann Rules by Using Changes in Electron Density". Chemistry: A European Journal. 13 (29): 8240–8247. doi:10.1002/chem.200700365. PMID17639522.
Corey, E. J.; Hortmann, Alfred G. (1963). "Total Synthesis of Dihydrocostunolide". Journal of the American Chemical Society. 85 (24): 4033. doi:10.1021/ja00907a030.
Corey, E. J.; Hortmann, Alfred G. (1965). "The total synthesis of dihydrocostunolide". Journal of the American Chemical Society. 87 (24): 5736–42. doi:10.1021/ja00952a037. PMID5845424.
Ayers, Paul W.; Morell, Christophe; De Proft, Frank; Geerlings, Paul (5 October 2007). "Understanding the Woodward–Hoffmann Rules by Using Changes in Electron Density". Chemistry: A European Journal. 13 (29): 8240–8247. doi:10.1002/chem.200700365. PMID17639522.
handle.net
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Geerlings, Paul; Ayers, Paul W.; Toro-Labbé, Alejandro; Chattaraj, Pratim K.; De Proft, Frank (2012). "The Woodward–Hoffmann Rules Reinterpreted by Conceptual Density Functional Theory". Accounts of Chemical Research. 45 (5): 683–95. doi:10.1021/ar200192t. hdl:10533/131820. PMID22283422.
Geerlings, Paul; Ayers, Paul W.; Toro-Labbé, Alejandro; Chattaraj, Pratim K.; De Proft, Frank (2012). "The Woodward–Hoffmann Rules Reinterpreted by Conceptual Density Functional Theory". Accounts of Chemical Research. 45 (5): 683–95. doi:10.1021/ar200192t. hdl:10533/131820. PMID22283422.
Geerlings, Paul; Ayers, Paul W.; Toro-Labbé, Alejandro; Chattaraj, Pratim K.; De Proft, Frank (2012). "The Woodward–Hoffmann Rules Reinterpreted by Conceptual Density Functional Theory". Accounts of Chemical Research. 45 (5): 683–95. doi:10.1021/ar200192t. hdl:10533/131820. PMID22283422.
Geerlings, Paul; Ayers, Paul W.; Toro-Labbé, Alejandro; Chattaraj, Pratim K.; De Proft, Frank (2012). "The Woodward–Hoffmann Rules Reinterpreted by Conceptual Density Functional Theory". Accounts of Chemical Research. 45 (5): 683–95. doi:10.1021/ar200192t. hdl:10533/131820. PMID22283422.
Ayers, Paul W.; Morell, Christophe; De Proft, Frank; Geerlings, Paul (5 October 2007). "Understanding the Woodward–Hoffmann Rules by Using Changes in Electron Density". Chemistry: A European Journal. 13 (29): 8240–8247. doi:10.1002/chem.200700365. PMID17639522.
Corey, E. J.; Hortmann, Alfred G. (1965). "The total synthesis of dihydrocostunolide". Journal of the American Chemical Society. 87 (24): 5736–42. doi:10.1021/ja00952a037. PMID5845424.
Ayers, Paul W.; Morell, Christophe; De Proft, Frank; Geerlings, Paul (5 October 2007). "Understanding the Woodward–Hoffmann Rules by Using Changes in Electron Density". Chemistry: A European Journal. 13 (29): 8240–8247. doi:10.1002/chem.200700365. PMID17639522.
Zimmerman, Howard E. (1971-08-01). "Moebius-Hueckel concept in organic chemistry. Application of organic molecules and reactions". Accounts of Chemical Research. 4 (8): 272–280. doi:10.1021/ar50044a002. ISSN0001-4842.
Dewar, M.J.S. (1966-01-01). "A molecular orbital theory of organic chemistry—VIII: romaticity and electrocyclic reactions". Tetrahedron. 22: 75–92. doi:10.1016/S0040-4020(01)82171-2. ISSN0040-4020.
Dewar, Michael J. S. (1971-11-01). "Aromaticity and Pericyclic Reactions". Angewandte Chemie International Edition in English. 10 (11): 761–776. doi:10.1002/anie.197107611. ISSN1521-3773.
Fukui, Kenichi (1965-01-01). "Stereoselectivity associated with noncycloaddition to unsaturated bonds". Tetrahedron Letters. 6 (28): 2427–2432. doi:10.1016/S0040-4039(00)90203-X. ISSN0040-4039.
Srinivasan, R. (1968-07-01). "Mechanism of the photochemical valence tautomerization of 1,3-butadienes". Journal of the American Chemical Society. 90 (16): 4498–4499. doi:10.1021/ja01018a080. ISSN0002-7863.
Winter, Rudolph Ernst K. (1965-01-01). "The preparation and isomerization of cis- and trans-3,4-dimethylcyclobutene". Tetrahedron Letters. 6 (17): 1207–1212. doi:10.1016/S0040-4039(01)83997-6. ISSN0040-4039.
Berson, Jerome A.; Nelson, George L. (1967-10-01). "Inversion of configuration in the migrating group of a thermal 1,3-sigmatropic rearrangement". Journal of the American Chemical Society. 89 (21): 5503–5504. doi:10.1021/ja00997a065. ISSN0002-7863.