{"id":222932,"date":"2025-06-01T06:01:40","date_gmt":"2025-06-01T06:01:40","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=222932"},"modified":"2025-06-01T06:01:43","modified_gmt":"2025-06-01T06:01:43","slug":"character-table-for-d3h-point-group-linear-253-quadratic-rotations-x-y-r-xz","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/01\/character-table-for-d3h-point-group-linear-253-quadratic-rotations-x-y-r-xz\/","title":{"rendered":"Character table for D3h point group, linear; 253 quadratic rotations x + y? R) (xz)"},"content":{"rendered":"\n

Character table for D3h point group, linear; 253 quadratic rotations x + y? R) (xz)
Character table for D3h point group, linear; 253 quadratic rotations x + y? R) (xz)<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Character Table for D3h Point Group<\/strong><\/h3>\n\n\n\n

The D3h<\/strong> point group is common in molecules with a trigonal planar shape, such as BF3<\/strong> or NO3\u2212<\/strong>. It contains symmetry elements including:<\/p>\n\n\n\n

    \n
  • E<\/strong> (identity)<\/li>\n\n\n\n
  • 2C3<\/strong> (rotation by \u00b1120\u00b0)<\/li>\n\n\n\n
  • 3C\u20322<\/strong> (rotation about axes in the molecular plane)<\/li>\n\n\n\n
  • \u03c3h<\/strong> (horizontal mirror plane)<\/li>\n\n\n\n
  • 2S3<\/strong> (improper rotations)<\/li>\n\n\n\n
  • 3\u03c3v<\/strong> (vertical mirror planes)<\/li>\n<\/ul>\n\n\n\n

    Here is the character table<\/strong> for the D3h point group:<\/p>\n\n\n\n

    D3h<\/th>E<\/th>2C3<\/th>3C\u20322<\/th>\u03c3h<\/th>2S3<\/th>3\u03c3v<\/th>Basis Functions<\/th><\/tr><\/thead>
    A\u20321<\/td>1<\/td>1<\/td>1<\/td>1<\/td>1<\/td>1<\/td>z<\/td><\/tr>
    A\u20322<\/td>1<\/td>1<\/td>\u20131<\/td>1<\/td>1<\/td>\u20131<\/td><\/td><\/tr>
    E\u2032<\/td>2<\/td>\u20131<\/td>0<\/td>2<\/td>\u20131<\/td>0<\/td>(x, y), (x\u00b2\u2013y\u00b2, xy)<\/td><\/tr>
    A\u20331<\/td>1<\/td>1<\/td>1<\/td>\u20131<\/td>\u20131<\/td>\u20131<\/td><\/td><\/tr>
    A\u20332<\/td>1<\/td>1<\/td>\u20131<\/td>\u20131<\/td>\u20131<\/td>1<\/td>Rz<\/td><\/tr>
    E\u2033<\/td>2<\/td>\u20131<\/td>0<\/td>\u20132<\/td>1<\/td>0<\/td>(Rx, Ry), (xz, yz)<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n
    \n\n\n\n

    Explanation <\/strong><\/h3>\n\n\n\n

    The D3h point group is of particular importance in chemistry for describing the symmetry of trigonal planar molecules. The character table above organizes the irreducible representations (irreps)<\/strong> of this group and associates them with symmetry-adapted basis functions<\/strong> like coordinates and orbital shapes.<\/p>\n\n\n\n

    The question asks about the symmetry of linear<\/strong>, quadratic<\/strong>, and rotational<\/strong> functions such as x + y<\/strong>, Rz<\/strong>, and xz<\/strong>.<\/p>\n\n\n\n

      \n
    • Linear functions (x, y, z):<\/strong>\n
        \n
      • x, y<\/strong> transform as the E\u2032<\/strong> irrep. This is a 2D degenerate representation<\/strong>.<\/li>\n\n\n\n
      • z<\/strong> transforms as A\u20321<\/strong>, since it remains unchanged under the main symmetry operations.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Rotational functions (Rx, Ry, Rz):<\/strong>\n
          \n
        • Rx and Ry<\/strong> belong to the E\u2033<\/strong> representation.<\/li>\n\n\n\n
        • Rz<\/strong>, rotation about the principal axis, transforms as A\u20332<\/strong>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
        • Quadratic functions:<\/strong>\n
            \n
          • x\u00b2 \u2013 y\u00b2 and xy<\/strong> are in the E\u2032<\/strong> irrep.<\/li>\n\n\n\n
          • xz and yz<\/strong> fall under E\u2033<\/strong>, as they mix in-plane and out-of-plane axes.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n

            As for x + y<\/strong>, it is a linear combination of x and y and still belongs to the E\u2032<\/strong> representation. The individual components don\u2019t affect the irreducible representation, only the symmetry behavior matters.<\/p>\n\n\n\n

            Thus:<\/p>\n\n\n\n

              \n
            • x + y \u2192 E\u2032<\/strong><\/li>\n\n\n\n
            • Rz \u2192 A\u20332<\/strong><\/li>\n\n\n\n
            • xz \u2192 E\u2033<\/strong><\/li>\n<\/ul>\n\n\n\n

              Understanding these assignments is essential in vibrational spectroscopy, molecular orbital theory, and predicting activity in IR or Raman spectra. By matching a molecular vibration or orbital with these symmetry types, chemists can predict which transitions are allowed and interpret experimental data more effectively.<\/p>\n\n\n\n

              \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

              Character table for D3h point group, linear; 253 quadratic rotations x + y? R) (xz)Character table for D3h point group, linear; 253 quadratic rotations x + y? R) (xz) The Correct Answer and Explanation is: Character Table for D3h Point Group The D3h point group is common in molecules with a trigonal planar shape, such […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-222932","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222932","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=222932"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/222932\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=222932"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=222932"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=222932"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}