{"id":9556,"date":"2013-02-17T09:30:06","date_gmt":"2013-02-17T09:30:06","guid":{"rendered":"http:\/\/www.ch.imperial.ac.uk\/rzepa\/blog\/?p=9556"},"modified":"2014-01-17T07:33:39","modified_gmt":"2014-01-17T07:33:39","slug":"linking-numbers-and-twist-and-writhe-components-for-two-extended-porphyrins","status":"publish","type":"post","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556","title":{"rendered":"Linking numbers, and twist and writhe components for two extended porphyrins."},"content":{"rendered":"
\n

My last comment as\u00a0appended to the previous post<\/a>\u00a0promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the\u00a0C\u00e3lug\u00e3reanu\/Fuller<\/a> theorem\u00a0 which decomposes the topology of a space curve into two components, its twist (Tw) and its writhe (Wr, this latter being the extent to which coiling of the central curve has relieved local twisting) and establishes a topological invariant called the linking number<\/a>[1]<\/a><\/span><\/p>\n

\u00a0Lk = Tw + Wr\u00a0<\/strong><\/p>\n\n\n\n
\n
\"Click

HIYTAL. Click for 3D.<\/p><\/div>\n<\/td>\n

\n
\"SELQUW.

SELQUW. Click for 3D.<\/p><\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Visual inspection of the models above (I really do encourage you to click on the images to load the 3D coordinates) reveals that HIYTAL[2]<\/a><\/span> has a major coil that forms one and a half helical turns in a clockwise direction, and a loop connecting the ends of the coil which forms a half-helical turn in an anti-clockwise direction. SELQUW[3]<\/a><\/span>\u00a0has a major coil comprising one and half helical turns in an anti-clockwise direction and a connecting loop which also coils anti-clockwise. So the former sustains a total of one<\/strong>\u00a0full (clockwise) helical turn and the latter two<\/strong>\u00a0full (anti-clockwise) helical turns.<\/p>\n

The nomenclature for helical molecules<\/a> includes a chiral descriptor P (for a positive helical turn) and M (for a negative helical turn). What such a descriptor does not do is quantify the total number of helices describing the topology. So I suggest we use instead the linking number Lk. Instead of P and M, we have positive and negative integers (in units of 2\u03c0) providing this quantitative information.<\/p>\n

The linking number analysis for these two molecules comes out as below.\u2021<\/sup> I have multiplied the linking number unit from 2\u03c0 to 1\u03c0 for a reason that I will explain shortly:<\/p>\n\n\n\n\n\n
\u00a0<\/td>\n\u03c0-electrons<\/td>\nLk<\/td>\nTw<\/td>\nWr<\/td>\n\u0394r<\/sub> (meso)<\/td>\n<\/tr>\n
SELQUW<\/td>\n56=4n<\/td>\n-4<\/td>\n-1.34\u00a0<\/td>\n-2.66<\/td>\n0.048<\/td>\n<\/tr>\n
HIYTAL<\/td>\n62=4n+2<\/td>\n+2\u00a0<\/td>\n+0.46<\/td>\n+1.54<\/td>\n0.045<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

You can see that the linking numbers (and their signs) correspond exactly to the visual analysis of the helical turns above. My reason for including the factor of 2 is that it enables us to make a further link to the H\u00fcckel aromaticity rule:<\/p>\n

    \n
  1. Cyclic conjugated systems are aromatic<\/strong> if they contain 4n+2 \u03c0-electrons and have an even or zero linking number (in units of 1\u03c0).\u00a0<\/span><\/li>\n
  2. Cyclic conjugated systems are aromatic<\/strong> if they contain 4n\u00a0\u03c0-electrons and have an odd linking number (in units of 1\u03c0).\u00a0<\/li>\n
  3. Cyclic conjugated systems are anti-aromatic<\/strong> if they contain 4n \u03c0-electrons and have an even or zero linking number (in units of 1\u03c0).\u00a0<\/li>\n
  4. Cyclic conjugated systems are anti-aromatic<\/strong> if they contain 4n+2 \u03c0-electrons and have an odd linking number (in units of 1\u03c0).\u00a0<\/li>\n<\/ol>\n

    By these rules, SELQUW contains (by the shortest path) 56 \u03c0-electrons, belongs to the 4n electron rule (n=14) and hence is formally anti-aromatic (rule 3 above). HIYTAL has a path of 62-electrons, belongs to the 4n+2 rule (n=15) and hence is formally aromatic (rule 1 above).\u00a0<\/p>\n

    For systems with so many (correlated) electrons, it is probably tenuous to make a connection between the bond-length alternation at the meso position and the aromaticity (or lack of it). I comment only that HIYTAL converts more of the coiling into writhing of the central curve than does SELQUW, and this destroys less\u00a0\u03c0-\u03c0 overlap by reducing the overall degree of twisting. I might also speculate that nevertheless a modest degree of twisting may impact upon the intrinsic distortivity of\u00a0\u03c0-electrons in cyclically conjugated systems (such as that in benzene[4]<\/a><\/span>), as noted in this earlier post<\/a>. Such effects may make the interpretation of bond-alternation in such helical systems more difficult.<\/p>\n

    \n

    \u2021 A program for calculating these components can be found here<\/a>. For a fun-packed journey through linking numbers and the association with valentine cards, go see this post<\/a> here!<\/sup><\/p>\n

    References<\/h2>\n
    1. S.M. Rappaport, and H.S. Rzepa, \"Intrinsically Chiral Aromaticity. Rules Incorporating Linking Number, Twist, and Writhe for Higher-Twist M\u00f6bius Annulenes\", Journal of the American Chemical Society<\/i>, vol. 130, pp. 7613-7619, 2008. https:\/\/doi.org\/10.1021\/ja710438j<\/a>\n\n<\/li>\n
    2. S. Shimizu, W. Cho, J. Sessler, H. Shinokubo, and A. Osuka, \"<i>meso<\/i>\u2010Aryl Substituted Rubyrin and Its Higher Homologues: Structural Characterization and Chemical Properties\", Chemistry \u2013 A European Journal<\/i>, vol. 14, pp. 2668-2678, 2008. https:\/\/doi.org\/10.1002\/chem.200701909<\/a>\n\n<\/li>\n
    3. https:\/\/doi.org\/<\/a>\n\n<\/li>\n
    4. S. Shaik, A. Shurki, D. Danovich, and P.C. Hiberty, \"A Different Story of \u03c0-DelocalizationThe Distortivity of \u03c0-Electrons and Its Chemical Manifestations\", Chemical Reviews<\/i>, vol. 101, pp. 1501-1540, 2001. https:\/\/doi.org\/10.1021\/cr990363l<\/a>\n\n<\/li>\n<\/ol>\n\n<\/div> ","protected":false},"excerpt":{"rendered":"

      My last comment as\u00a0appended to the previous post\u00a0promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the\u00a0C\u00e3lug\u00e3reanu\/Fuller theorem\u00a0 which decomposes the topology of a space curve into two components, its twist (Tw) and its writhe (Wr, this latter being the extent to which coiling of the central curve has relieved […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"activitypub_content_warning":"","activitypub_content_visibility":"","activitypub_max_image_attachments":5,"activitypub_interaction_policy_quote":"anyone","activitypub_status":"","footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[4],"tags":[999,990],"ppma_author":[2661],"class_list":["post-9556","post","type-post","status-publish","format-standard","hentry","category-interesting-chemistry","tag-conjugated-systems","tag-helical-systems"],"yoast_head":"\nLinking numbers, and twist and writhe components for two extended porphyrins. - Henry Rzepa's Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Linking numbers, and twist and writhe components for two extended porphyrins. - Henry Rzepa's Blog\" \/>\n<meta property=\"og:description\" content=\"My last comment as\u00a0appended to the previous post\u00a0promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the\u00a0C\u00e3lug\u00e3reanu\/Fuller theorem\u00a0 which decomposes the topology of a space curve into two components, its twist (Tw) and its writhe (Wr, this latter being the extent to which coiling of the central curve has relieved […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556\" \/>\n<meta property=\"og:site_name\" content=\"Henry Rzepa's Blog\" \/>\n<meta property=\"article:published_time\" content=\"2013-02-17T09:30:06+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2014-01-17T07:33:39+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2013\/02\/hiytal.jpg\" \/>\n<meta name=\"author\" content=\"Henry Rzepa\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Henry Rzepa\" \/>\n\t<meta name=\"twitter:label2\" content=\"Estimated reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"3 minutes\" \/>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Linking numbers, and twist and writhe components for two extended porphyrins. - Henry Rzepa's Blog","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556","og_locale":"en_GB","og_type":"article","og_title":"Linking numbers, and twist and writhe components for two extended porphyrins. - Henry Rzepa's Blog","og_description":"My last comment as\u00a0appended to the previous post\u00a0promised to analyse two so-called extended porphyrins for their topological descriptors. I start with the\u00a0C\u00e3lug\u00e3reanu\/Fuller theorem\u00a0 which decomposes the topology of a space curve into two components, its twist (Tw) and its writhe (Wr, this latter being the extent to which coiling of the central curve has relieved […]","og_url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556","og_site_name":"Henry Rzepa's Blog","article_published_time":"2013-02-17T09:30:06+00:00","article_modified_time":"2014-01-17T07:33:39+00:00","og_image":[{"url":"http:\/\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2013\/02\/hiytal.jpg","type":"","width":"","height":""}],"author":"Henry Rzepa","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Henry Rzepa","Estimated reading time":"3 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556#article","isPartOf":{"@id":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556"},"author":{"name":"Henry Rzepa","@id":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/#\/schema\/person\/2b40f7b9c872a4dc1547e040a11b6281"},"headline":"Linking numbers, and twist and writhe components for two extended porphyrins.","datePublished":"2013-02-17T09:30:06+00:00","dateModified":"2014-01-17T07:33:39+00:00","mainEntityOfPage":{"@id":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556"},"wordCount":637,"commentCount":0,"image":{"@id":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556#primaryimage"},"thumbnailUrl":"http:\/\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2013\/02\/hiytal.jpg","keywords":["conjugated systems","helical systems"],"articleSection":["Interesting chemistry"],"inLanguage":"en-GB","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556","url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9556","name":"Linking numbers, and twist and writhe components for two extended porphyrins. - 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It has its origins in a project to make odd-numbered cyclocarbons on STM (scanning tunnelling microscope) surfaces. I had discussed even-numbered cyclocarbons in another post, where I also happened to include several odd-numbered examples,\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":2046,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=2046","url_meta":{"origin":9556,"position":1},"title":"Chemistry with a super-twist: A molecular trefoil knot, part 1.","author":"Henry Rzepa","date":"May 31, 2010","format":false,"excerpt":"Something important happened in chemistry for the first time about 100 years ago. A molecule was built (nowadays we would say synthesized) specifically for the purpose of investigating a theory. It was cyclo-octatetraene or (CH)8, and it was made by Willst\u00e4tter and Waser to try to find out if benzene,\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2010\/05\/metallatrefoil.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":24503,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=24503","url_meta":{"origin":9556,"position":2},"title":"Molecule of the year 2021: Infinitene.","author":"Henry Rzepa","date":"December 16, 2021","format":false,"excerpt":"The annual \"molecule of the year\" results for 2021 are now available ... and the winner is Infinitene., This is a benzocirculene in the form of a figure eight loop (the infinity symbol), a shape which is also called a lemniscate after the mathematical (2D) function due to Bernoulli. The\u2026","rel":"","context":"In "Chiroptics"","block_context":{"text":"Chiroptics","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=2644"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.ic.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2021\/12\/infinitene.jpg?resize=350%2C200&ssl=1","width":350,"height":200},"classes":[]},{"id":21250,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=21250","url_meta":{"origin":9556,"position":3},"title":"The Kekul\u00e9 vibration as a function of aromatic ring size. A different perspective using lemniscular rings.","author":"Henry Rzepa","date":"September 27, 2019","format":false,"excerpt":"In the previous posts, I tried to track down the onset of bond length alternation (BLA) as a function of ring size in aromatic cyclocarbons, finding the answer varied dramatically depending on the type of method used to calculate it. So here I change the system to an unusual kind\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":8398,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=8398","url_meta":{"origin":9556,"position":4},"title":"Di-imide reduction with a twist: A M\u00f6bius version.","author":"Henry Rzepa","date":"November 26, 2012","format":false,"excerpt":"I was intrigued by one aspect of the calculated transition state for di-imide reduction of an alkene; the calculated NMR shieldings indicated an diatropic ring current at the centre of the ring, but very deshielded shifts for the hydrogen atoms being transferred. This indicated, like most thermal pericyclic reactions, an\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"","src":"https:\/\/i0.wp.com\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2012\/11\/GaussViewScreenSnapz004.jpg?resize=350%2C200","width":350,"height":200},"classes":[]},{"id":9512,"url":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?p=9512","url_meta":{"origin":9556,"position":5},"title":"Helically conjugated molecules. A follow-up to [144]-annulene.","author":"Henry Rzepa","date":"February 12, 2013","format":false,"excerpt":"An extensive discussion developed regarding my post on a fascinating helical [144]-annulene. Topics included the nature of the ring current sustained by the\u00a0\u03c0-electrons and in particular the bond-length alternation around the periphery and whether this should alter if the electron count were to be changed to that of a 4n+2\u2026","rel":"","context":"In "Interesting chemistry"","block_context":{"text":"Interesting chemistry","link":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/?cat=4"},"img":{"alt_text":"SELQUW. Click for 3D.","src":"https:\/\/i0.wp.com\/www.ch.imperial.ac.uk\/rzepa\/blog\/wp-content\/uploads\/2013\/02\/SELQUW.jpg?resize=350%2C200","width":350,"height":200},"classes":[]}],"jetpack_likes_enabled":false,"authors":[{"term_id":2661,"user_id":1,"is_guest":0,"slug":"admin","display_name":"Henry Rzepa","avatar_url":"https:\/\/secure.gravatar.com\/avatar\/897b6740f7f599bca7942cdf7d7914af5988937ae0e3869ab09aebb87f26a731?s=96&d=blank&r=g","0":null,"1":"","2":"","3":"","4":"","5":"","6":"","7":"","8":""}],"_links":{"self":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/9556","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=9556"}],"version-history":[{"count":13,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/9556\/revisions"}],"predecessor-version":[{"id":11920,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=\/wp\/v2\/posts\/9556\/revisions\/11920"}],"wp:attachment":[{"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=9556"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=9556"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=9556"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/www.ch.ic.ac.uk\/rzepa\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fppma_author&post=9556"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}