View source for Multiplier (Fourier analysis)

[[Image:Water molecule dimensions.svg|thumb|200px|Geometry of the [[water (molecule)|water molecule]]]]

'''Molecular geometry''' is the [[three-dimensional space|three-dimensional]] arrangement of the [[atom]]s that constitute a [[molecule]]. It determines several properties of a substance including its [[Reactivity (chemistry)|reactivity]], [[Chemical polarity|polarity]], [[Phase (matter)|phase of matter]], [[color]], [[magnetism]], and [[biological activity]].<ref>{{McMurray}}</ref><ref>{{Cotton&Wilkinson6th}}</ref> The angles between bonds that an atom forms depend only weakly on the rest of molecule, i.e. they can be understood as approximately local and hence [[Transferability (chemistry)|transferable properties]].

== Molecular geometry determination ==
The molecular geometry can be determined by various [[spectroscopy|spectroscopic methods]] and [[diffraction]] methods. [[Infrared spectroscopy|IR]], [[Rotational spectroscopy|microwave]] and [[Raman spectroscopy]] can give information about the molecule geometry from the details of the vibrational and rotational absorbance detected by these techniques. [[X-ray crystallography]], [[neutron diffraction]] and [[electron diffraction]] can give molecular structure for crystalline solids based on the distance between nuclei and concentration of electron density. [[Gas electron diffraction]] can be used for small molecules in the gas phase. [[nuclear magnetic resonance|NMR]] and [[Förster resonance energy transfer|FRET]] methods can be used to determine complementary information including relative distances,
<!-- 
References for NMR and FRET distances 
--><ref>[http://dwb.unl.edu/Teacher/NSF/C08/C08Links/pps99.cryst.bbk.ac.uk/projects/gmocz/fret.htm FRET description]</ref><ref>{{ cite journal| last1=Hillisch| doi=10.1016/S0959-440X(00)00190-1| first1=A| last2=Lorenz| first2=M| last3=Diekmann| first3=S |title=Recent advances in FRET: distance determination in protein–DNA complexes| pmid=11297928 | journal=Current Opinion in Structural Biology| year=2001| volume=11|issue=2 | pages=201–207}}</ref><ref>[http://www.fretimaging.org/mcnamaraintro.html FRET imaging introduction]</ref>
<!-- End References for NMR and FRET distances -->dihedral angles,
<!-- References for NMR determination of dihedral angles (through <sup>3</sup>J coupling constants) --><ref>[http://www.jonathanpmiller.com/Karplus.html obtaining dihedral angles from <sup>3</sup>J coupling constants]</ref><ref>[http://www.spectroscopynow.com/FCKeditor/UserFiles/File/specNOW/HTML%20files/General_Karplus_Calculator.htm Another Javascript-like NMR coupling constant to dihedral]</ref>
<!-- End references for NMR determination of dihedral angles (through <sup>3</sup>J coupling constants) --> 
angles, and connectivity.  Molecular geometries are best determined at low temperature because at higher temperatures the molecular structure is averaged over more accessible geometries (see next section). Larger molecules often exist in multiple stable geometries ([[conformational isomerism]]) that are close in energy on the [[potential energy surface]].  Geometries can also be computed by [[ab initio quantum chemistry methods]] to high accuracy.  The molecular geometry can be different as a solid, in solution, and as a gas.

The position of each atom is determined by the nature of the [[chemical bond]]s by which it is connected to its neighboring atoms. The molecular geometry can be described by the positions of these atoms in space, evoking [[bond length]]s of two joined atoms, [[bond angle]]s of three connected atoms, and [[torsion angle]]s ([[dihedral angle]]s) of three [[path graph|consecutive]] bonds.

== The influence of thermal excitation ==
Since the motions of the atoms in a molecule are determined by quantum mechanics, one
must define “motion” in a quantum mechanical way. The overall (external) quantum mechanical motions translation and rotation hardly change the geometry of the molecule. (To some extent rotation influences
the geometry via [[Coriolis force]]s and [[rotational spectroscopy|centrifugal distortion]], but this is negligible for the present discussion.)
A third type of motion  is vibration, which is the  internal motion of the atoms in a molecule. The molecular vibrations are harmonic (at least to good approximation), which means that the atoms  oscillate about their equilibrium, even at the absolute zero of temperature. At absolute zero all atoms are in their vibrational ground state and show zero point quantum mechanical motion, that is, the wavefunction of a single vibrational mode is not a sharp peak, but an exponential of finite width.{{clarification needed|date=June 2013}}  At higher temperatures the vibrational modes may be thermally excited (in a classical interpretation one expresses this by stating that “the molecules will vibrate faster”), but they oscillate still around the recognizable geometry of the molecule. 

To get a feeling for the probability that the vibration of  molecule may be thermally excited,
we inspect the [[Boltzmann distribution|Boltzmann factor]] <math>\exp\left( -\frac{\Delta E}{kT} \right) </math>,
where <math>\Delta E</math> is the excitation energy of the vibrational mode, <math>k</math> the [[Boltzmann constant]] and <math>T</math> the absolute temperature. At 298&nbsp;K (25&nbsp;°C), typical values for the Boltzmann factor are:
0.089 for ΔE = 500&nbsp;cm<sup>−1</sup> ; ΔE = 0.008 for 1000&nbsp;cm<sup>−1</sup> ; 7×10<sup>−4</sup>  for ΔE = 1500&nbsp;cm<sup>−1</sup>.  That is, if the excitation energy is 500&nbsp;cm<sup>−1</sup>, then about 9 percent of the
molecules are thermally excited at room temperature. The lowest excitation vibrational energy in water is the bending mode (about 1600&nbsp;cm<sup>−1</sup>). Thus, at room temperature less than 0.07 percent of all the  molecules of a given amount of water will vibrate faster than at absolute zero.

As stated above, rotation hardly influences the molecular geometry. But, as a quantum mechanical motion, it is thermally excited at relatively (as compared to vibration) low temperatures. From a classical point of view it can be stated that at higher temperatures more molecules will rotate faster,
which implies higher [[angular velocity]] and [[angular momentum]]. In quantum mechanically language: more eigenstates of higher angular momentum  become [[Boltzmann distribution|thermally populated]] with rising temperatures. Typical rotational excitation energies are on the order of a few cm<sup>−1</sup> of corresponding [[wave number]]. 

The results of many spectroscopic experiments are broadened because they involve an averaging over rotational states. It  is often difficult to extract geometries from spectra at high temperatures, because the number of rotational states probed in the experimental averaging increases with increasing temperature.  Thus, many spectroscopic observations can only be expected to yield reliable molecular geometries  at temperatures close to absolute zero, because at higher temperatures too many higher rotational states are thermally populated.

== Bonding ==

Molecules, by definition, are most often held together with [[covalent bond]]s involving single, double, and/or triple bonds, where a "bond" is a [[shared pair]] of electrons (the other method of bonding between atoms is called [[ionic bonding]] and involves a positive [[cation]] and a negative [[anion]]).

Molecular geometries can be specified in terms of '''bond lengths''', '''bond angles''' and '''torsional angles'''. The bond length is defined to be the average distance between the centers of two atoms bonded together in any given molecule. A bond angle is the angle formed between three atoms across at least two bonds. For four atoms bonded together in a chain, the [[Dihedral angle|torsional angle]] is the angle between the plane formed by the first three atoms and the plane formed by the last three atoms.

There exists a mathematical relationship among the bond angles for one central atom and four peripheral atoms (labeled 1 through 4) expressed by the following determinant.  This constraint removes one degree of freedom from the choices of (originally) six free bond angles to leave only five choices of bond angles.  (Note that the angles <math>\theta_{11}</math>, <math>\theta_{22}</math>, <math>\theta_{33}</math>, and <math>\theta_{44}</math> are always zero.)

::<math>0 = \begin{vmatrix}

\cos \theta_{11} & \cos \theta_{12} & \cos \theta_{13} & \cos \theta_{14} \\
\cos \theta_{21} & \cos \theta_{22} & \cos \theta_{23} & \cos \theta_{24} \\
\cos \theta_{31} & \cos \theta_{32} & \cos \theta_{33} & \cos \theta_{34} \\
\cos \theta_{41} & \cos \theta_{42} & \cos \theta_{43} & \cos \theta_{44} \end{vmatrix}</math>

Molecular geometry is determined by the [[quantum mechanics|quantum mechanical]] behavior of the electrons. Using the [[valence bond theory|valence bond approximation]] this can be understood by the type of bonds between the atoms that make up the molecule. When atoms interact to form a [[chemical bond]], the atomic orbitals are said to mix in a process called [[orbital hybridisation]].  The two most common types of bonds are [[sigma bond]]s and [[pi bond]]s.  The geometry can also be understood by [[molecular orbital theory]] where the electrons are delocalised.

An understanding of the wavelike behavior of electrons in atoms and molecules is the subject of [[quantum chemistry]].

== Isomers ==

[[Isomer]]s are types of molecules that share a chemical formula but have different geometries, resulting in very different properties:

* A '''pure''' substance is composed of only one type of isomer of a molecule (all have the same geometrical structure).

* [[Structural isomerism|Structural isomers]] have the same chemical formula but different physical arrangements, often forming alternate molecular geometries with very different properties. The atoms are not bonded (connected) together in the same orders.
** [[Functional isomer]]s are special kinds of structural isomers, where certain groups of atoms exhibit a special kind of behavior, such as an ether or an alcohol.

* [[Stereoisomer]]s may have many similar physicochemical properties (melting point, boiling point) and at the same time very different [[biochemistry|biochemical]] activities. This is because they exhibit a [[handedness]] that is commonly found in living systems. One manifestation of this [[Chirality (chemistry)|chirality]] or handedness is that they have the ability to rotate polarized light in different directions.

* [[Protein folding]] concerns the complex geometries and different isomers that [[protein]]s can take.

==Types of molecular structure==
{{main|VSEPR theory #AXE method}}

Some common shapes of simple molecules include:
*'''[[Linear molecular geometry|Linear]]:''' In a linear model, atoms are connected in a straight line.  The bond angles are set at 180°.  A bond angle is very simply the geometric angle between two adjacent bonds.  For example, carbon dioxide and [[nitric oxide]] have a linear molecular shape.
*'''[[Trigonal planar molecular geometry|Trigonal planar]]:''' Just from its name, it can easily be said that molecules with the trigonal planar shape are somewhat triangular and in one [[Plane (geometry)|plane (flat)]].  Consequently, the bond angles are set at 120°.  An example of this is [[boron trifluoride]].
*'''[[Bent molecular geometry|Bent]]:''' Bent or angular molecules have a non-linear shape. A good example is water, or H<sub>2</sub>O, which has an angle of about 105°. A water molecule has two pairs of bonded electrons and two unshared lone pairs.
*'''[[Tetrahedral molecular geometry|Tetrahedral]]:''' ''Tetra-'' signifies four, and ''-hedral'' relates to a face of a solid, so "[[tetrahedral]]" literally means "having four faces".  This shape is found when there are [[star (graph theory)|four bonds all on one central atom]], with no extra unshared [[electron]] pairs.  In accordance with the [[VSEPR]] (valence-shell electron pair repulsion theory), the bond angles between the electron bonds are [[Inverse trigonometric functions|arccos]](−1/3) = 109.47°.  An example of a tetrahedral molecule is [[methane]] (CH<sub>4</sub>). <!-- what about molecules without a central atom, which form a complete graph? -->
*'''[[Octahedral molecular geometry|Octahedral]]:''' ''Octa-'' signifies eight, and ''-hedral'' relates to a face of a solid, so "[[octahedral]]" literally means "having eight faces".  The bond angle is 90 degrees. An example of an octahedral molecule is [[sulfur hexafluoride]] (SF<sub>6</sub>).
*'''Pyramidal:''' Pyramidal-shaped molecules have [[Pyramid (geometry)|pyramid-like shapes]].  Unlike the linear and trigonal planar shapes but similar to the tetrahedral orientation, pyramidal shapes require three dimensions in order to fully separate the electrons.  Here, there are only three pairs of bonded electrons, leaving one unshared lone pair. Lone pair – bond pair repulsions change the angle from the tetrahedral angle to a slightly lower{{fact|date=June 2013|reason=ORLY?}} value. An example is NH<sub>3</sub> ([[ammonia]]).

===VSEPR table===

The bond angles in the table below are ideal angles from the simple [[VSEPR theory]], followed by the actual angle for the example given in the following column where this differs. For many cases, such as trigonal pyramidal and bent, the actual angle for the example differs from the ideal angle, but all examples differ by different amounts. For example, the angle in H<sub>2</sub>S (92°) differs from the tetrahedral angle by much more than the angle for H<sub>2</sub>O (104.5°) does.
{| class="wikitable sortable"
|-
! Bonding electron pairs
! Lone pairs
! Electron domains (Steric #)
! Shape
! Ideal bond angle (example's bond angle)
! Example
! Image
|-
| <center>2</center>
| <center>0</center>
| <center>2</center>
| [[linear molecular geometry|linear]]
| <center>180°</center>
| <center>[[carbon dioxide|CO<sub>2</sub>]]</center>
| <center>[[Image:Linear-3D-balls.png|50px]]</center>
|-
| <center>3</center>
| <center>0</center>
| <center>3</center>
| [[trigonal planar molecular geometry|trigonal planar]]
| <center>120°</center>
| <center>[[Boron trifluoride|BF<sub>3</sub>]]</center>
| <center>[[Image:Trigonal-3D-balls.png|50px]]</center>
|-
| <center>2</center>
| <center>1</center>
| <center>3</center>
| [[bent molecular geometry|bent]]
| <center>120° (119°) </center>
| <center>[[Sulfur dioxide|SO<sub>2</sub>]]</center>
| <center>[[Image:Bent-3D-balls.png|50px]]</center>
|-
| <center>4</center>
| <center>0</center>
| <center>4</center>
| [[tetrahedral molecular geometry|tetrahedral]]
| <center>109.5°</center>
| <center>[[methane|CH<sub>4</sub>]]</center>
| <center>[[Image:AX4E0-3D-balls.png|50px]]</center>
|-
| <center>3</center>
| <center>1</center>
| <center>4</center>
| [[trigonal pyramidal molecular geometry|trigonal pyramidal]]
| <center>107° </center>
| <center>[[ammonia|NH<sub>3</sub>]]</center>
| <center>[[Image:Pyramidal-3D-balls.png|50px]]</center>
|-
| <center>2</center>
| <center>2</center>
| <center>4</center>
| bent
| <center>109.5° (104.5°) </center>
| <center>[[H2O|H<sub>2</sub>O]]</center>
| <center>[[Image:Bent-3D-balls.png|50px]]</center>
|-
| <center>5</center>
| <center>0</center>
| <center>5</center>
| [[trigonal bipyramidal molecular geometry|trigonal bipyramidal]]
| <center>90°, 120°, 180°</center>
| <center>[[phosphorus pentachloride|PCl<sub>5</sub>]]</center>
| <center>[[Image:Trigonal-bipyramidal-3D-balls.png|50px]]</center>
|-
| <center>4</center>
| <center>1</center>
| <center>5</center>
| [[seesaw molecular geometry|seesaw]]
| <center>180°, 120°, 90° (173.1°, 101.6°) </center>
| <center>[[sulfur tetrafluoride|SF<sub>4</sub>]]</center>
| <center>[[Image:Seesaw-3D-balls.png|50px]]</center>
|-
| <center>3</center>
| <center>2</center>
| <center>5</center>
| [[T-shaped molecular geometry|T-shaped]]
| <center>90°, 180° (87.5°, < 180°) </center>
| <center>[[chlorine trifluoride|ClF<sub>3</sub>]]</center>
| <center>[[Image:T-shaped-3D-balls.png|50px]]</center>
|-
| <center>2</center>
| <center>3</center>
| <center>5</center>
| linear
| <center>180°</center>
| <center>[[xenon difluoride|XeF<sub>2</sub>]]</center>
| <center>[[Image:Linear-3D-balls.png|50px]]</center>
|-
| <center>6</center>
| <center>0</center>
| <center>6</center>
| [[octahedral molecular geometry|octahedral]]
| <center>90°, 180°</center>
| <center>[[sulfur hexafluoride|SF<sub>6</sub>]]</center>
| <center>[[Image:AX6E0-3D-balls.png|50px]]</center>
|-
| <center>5</center>
| <center>1</center>
| <center>6</center>
| [[square pyramidal molecular geometry|square pyramidal]]
| <center>90° (84.8°), 180° </center>
| <center>[[bromine pentafluoride|BrF<sub>5</sub>]]</center>
| <center>[[Image:Square-pyramidal-3D-balls.png|50px]]</center>
|-
| <center>4</center>
| <center>2</center>
| <center>6</center>
| [[square planar molecular geometry|square planar]]
| <center>90°, 180°</center>
| <center>[[xenon tetrafluoride|XeF<sub>4</sub>]]</center>
| <center>[[Image:Square-planar-3D-balls.png|50px]]</center>
|-
| <center>7</center>
| <center>0</center>
| <center>7</center>
| [[pentagonal bipyramidal molecular geometry|pentagonal bipyramidal]]
| <center>90°, 72°, 180°</center>
| <center>[[iodine heptafluoride|IF<sub>7</sub>]]</center>
| <center>[[Image:Pentagonal-bipyramidal-3D-balls.png|50px]]</center>
|-
| <center>6</center>
| <center>1</center>
| <center>7</center>
| [[pentagonal pyramidal molecular geometry|pentagonal pyramidal]]
| <center>72°, 90°, 144°</center>
| <center>XeOF<sub>5</sub><sup>−</sup></center>
| <center>[[Image:Pentagonal-pyramidal-3D-balls.png|50px]]</center>
|-
| <center>5</center>
| <center>2</center>
| <center>7</center>
| [[pentagonal planar molecular geometry|planar pentagonal]]
| <center>72°, 144°</center>
| <center>[[Tetramethylammonium pentafluoroxenate|XeF<sub>5</sub><sup>−</sup>]]</center>
| <center>[[Image:Pentagonal-planar-3D-balls.png|50px]]</center>
|-
| <center>8</center>
| <center>0</center>
| <center>8</center>
| [[square antiprismatic molecular geometry|square antiprismatic]]
| <center> </center>
| <center>[[Nitrosonium octafluoroxenate(VI)|XeF<sub>8</sub><sup>2−</sup>]]</center>
| <center>[[Image:Square-antiprismatic-3D-balls.png|50px]]</center>
|-
| <center>9</center>
| <center>0</center>
| <center>9</center>
| [[tricapped trigonal prismatic molecular geometry|tricapped trigonal prismatic]]
| <center> </center>
| <center>[[Potassium_nonahydridorhenate|ReH<sub>9</sub><sup>2−</sup>]]</center>
| <center>[[Image:AX9E0-3D-balls.png|50px]]</center>
|}

==Other molecular structures==
{{main|Orbital hybridisation}}
For transition metals as well as a more comprehensive quantum chemical description for both, [[Orbital hybridisation]] is used.

==3D representations==
*'''Line''' or '''stick''' – atomic nuclei are not represented, just the bonds as sticks or lines. As in 2D molecular structures of this type, atoms are implied at each vertex.
	
 :{| class=wikitable
 |-
 |<!--col1-->[[Image:Formic-acid-3D-stick.png|center|110px]]
 |[[Image:L-aspartic-acid-3D-sticks.png|center|110px]]
 |[[Image:ATP-xtal-3D-sticks.png|center|110px]]
 |[[Image:Endohedral fullerene.png|center|110px]]
 |}

*'''Electron density plot''' – shows the electron density determined either [[crystallography|crystallographically]] or using [[quantum mechanics]] rather than distinct atoms or bonds
:{| class=wikitable
 |-
 |[[Image:NorbornylCation ElectronDensity.jpg|center|110px]]
 |[[Image:WinsteinYellow.jpg|center|110px]]
 |}
*'''Ball and stick''' – atomic nuclei are represented by spheres (balls) and the bonds as sticks
 :{| class=wikitable
 |-
 |<!--col1-->[[Image:Methanol-3D-balls.png|center|110px]]
 |[[Image:Methanol struktur.png|center|110px]]
 |[[Image:PropyleneGlycol-stickAndBall.png|center|110px]]
 |[[Image:3LRI SolutionStructureAndBackboneDynamicsOfHumanLong arg3 insulin-Like Growth Factor 1 02.png|center|170px]]
 |}
*[[Spacefilling model]]s or [[CPK model]]s (also an [[CPK coloring|atomic coloring scheme]] in representations) – the molecule is represented by overlapping spheres representing the atoms

 :{| class=wikitable
 |-
 |<!--col1-->[[Image:Methanol.pdb.png|center|110px|]]
 |[[Image:Ubiquitin spheres.png|center|110px]]
 |[[Image:P-cresol-spaceFilling.png|center|110px]]
 |[[Image:3GF1 Insulin-Like Growth Factor Nmr 10 01.png|center|170px]]
 |}
*'''Cartoon''' – a representation used for proteins where loops, beta sheets, alpha helices are represented diagrammatically and no atoms or bonds are represented explicitly just the protein backbone as a smooth pipe
:{| class=wikitable
 |-
 |[[Image:Beta-meander1.png|center|110px]]
 |[[Image:MreB.png|center|110px]]
 |[[Image:Anthrax toxin protein key motif.svg|center|110px]]
 |[[Image:8tim TIM barrel.png|center|170px]]
 |}
The greater the amount of lone pairs contained in a molecule the smaller the angles between the atoms of that molecule. The [[VSEPR theory]] predicts that lone pairs repel each other, thus pushing the different atoms away from them.

==See also==
* [[Molecular graphics]]
* [[Molecular modelling]]
* [[Molecular mechanics]]
* [[Molecule editor]]
* [[Molecular design software]]
* [[Quantum chemistry]]
* [[Polyhedral skeletal electron pair theory]]
* [[Topology (chemistry)]]
* [[Jemmis mno rules]]

==References==
{{Reflist}}

==External links==
*[http://www.tutor-pages.com/Chemistry/Molecular_Geometry/Polar_Or_Nonpolar.html Molecular Geometry & Polarity Tutorial] 3D visualization of molecules to determine polarity.

{{MolecularGeometry}}

{{DEFAULTSORT:Molecular Geometry}}
[[Category:Molecular geometry| ]]