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Water structure: Methods

Why different methods give different water structures?
Dielectric spectroscopy
Diffraction methods
Modeling
Nuclear Magnetic Resonance (NMR)
Osmotic stress
Physical properties
Vibrational spectra (Raman and infrared)
Terahertz (THz) absorption spectroscopy
Microwave (GHz) absorption spectroscopy
Sum frequency generation
X-Ray spectroscopy

Gaussian and Lorentzian curves

 

 

Several independent methods have been used to investigate the structure of liquid water and aqueous and biomolecular [2729] solutions. Although each method is often promoted as producing accurate results, the different methods produce conflicting conclusions. This not only generates academic disputes but also, and more importantly, causes general confusion and bewilderment leading to scientific disorientation.

 

Why different methods give different water structures?

 

Each method produces data but this data does not directly give the structure of water. It has to be interpreted. Any analysis depends on a theoretical base to interpret the data. More importantly, the results are only as good as this theoretical base, however excellent the quality of the data collected. In addition, different methods give information at particular time-, or size-scales and so may naturally differ. ^a Here I show the principles behind the most important analytical methods (Dielectric spectroscopy, Diffraction, Models, Nuclear magnetic resonance, Physical properties, Vibrational spectra, X-Ray spectroscopy) and the main assumptions made to interpret their resulting data.

The structural forms and timescales for the different techniques are given below:

 

Approximate timescales for different methods

Structure	Method(s)	Time and size scales
Instantaneous	X-ray absorption	≈ 10-15 s, single molecule
Almost instantaneous	Infrared, Raman	≈ 10-14- 10-13 s, single molecule - local cluster
Vibrationally averaged	Models, Pump-probe laser	≈ 10-12- 10-10 s, single molecule - local cluster
Diffusionally averaged	Neutron scattering, NMR shift	≈ 10-9-10-6 s, local cluster
Probability	X-ray diffraction, thermodynamics	> s, local cluster

 

The structuring found depends on the time-scale of the investigating methodology [2712]. Instantaneous structures show the actual positions and orientations of molecules, but these will be disordered relative to each other due to vibrational and linear and rotational diffusional motions; thus they will not show, or will show only poorly, any long-term or extensive order in the system. Slow processes, such as diffraction, show 'averaged' structures that may be a very poor representation of any structuring consisting of two rapidly interchanging structures. ^e Diffraction data shows long-term order but cannot determine whether the order is due to the same molecules being present in the same positions throughout or whether there are simply 'favored' positions with the molecules exchanging rapidly between them.

 

Attempts are being made to combine different techniques (X-ray absorption and emission, NMR shielding and X-ray diffraction), with different time dependencies, to best fit with various models [3716]. [Back to Top ]

Dielectric spectroscopy

Dielectric spectroscopy [950, 3359] measures water tumbling and can detect water quantitatively in different clusters if they are held by different average hydrogen bond strengths. The dielectric loss is determined over a wide frequency range (kHz-GHz). Slow tumbling (≈ ns) is due to tightly bound water whereas fast tumbling (≈ ps) is due to loosely bound water. Shown opposite is an example spectrum of a protein solution [785], where the non-bulk water dielectric loss spectrum (red) can be divided into three underlying Gaussian peaks. ^c Difficulty remains in unambiguously assigning the molecular structures that produce these peaks. Dielectric spectroscopy does seem to show that long-range interactions (≈ 0.1 mm) are present in water [1630]. [Back to Top ]

Diffraction methods

Diffraction methods make use of Bragg's law, involving the constructive interference of scattering (wavelength λ ≈ 1 Å, integer n) from regularly spaced atoms (shown as blue dots with spacing d, see right). Neutron diffraction determines nuclear atomic coordinates whereas X-ray diffraction refers to the maxima of the electron density distribution. The timescale of X-ray and neutron scattering is very brief. Thus, diffraction methods [2553] involve the sum of a large number of instantaneous data from the many effectively frozen configurations of the sample. Thus, they measure the time-averaged distances between the atoms of water molecules. However, they cannot directly determine the cluster geometry. Also, the long-range hydrogen-bonded structure and correlations in liquid water cannot be determined by diffraction as there is no long-range positional order of molecules in a liquid, so any such conclusions drawn are meaningful only on the short scale (< ≈ nm).

 

X-ray diffraction (~10 keV) is sensitive to the concentration of electrons (so not sensitive to isotopes). For disordered materials, like liquid water, the diffraction pattern is first given in terms of the scattering vector (see left) and then translated into a derived radial distribution function. Even though there are twice as many hydrogen atoms as oxygen atoms in water, these electron-poor atoms contribute little (hydrogen atoms have ≈ 1/13 the electron density of the oxygen atoms and 2 ˣ 1/13 ≈ = 0.15), with ≈ 85% of the data being provided by the g_OO(r) radial distribution function and the remainder from the g_OH(r) radial distribution function [1024] (H - H interactions being less than 0.6 %). Recently methodology for detecting hydrogen atoms and X-H bonds has been developed, and this may be of great use in the future so long as the temperature is below 140 K, and that the resolution is at least 0.6 Å [2604].

 

Neutron diffraction utilizes the wave property of slow neutrons produced in an atomic reactor (≈ 0.1 eV, ≡ ≈ 1 Å). It is sensitive to nuclei (including isotopes) and can be used to determine the positions of light atoms, such as hydrogen.The strength of the interaction between the neutron and a nucleus is called the scattering length which is specific to each isotopic nucleus (b, in femtometers, fm). Interference terms in the scattering from pairs of atoms give rise to coherent scattering, whereas scattering from all individual atoms gives rise to incoherent scattering. For water, neutron diffraction needs mixtures of D₂O, HDO, and H₂O as H atoms are strong incoherent scatterers whereby the scatter contains no structural information. Such mixtures produce data with less precision and accuracy as such mixtures structure water differently from pure H₂O.

Selected neutron scattering lengths and cross sections ^g

Isotopic nucleus	bcoherent , fm	bincoherent , fm	cross sectioncoherent , b	cross sectionincoherent , b
1H	-3.7406	25.274	1.7583	80.27
2H	6.671	4.04	2.05	7.64
12C	6.6511	0	5.559	0
13C	6.19	-0.52	4.81	0.034
14N	9.37	2.0	11.03	0.5
16O	5.803	0	4.232	0
31P	5.13	0.2	3.307	0.005
32S	2.804	0	0.9880	0

 

Neutron diffraction gives g_OO(r), g_OD(r) and g_DD(r) radial distribution data, which does allow some, if incomplete, orientation information to be extracted for neighboring water molecules. The results generally need much interpretation and help from modeling [2405]. Thus even crystalline ice Ih gives rise to artifactual peaks, see the O···O peaks at 2.43 Å and 3.45 Å opposite. Also, the widths of the peaks are greater than might be expected for a crystalline structure. The major drawback of neutron diffraction is the cost of the nuclear reactor facility with its intrinsic risk and poor accessibility for research.

 

Electron diffraction can only be used on thin films due to its strong scattering. Recent wide-angle X-ray diffraction has confirmed the existence of two distinctly different hydrogen-bonded environments in liquid water [1755].

 

The use of diffraction methods for studying the hydration structure of alkali ions has been reviewed [1062]. Diffraction data may be refined by fitting water model simulations [1245]. The resulting fit, however, is insensitive to the model chosen [888] such that optimizing the model to the fit does not, perforce, produce a better model. Diffraction results are discussed further elsewhere. Diffraction methods give data little different from bulk water beyond the first hydration shell where other methods indicate changed structure [1427].

 

Diffraction methods are excellent for testing models of water structure but less useful for delivering the real structure of water due to the difficulty in obtaining orientation information and correlations at intermediate distances. However, recently as the instrumentation has improved, the local dynamics of water in real space and time, such as viscosity and diffusion, has been visualised by inelastic X-ray and neutron scattering measurements. It makes use of the Van Hove function (VHF)

 

where r_i(t) is the position of the atom i at time t, i = 1, 2, . . . N, and ρ₀ is the atomic number density of the system [3741]. [Back to Top ]

Modeling

Many molecular models for water are described on another page of this site. An underlying assumption of these models is that the structure of liquid water may be modeled using just two-body effects (that is, molecule A affects molecule B independently of molecule C affecting molecule B) and do not include any contribution from any covalency in their interaction. Models for liquid water have, so far, proved to be poor predictors for physical and molecular properties outside those used in their development and parameterization. Modeling studies may result in a picture of an 'average' liquid water molecule, which may be misleading as such 'average' structures do not exist in most ice phases and may not be relevant to liquid water. [Back to Top ]

Nuclear Magnetic Resonance (NMR)

Water contains three isotopes of use in NMR. ^d

 

Properties of hydrogen nuclei

NMR, H2O atoms	1H	2H	17O
Spin states	+½, -½	1, 0, -1	5/2, 3/2, 1/2, -1/2, -3/2, -5/2
Relative sensitivity at natural abundance	1	1.4 ˣ 10-6	1.1 ˣ 10-5
Gyromagnetic ratio, rad ˣ T-1 ˣ s-1 *	26.7522 ˣ 107	4.1066 ˣ 107	-3.6281 ˣ 107
Frequency at 2.3488 T (MHz) *	100	15.3506	13.5565

 

The ¹H and ²H NMR spectra can be used to quantify the deuterium atom content of H₂O, HDO, D₂O mixtures from the natural abundance to nearly 100% [1817]. As it has been reported that magnetic fields may alter the clustering of water ([192, 485, 647]), then the results of NMR must be considered with this possibility in mind. Another possible weakness to the use of NMR is that only ortho-water is NMR-active, para-water having no intrinsic magnetic moment [835]. However, all unpaired protons are NMR-active (not just ortho-water), and the exchange of protons may be a rapid compared with gathering the NMR signal [2118].

 

NMR analysis generally gives reduced information due to proton exchange and movement of water between environments. The average proton residence time on a water molecule is about a millisecond at 25 °C and pH 7, being less than that at other pHs or higher temperatures. Therefore, only single, population-weighted averaged, ¹H, ²H or ¹⁷O NMR peaks are generally seen in aqueous solutions. The exception to this is where separate non-exchanging or very slowly exchanging pools of water molecules are found.

 

Water's NMR chemical shifts (¹⁷O and ¹H) are affected by ions in solution with increasing ionic radius generally causing an increased deshielding effect on the water oxygen atoms but a decreased deshielding effect on the water's protons [781]. The effects are linearly concentration-dependent with a change in the slope found above the concentration that causes a minimum in the specific heat (≈ 2-3 M) [556]. Here, the charge density effect (greater charge density causing greater deshielding), as seen in the division of ions into kosmotropes and chaotropes is less important than the ionic size effect [781]. As larger ionic size results in an increased number of first shell water molecules interacting with the ion's field within the clathrate surroundings, the apparent shielding change may be due to an increased effect on the average shift.

 

NMR spectroscopy is one of the most useful methods for experimentally characterizing H-bond interactions in biological systems [3347] and often provides the strongest evidence for any partial covalent character. The ¹H-shielding values of H-bond proton are determined by the depletion of electronic density around the bonding hydrogen atom, which stems from the H-bond formation. Both ¹H and ¹⁷O NMR peaks shift to downfield (greater ppm) with increasing hydrogen bond strength; in the case of ¹⁷O, the donor O shift increasing greater than the acceptor O reduces. However, only ‘averaged’ water environments may be obtained. Where several aqueous environments exist, the ¹H NMR spectrum (mainly just one broad peak) may be deconvoluted into Gaussian sub-bands due to differing degrees of hydrogen-bonded water with greater chemical shift (that is, ppm) indicating stronger hydrogen bonding and greater tetrahedrality of the water clustering [851].

 

The changes in the motion of interfacial water molecules on thawing their ices, detected using NMR, can characterize the dynamic structure of proteins so adding considerably to the information available from protein 3-D structural determination [2249]. NMR has been used to estimate changes in water's hydrogen bond lengths and strengths [458]. Such investigation makes use ab initio calculations to interpret changes in the longitudinal relaxation τ₁ (see below) of HDO in D₂O solution by assuming (somewhat incorrectly) that this mixture has a similar structure to pure H₂O.

 

The NMR signal undergoes two relaxation processes; spin-lattice (longitudinal) relaxation resulting in the decay of the spin distribution (τ₁) and spin-spin (transverse) relaxation (τ₂) resulting in the loss of the spin coherence. τ₂ is inversely proportional to the width (at half height) of the NMR peak. τ₁ is 3.6 s at 25 °C due mostly to dipole-dipole couplings [430]. As τ₁ reduces with increasing dissolved molecular oxygen concentration (due to its paramagnetic nature) [782] and dissolved oxygen concentrations are expected to be greater in more hydrophobic environments near hydrophobic surfaces, dissolved oxygen is capable of giving rise to artifacts in the NMR spectra of water. Such effects may change over a period of several days [294].

 

Shorter τ₂ values indicate less molecular mobility. The ¹H τ₂ of water at 25 °C, viscous water and ice at 0 °C are about 3 s, ≈ 1 ms, and ≈ 5 μs respectively. However, the T₂ relaxation times are also reduced by exchange reactions with, for example, proteins carboxyl, hydroxyl, and amine groups. Generally only single, population-weighted averaged, ¹H, ²H or ¹⁷O relaxation rates are seen in aqueous solutions. The exception to this is where separate non- (or very slowly) exchanging pools of water molecules exist such as occur in porous media including biological tissue and many foodstuffs. τ₁ and τ₂ are approximately equal for liquid water, both dropping with increased hydrogen-bonded clustering (for example, increased viscosity) under normal conditions. However, as the degree of tetrahedral clustering increases further, τ₁ increases asτ₂ decreases [581]. τ₁ thus shows a minimum at intermediate viscosity (rotational correlation time), ^f at the NMR resonant frequency, whereas τ₂ decreases continuously with increasing viscosity. Note that NMR relaxation is a non-equilibrium kinetic event and cannot give thermodynamic quantities such as water activity.

 

 

Nuclear Overhauser effects (NOEs) may be used to identify relatively static water molecules within the first hydration shell (<≈ 0.5 nm) around biological molecules [784].

 

Apparent diffusion rates of water are measured using magnetic field gradients [652,2112] and may be incorporated into magnetic resonance images (MRI). As diffusion is restricted in more extensive aqueous clusters, this method can differentiate different pools of water by their mean diffusion rates. NMR relaxation measurements can give the rotational diffusion coefficients using the natural ¹⁷O abundance [652, 2112]. [Back to Top ]

Osmotic stress

Osmotic stress [2006, 2113] is complementary to other physical techniques in that it probes the changes in hydration that accompany biologically relevant processes from their dependence on osmotic pressure (water activity). For example, the number of thermodynamically unique water molecules associated with the double helix and released from the single strands upon melting may be determined from the depression of melting temperature upon decreasing the water activity [2113]. [Back to Top ]

Physical properties

Physical properties such as viscosity, compressibility, the speed of sound, heat capacity, refractive index, thermal conductivity, and surface tension can all be used to gather information concerning the structure of water. The interpretation of this data is not clear-cut, however. Note that only the surface properties, not the bulk properties, are discovered using surface tension. The surface structure of water is very different from the bulk structure and may mislead. [Back to Top ]

Further confusion of the Raman spectroscopy is that the peaks in the 3000-3700 cm^-1 range are affected by the Raman excitation wavelength [1536]. In particular, the component at ≈ 3200 cm^-1 is in resonance with visible red light from a vibrational overtone.

 

Assignments from different authors

Peak a	Water's assigned hydrogen bond molecular linkages
0	unlinked	1 bifurcated b, 1 trifurcated c	2 H-bond, donors broken	3636 cm-1, unlinked
1	1 H-bond	2 bifurcated	2 H-bond, acceptors broken	3572 cm-1, 3 H-bond, acceptor broken
2	2 H-bond	1 O-H H-bonded, 1 bifurcated	3 H-bond, donor broken	3430 cm-1,2 H-bond, acceptor and donor broken
3	3 H-bond	Both O-H are weakly H-bonded	3 H-bond, acceptor broken	3220 cm-1, 4 H-bond
4	4 H-bond	Both O-H are strongly H-bonded	4 H-bond	3014 cm-1, 3 H-bond, donor broken
Ref.	[786]	[573]	[699b]	[1925]

 

A thorough and careful analysis of the combination band centered at about 5260 cm^-1 failed to distinguish the substructures that correlate with the vibrational sub-bands [909].

 

The use of a pump-probe laser using 70-fs pulses at 3350 cm^-1 (O-H stretch) and detecting the resultant molecular vibrations within the hydrogen-bonded clusters over a range of frequencies with time shows efficient energy redistribution [750]. However, this method is not yet able to show structuring details. Stimulated Raman excited fluorescence (SREF) microscopy using an IR fiber laser at 1031.2 nm has been developed into a water-sensing tool by coupling it with vibrational solvatochromism using Rhodamine 800. This technique has allowed the direct visualization of the spatially-resolved distribution of water states inside single mammalian cells [3787].

 

Two-dimensional Raman spectroscopy has been shown to provide information concerning water's intermolecular dynamics and may eventually provide data on the hydrogen bond network rearrangements responsible for water's anomalies [1097].

 

Care must be taken if the spectra of solutions are obtained by subtracting baseline Gaussian-like curves. This may give rise to peaks and troughs in unexpected non-physical positions (see below). In (A) the base peak (solid black line, given at 3000 cm^-1 is shifted by the new conditions (solid red line) by -50 cm^-1. The result of the subtraction of these peaks (dashed red line) gives a new peak (at -70 cm^-1) in a position different from the peak, and a trough at +20 cm^-1 with no obvious physical cause. In (B) the base peak (solid black line, given at 3000 cm^-1 is broadened (solid blue line) or narrowed (solid red line) by the new conditions. The results of subtraction of these peaks (dashed lines) gives new peaks and troughs with no obvious physical cause. The position of the peak has not shifted, but its subtracted curve may give a peak or a trough in this position plus neighboring peaks or troughs (see below).

 

The results of subtracting Gaussian curves.

Terahertz (THz) absorption spectroscopy

Terahertz (THz) spectroscopy, two-dimensional Raman-THz spectroscopy [2050] (0.3 - 20 THz ≡ 10 - 600 cm^-1), and THz-FTIR spectroscopy may be used to investigate hydrogen bond network resonances, hydration dynamics, and water dynamics on the picosecond timescale. THz absorption spectroscopy probes the reorientation of permanent and induced dipole moments of clusters and, from the time-dependence of these fluctuations, it is able to determine the frequency and strength of the cluster motions [1783], and the effect of a solute on the collective motions of the hydrogen-bonded network. In particular, the librational resonances of liquid water, that reflect water's hydrogen-bonding, can be studied. Such spectroscopy can also be used to investigate the hydration layers around biomolecules [1196, 1427] and how they change during biological processes [2237]. This spectroscopy is powerful for examining changes in water mobility and shows changes in the water structuring at greater distances from biomolecules than those found using NMR or diffraction methods [1368, 1811]. The hydration state of solutions may be examined using terahertz time-domain attenuated total reflection spectroscopy, which makes use of the complete complex dielectric function; both the dielectric loss, due to relaxation motion of water, and the changes in the real part of the relative permittivity (dielectric constant) [1437]. [Back to Top ]

Microwave (GHz) absorption spectroscopy

Rotational spectroscopy can be used in the gas phase for small water clusters. For example, broadband rotational spectroscopy has unambiguously identified the cage, prism, and book isomers of the water hexamer (H₂O)₆ [2260].

Sum frequency generation

Sum frequency generation (SFG) and second harmonic generation (SHG) surface spectra are spectroscopic methods for investigating the surface of water without interference from the bulk liquid [2627]. The output spectra derive from a summed combination of input beams when the infrared frequency is in resonance with a vibration within the interface (for example the O-H vibration of water, see right). As both the bulk aqueous solution and the gas phase are isotropic, they are not active in this spectroscopy, leaving just the anisotropic interfacial layer to provide the spectra, due to a net orientation of surface molecular dipoles.

 

 

 

 

The depth of this anisotropy governs the depth probed but is not independently determined. It may extend out to the limit of the wavelength or Nernst layer although mostly affected within nanometers of the interface with a definite bias towards the very outside of an aqueous interface (containing a lower density of water molecules) rather than towards the bulk. As the spectra are acquired over periods of seconds, it is of interest that the surface monolayer may be exchanged millions of time during this period [2408]. The result is that any spectral features are formed from a complex combination of spectra from within this ever-changing environment. The surface has the largest effect as can be seen from the SFG spectrum of the water-vapor interface shown left where the surface-dangling OH groups give the largest peak at 3700 cm^-1.

 

A version of this second harmonic generation methodology uses soft x-rays (between 100 eV and 1 keV) [3305]. This gives high interfacial specificity of the first few molecular layers of an exposed bulk sample or either side of a buried interface.

X-Ray spectroscopy (≈ 500 eV)

X-ray-based techniques can be used to study the hydrogen-bond network of liquid water [1965, 2623]. As the techniques have attosecond timescales, they reflect the instantaneous molecular configuration of the water molecules. Difficulties in the interpretation of these techniques are due to the lack of a robust theoretical understanding [2623].

 

X-Ray absorption spectroscopy (XAS) involves the absorption of high-energy photons to excite core (1s, 2s) electrons to unoccupied p-character valence levels. (see the molecular orbital energy levels left). The energies required are sensitive to the local environment mostly providing information concerning the hydrogen bond donating orbitals of the absorbing molecule at very short timescales (≈ 10^-18s) [373, 690a , 1757]. The experimental XAS transition energies [3370] are

1a₁ 4a₁      534.0 ev

1a₁ 2b₂      535.9 ev

 

The energies given left are those calculated for isolated molecules using the Restricted Hartree-Fock wave function (RHF) using the 6-31G** basis set. Some electron movements are forbidden, such as 2a₁ to half-empty1a₁ hydrogen-bonding affects the energy levels.

 

XAS has been used to investigate the hydration of concentrated salt solutions [2170]. At high photon energy, expelled electrons have sufficient energy to escape from their atom (to the continuum) and at higher energies, these escaped electrons may produce signals due to backscattering. Care must be taken to avoid artifacts at the water-air surface due to desorption [1440]. Soft X‑ray (between 100 eV and 1 keV) absorption spectroscopy has been used on liquid microjets and flow-cells. This has enabled the general study of liquids and solutions [3306].

 

X-ray Raman scattering (XRS) gives similar results to XAS and involves the inelastic scattering of x-rays from the core electrons.

 

Shown right is the O K-edge (1s) XAS of liquid water (red) with possible underlying Gaussian (blue) and continuum (green) contributions.

 

The first peak in the X-Ray absorption spectrum (right) at about 535 eV (pre-edge) is due to excitation from the 1s orbital to the vacant O-H 4a₁ antibonding orbital. It concerns water molecules with broken and weakened H-bonds. The post-edge bulge (~541 eV) is due to tetrahedral water molecules with intact hydrogen-bonds. Assignment of these Gaussian peaks to electron transitions depends heavily on the underlying theory (for example, see the current dispute) as there is little structure in the experimental spectrum, this spectrum varies from run to run [690c], between methods and between different energies of the X-rays.

 

Thus, data fitting is error-prone. An interesting discussion of these differences has been published [834].

 

Related to XAS is the X-ray emission spectra (XES, shown right, [1757]) involving transitions from the three outermost occupied molecular orbitals 1b₁, 3a₁ and 1b₂ to fill the vacancy (produced as above) in the 1a₁ (1s) orbital [411]; peaks occurring in the 520-528 eV range and corresponding to instantaneous local structuring (≈ 3-4 ˣ 10^-15 s). Recent X-ray emission spectroscopy shows the presence within liquid water of two well-separated peaks (split from the lone-pair non-bonding 1b₁ peak) that interconvert but do not broaden with changes in temperature [1639]. This strongly supports a two-state network model such as presented here, although it has been proposed that it may be alternatively due to ultrafast dynamics during the core-hole lifetime [2196].

 

Another recently improved X-ray technique is small-angle X-ray scattering (SAXS, [1757]) which is sensitive to density inhomogeneities in the nm-range has confirmed the existence of two distinctly different hydrogen-bonded environments in liquid water [1757].

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Footnotes

^a Consider, for example, the very different photographs obtained at night of cars on the road between the use of fast flash and a natural light long exposure. Short exposures show the cars but not the roads; long exposures show the roads but not the cars. [Back]

 

^b There is evidence, however, that the spectra may be due to a continuous variation in the hydrogen bonding and not due to discrete but overlapping populations [875]. [Back]

 

^c Gaussian curve. A Gaussian curve (normal distribution curve) may be represented by the relationship:

 

 

where σ is the standard deviation and b is the offset.

 

therefore, for vibrational peaks,

 

where A is the amplitude, A_max is the peak height, λ is the wave number, λ_max is the wavenumber at the peak and W is the width of the peak at half height, and the standard deviation (s) of the distribution s = W/(2 ˣ Ö(2 ˣ Ln (2)). Spectroscopic peaks can therefore be reduced to just three variable parameters, the position, the height, the width at half height.

 

The Gaussian curve assumes there are no outliers. It should be noted that although there is no theoretical reason why the actual sub-peaks should correspond exactly with Gaussian-shaped peaks, they generally fit well.

 

Lorentz distribution. (sometimes called the Cauchy distribution) The spectral distribution is sometimes better represented as a Lorentzian function

where A_max is the maximum peak height, λ is the wavenumber, λ_max is the position of the maximum wavenumber, and ω_L is the full width at half maximum height.*

 

* R. J. Meier, On art and science in curve-fitting vibrational spectra, Vibrational Spectroscopy, 39 (2005) 266-269.

[Back, 2]

 

^{d 16}O and ¹⁸O have zero nuclear spins. [Back]

 

^e As an example of this, consider the 'average' structure of a person; nothing like a man or a woman, but somewhere in between. [Back]

 

^f The rotational correlation time is the time it takes the average molecule to rotate one radian. It may be determined by dielectric spectroscopy. [Back]

 

^g V. F. Sears, Neutron scattering lengths and cross sections, Neutron News, 3 (1992.) 26-37. [Back]

 

^h Raman vs Infrared spectroscopy

 

Infrared spectroscopy requires that the molecular dipole moment is affected by the vibration. In Infrared (IR) spectroscopy the electric vector of the rediation field oscillates with the same frequency as does the molecular dipole moment.

 

Raman spectroscopy requires that the molecular polarizability is affected by the vibration. In Raman spectroscopy the molecule is exposed to an electric field so forcing the electrons and nuclei to move in opposite directions. and inducing a dipole moment. This dipole moment is proportional to the electric field strength and to the molecular polarizability. [Back]

 

 

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