Specifying Molecular Electronic and Geometrical Structures

Parts I: Cartesian Coordinates

How to Construct Cartesian coordinates of an Atom in a Molecule Using Cartesian coordinates of H2 as an example Syntax for the Cartesian coordinate format: Atomic label, x-coordinate, y-coordinate, z- coordinate O H1H2 x y e.g., C 2v H 2 O H  sp 3 hybridization of the O center

Molecular Structure Specification for Water O H1H2 x y Here is the molecular structure of water that is given in Cartesian coordinates 0 1 O H H Spin multiplicity Total charge on this molecule Atomic symbol x-Coordinatey-Coordinatez-Coordinate e.g., C 2v H 2 O (Unit used : angstrom for lengths)  The Cartesian coordinate style is not intuitive to give us bond lengths, bond angles, and dihedral angles of atoms of interest!

Parts II: Z-matrix (Internal Coordinates)

Using H4 internal coordinates as an example H e.g., C 2 H 2 O Syntax for The Z-matrix format: Atomic label, Atom 2, Bond length, Atom 3, Bond angle, Atom 4, Dihedral angle Atomic label: For current atom Atoms 1-3: Previously specified atoms, namely reference atoms Bond length: For the bond joining the current atom to atom 2 Bond angle: Formed by this bond and the bond joining atom 1, and atom 2 Dihedral angle: Formed by t he plane containing atoms 2-4 with the plane containing the current atom, atom 2, and atom 1 Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format Without Variables 4 3

Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format Without Variables Here is the molecular structure of hydrogen peroxide in the Z-matrix format 0 1 O O11.4 H H Spin multiplicity Total charge on the molecule Oxygen atom #2: O2-O1 = 1.4 Ǻ Oxygen atom #1 e.g., C 2 H 2 O 2 Hydrogen #3: H3-O1 = 0.9 Ǻ; H3-O1-O2 = 105  Hydrogen #4: H4-O2 = 0.9 Ǻ; H4-O2-O1 = 105  ; H4-O2- O1-H3 = 120  (Positive dihedral angles correspond to clockwise rotation in Newman projections) Unit: Angstrom for lengths and degrees for angles 4 3

Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format With Variables Here is another version of the hydrogen peroxide molecular specification 0 1 O O1R1 H1R22A H2R21A3D Variables: R1 1.4 R2 0.9 A 105. D 120. Oxygen atom #2: O2-O1 = R1 = 1.4 Ǻ Oxygen atom #1 e.g., C 2 H 2 O 2 Hydrogen #1: H3-O1 = R2 = 0.9 Ǻ; H3-O1-O2 = A = 105  Hydrogen #2: H4-O2 = R2 = 0.9 Ǻ; H4-O2-O1 = A = 105  ; H4-O2-O1-H3 = D = 120 

Molecular Structure Specification for Hydrogen Peroxide in Z-matrix Format With Variables and Constants Here is the third version of the hydrogen peroxide molecular specification 0 1 O O1R1 H1R22A H2R21A3D Variables: R1 1.4 R2 0.9 Constants: A 105. D 120. Spin multiplicity Total charge on the molecule Oxygen atom #2: O2-O1 = R1 = 1.4 Ǻ Oxygen atom #1 e.g., C 2, H 2 O 2 Hydrogen #1: H3-O1 = R2 = 0.9 Ǻ; H3-O1-O2 = A = 105  Hydrogen #2: H4-O2 = R2 = 0.9 Ǻ; H4-O2-O1 = A = 105  ; H4-O2-O1-H3 = D = 120 

Construct a Z-matrix for a More Complex Molecule The atoms inside the redline all lies in a plane e.g., C s, Propene (CH 2 CHCH 3 ) Step 1: Specify carbon atoms 0 1 C C1C2C C2C3C1A3 Variables: C2C 1.34 C3C 1.52 A Charge and multiplicity C: C1 at the end of the double bond C: C2 on the other end of the double bond C: C3 Third carbon A3: C3-C2-C1 bond angle C3C: C-C single bond length C2C: C-C double bond length  1.09

e.g., C s, Propene (CH 2 CHCH 3 ) Step 2: Specify the hydrogens on C1 and C2 H 1 H4C 2 A4 3 D4 H 1 H5C 2 A5 3 D5 H 2 H6C 1 A6 3 D5 Variables: H4C 1.09 H5C 1.09 H6C 1.09 A A A Constants: D4 0. D H: H1 H: H2 H: H3 D4: The H1-C1-C2-C3 dihedral H4C: H1-C1 bond length H5C: H5-C1 bond length H6C: H6-C2 bond length D5: The H2(or H3)-C1-C2-C3 dihedral Note that the decimal points in D4 and D5 must be included !  1.09 A4: H1-C1-C2 bond angel A5: H2-C1-C2 bond angel A6: H3-C2-C1 bond angel Construct a Z-matrix for a More Complex Molecule

e.g., C s, Propene (CH 2 CHCH 3 ) Step 3: Specify the planar hydrogen on C3 H 3 H7C 2 A7 1 D5 Variables: H7C 1.09 A Constants: D  H: H4 H7C: H4-C3 bond length A7: H4-C3-C2 bond angle D5: D9 = D5, H4-C3-C2-C1 dihedral 1.09 Newman Projections are often used to visualize dihedral angles 1.09 Construct a Z-matrix for a More Complex Molecule

e.g., C s, Propene (CH 2 CHCH 3 ) Step 4: Specify the non-planar hydrogens on C3 H 3 H8C 2 A8 1 D8 H 3 H9C 2 A9 1 –D8 Variables: H8C 1.09 H9C 1.09 A A D8 60. H: H5 H: H6 H8C: H5-C3 bond length H9C: H6-C3 bond length A8: H5-C3-C2 bond angle D8: D9 = -D8, the H5-C3-C2-C1 dihedral The geometry of C3 is tetrahedral, and thus the bond angle of each of the hydrogens with respect to the C3-C2 bond is  Newman Projections are often used to visualize dihedral angles A9: H6-C3-C2 bond angle  1.09 Construct a Z-matrix for a More Complex Molecule

e.g., C s, Propene (CH 2 CHCH 3 ) Step 5: List all created internal coordinates in a Z-matrix  C C 1 C2C C 2 C3C 1 A3 H 1 H4C 2 A4 3 D4 H 1 H5C 2 A5 3 D5 H 2 H6C 1 A6 3 D5 H 3 H7C 2 A7 1 D5 H 3 H8C 2 A8 1 D8 H 3 H9C 2 A9 1 -D8 Variables: C2C 1.34 C3C 1.52 H4C 1.09 H5C 1.09 H6C 1.09 H7C 1.09 H8C 1.09 H9C 1.09 A A A A A A A D8 60. Constants: D4 0. D Construct a Z-matrix for a More Complex Molecule

Parts III: Mixed Internal and Cartesian Coordinates

e.g., O h, Cr(CO) 6 Specify the complete molecular structure 0 1 Cr C O C O C O C O C O C O Cartesian Coordinates for Cr(CO) Electron configuration: (Ar)3d 5 4s 1

Mixed Cartesian and Internal Coordinates for Cr(CO) 5 NH 3 e.g., Cr(CO 5 )NH Cr C O C O C O C O C O N H 12 HN 1 HNCr 2 0. H 12 HN 1 HNCr 13 D H 12 HN 1 HNCr 13 –D Variables HN 1.02 HNCr 115. D Cartesian coordinates are included in a Z- matrix by specifying the bonded-to atom as 0

Mixed Cartesian and Internal Coordinates for Cr(CO) 5 NH 3 e.g., C s, Cr(CO 5 )NH Cr C 0 CCr O C 0 -CCr O C 0 0. CCr 0. O C CCr 0. O C CCr O N H 12 HN 1 HNCr 2 0. H 12 HN 1 HNCr 13 D H 12 HN 1 HNCr 13 –D Variables CCr 1.93 HN 1.02 HNCr 115. D The variable names for Cartesian coordinates are given symbolically in the same manner as for internal coordinates

Parts IV: Using Dummy Atoms in Z-matrices

Use of Dummy Atom X to Fix the Three-fold Axis in C 3v Ammonia e.g., C 3v, NH 3 Cs:Cs: N X 1 1. H 1 nh 2 hnx H 1 nh 2 hnx H 1 nh 2 hnx Variables nh 1.0 hnx H2 H3 H1 N(X) N H 1 nh H 1 nh 2 hnx H 1 nh 2 hnx Variables nh 1.0 hnx X C 3v :  The use of dummy atoms within Z- matrices, which are represented by the pseudo atomic symbol X, is to fix a symmetric axis

J. Phys. Chem. B 113 (2009) 5290 X In the calculations, a dummy atom X is placed in the six-atom cycle Use of Dummy Atom X to Contact Nonbonding Molecular Fragments

Sources for Geometrical Structural Parameters  Periodic table of elements  Standard references like the CRC series  Published experiments and calculations hybridizationBond angleGeometryExamples sp 180  LineCHCH, BeCl 2, Hg(NH 3 ) 2+ sp  Planar triangleCH 2 CH 2, BF 2, [CuCl 3 ] 2- sp  TetrahedronCH 4, BF 2, [Ni(NH3) 4 ] 2+ dsp 2 90  SquareNi(CN) 4 2- dsp 3 90 , 120  Trigonal bipyramidPCl 5, Fe(CO) 5 d 2 sp 3 90  OctahedronSF 6, Co(CN) 6  Hybridization of central ions or atoms

Parts V: Summarize Geometry Specification

1. Cartesian coordinates: atomic symbol, x, y, z coordinates of each nucleus Gaussian expects values in Angstroms convenient because most molecular building programs will output Cartesian coordinates 2. Z-matrix coordinates: specify positions of atoms relative to one another using bond lengths, angles and dihedral angles (3N-6 variables) Gaussian expects values in Angstroms and degrees also called internal coordinates one section specifies connectivity, second section specifies values of variables corresponding to bond lengths, etc. convenient for PES scans because bonds and angles are defined explicitly Cartesian & Z-matrix Styles

C1C2 H3 H4H6 H5 C C 1 B1 H 1 B2 2 A1 H 1 B2 2 A1 3 D1 H 2 B2 1 A1 3 D2 H 2 B2 1 A1 5 D3 B4 A3 D2 variables: B1=1.5 B2=1.1 A1=120.0 D1=0.0 D2=0.0 D3=180.0 can simplify by taking advantage of symmetry expect C-H bonds to be same lengths  use variable B2 for all C-H bonds expect H-C-C angles to be the same  use variable A1 for all H-C-C angles careful, though assigning the same label to two or more geometric variables means they have to remain equal throughout entire calculation Geometrical connectivity Atom1 Atom2 Bond orders formed between Atoms 1, 2 Connectivity Specification