Lab 9
NMR Spectra, EPR Spectra, Other Properties and Newer ADF

9.1 Overview

Note: for this lab you will be using ADF 2008. To use this version of the program you will need to use the command
source /home/programs/ADF/adf2008/profile.01.sh
rather than the usual source command involving ADF 2006

NMR and EPR spectroscopy involve the interaction between spin angular momentum and a magnetic field. In the case of NMR spectroscopy it is a nuclear spin and in the case of EPR an electron spin. The interactions between the spins and the magnetic field are in themselves not so interesting. This kind of Zeeman interaction has been known for a long time. What is interesting is that the spin-magnetic field interaction is very sensitive to the chemical environment that the spin finds itself in. Thus NMR and EPR spectra can provide a wealth of information about the electronic structure, the geometry and more of the molecule of interest.

In recent years a lot of work has gone into figuring out theories of how NMR and EPR spectroscopy can be described with DFT. These theories have been included into a number of computational chemistry programs including ADF. In this lab we will look at several aspects of NMR including shielding constants, chemical shifts and spin-spin coupling. On the EPR side, we will consider the EPR g-value and the magnetic hyperfine coupling constant A.

In addition the important properties considered in this lab and earlier labs theoreticians have developed methods for calculating other molecular properties and spectroscopic techniques that are perhaps less well known but are still of interest. In this lab we will also consider some of the other properties/spectroscopic techniques available in ADF.

9.2 NMR Spectroscopy

It is beyond the scope of this lab to consider the theory of NMR (or EPR). It is assumed that you have some idea of what is going on in an NMR experiment and that you are familiar with chemical shifts, spin-spin coupling, the g-factor and the hyperfine splitting of an EPR spectrum. A few points about NMR and EPR spectroscopy will be made when needed.

9.2.1 NMR Shielding Constants

Many nuclei have spin and therefore can interact with a magnetic field. When a molecule is placed in a magnetic field its electron distribution reacts to that field and shields the nuclei. This shielding of a particular nucleus depends on the chemical environment that nucleus is in making the chemical shielding a very useful quantity to measure.

9.2.2 NMR Shielding Constants and ADF

Figure 9.1: The ADFINPUT options for NMR shielding constants.

The options to control the calculation of NMR properties can be accessed through the Properties: NMR menu option. Doing so will produce a series of options (see figure 9.1). By default, nothing is selected (don't calculate any NMR properties). To calculate NMR shielding constants, one or both of the first two buttons must be selected. For the moment we will only be concerned with the isotropic part (figure 9.1).

All nuclei in a molecule experience the applied magnetic field and will have an associated shielding constant. The amount of effort required to calculate shielding constants increases with the number of nuclei that we are interested in. Often, we will be interested in the shielding constant of a subset of the nuclei since an experiment is set up so that only the part of the spectrum due to one element is measured (proton NMR, ¹³C NMR, ³¹P NMR etc). Thus, you must choose which nuclei you will be investigating. By default, the program will calculate the shielding for no nuclei. You need to choose at least one nucleus to get something. To choose a nucleus or nuclei you first select them in the build window of ADFINPUT. You will then be able to add them to your list of nuclei for which to calculate shielding constants by clicking on the Add button above the NMR shielding window (figure 9.1). If you made an error the Remove button will help you fix it up. Once this is done you are ready to calculate shielding constants.

Figure 9.2: NMR shielding constants in the logfile.

Unlike most the other spectroscopic methods we have considered so far we cannot visualize the results of an NMR shielding constants (or any NMR or EPR related calculation for that matter) with the SPECTRA program. We simply must look at the numbers calculated and go from there.

The results of an NMR shielding constant calculation can be found in two ways. The shielding constants can be found at the end of the logfile that is printed out once your calculation is finished (figure 9.2). Alternatively, you can open up the detailed output with the SCM: Output command and then go to the NMR shielding constants with the Other Properties: NMR Shielding (NMR Program) menu item. This section includes some analysis of the shielding constant but the final answer for each nucleus is labelled "Total Isotropic NMR Shielding." (figure 9.3)

Figure 9.3: NMR shielding constants in the detailed output.

Note that many of the NMR and EPR properties are not calculated by ADF itself but by other related programs that receive information from ADF (orbitals, geometry etc) and calculate the spectroscopic constants. This will actually make very little difference to you as ADFINPUT will handle the running of all required programs as soon as you choose a property for calculation. You should keep this point in mind when looking for the results however.

Exercise 9.1: Calculate the shielding constants at all nuclei in the water molecule and the methane molecule. Use the BLYP functional and a TZ2P basis set. Compare your results with the experimental values in table 9.1 Is is necessary to select all nuclei in H₂O and CH₄? Why or why not?

Molecule	Nucleus	Shielding Constant (ppm)
H₂O	O	344.0
	H	30.1
CH₄	C	195.1
	H	30.6

Table 9.1: Experimental shielding constants for H₂O and CH₄.

9.2.3 Shielding Anisotropy

Figure 9.4: The NMR shielding tensor in the detailed output.

NMR spectra are most often run on compounds in solution. When this is done the molecules of the compound are tumbling over and over in the solution and the measured NMR shielding constant is an average of the value of the constant at all possible orientations of the molecule with respect to the applied magnetic field. We see an isotropic shielding constant.

The shielding of a nucleus will change if the orientation of a molecule changes with respect to the applied magnetic field. Sometimes the molecules are not free to rotate, most notably when they are in the solid state. If the substance is a crystal then all of the molecules present will be pointing in the same direction and it is possible to orient that crystal in the magnetic field and measure the shielding as a function of orientation. If the substance is a powder then all possible orientations are present but the molecules are not rotating and a wide powder pattern is observed. The dependence of the shielding on orientation is described by a special type of 3 by 3 matrix called a tensor (or, more precisely, a first rank tensor).

Often what is measured experimentally is the shielding anisotropy or the difference between the largest and the smallest shielding constants observed as the orientation is changed (or from the powder pattern).

If a molecule has a threefold or higher rotational axis then the extreme values of the shielding constant will correspond to along the threefold or higher axis (the parallel component) and perpendicular to the threefold-or-higher axis (the perpendicular component).

This discussion concerning orientation dependence also applies to several of the other spectroscopic parameters that we consider in this lab (chemical shifts, spin-spin coupling parameters, g-factors and A_iso). All of these properties are orientation dependent and can be described by a tensor. Except for shielding constants we will only consider the isotropic values to prevent the length of the lab getting too out of control.

9.2.4 The Shielding Tensor and ADF

Choosing to run a calculation where you evaluate the shielding tensor is similar to a calculation of the isotropic value except that the second button is pressed in the Properties: NMR options (see figure 9.1). Note that for most of the other spectroscopic parameters considered here (spin-spin coupling constants, g-factors and A_iso we always calculate both the isotropic and tensor values.

The shielding tensor is not printed out in the logfile so you must access the detailed output to find it. In the output you will find the shielding tensor through the Other Properties: NMR Shielding (NMR program) again. (figure 9.4). The tensor depends on the orientation of the molecule relative to the coordinate system. There is always one orientation, called the principle axis representation where the tensor is diagonal. This diagonal representation is the one we want for evaluating anisotropies.

The anisotropy is defined as the difference in shielding between the largest and the smallest of the diagonal values of the shielding tensor.

Exercise 9.2: Evaluate the shielding anisotropy at the carbon atom of CO₂. Evaluate the anisotropy at the carbon and nitrogen atoms of HCN. Use a TZ2P basis set and the BLYP functional. Compare your results to the experimental data in table 9.2.

Molecule	Nucleus	Shielding Anisotropy (ppm)
CO₂	C	344.0
HCN	C	280
	N	564

Table 9.2: Experimental shielding anisotropies for H₂O and CH₄.

9.2.5 Chemical Shifts

It is all very well to be able to calculate shielding constants but what is normally talked about in NMR experiments are chemical shifts. A chemical shift is the shielding constant relative to the shielding constant of a reference molecule and nucleus rather than the absolute values that we have been using so far. Experiments are carried out this way simply because it is much easier to measure chemical shifts than shielding constants. A shielding constant is usually written as s and the chemical shift as d. The chemical shift is defined as

d = s_ref-s

(9.1)

where s_ref is the shielding constant of the reference compound and s is the shielding constant of the molecule of interest. It's worth noting that this definition means that the larger the shielding constant a nucleus has the smaller its chemical shift will be.

Exercise 9.3: Tetramethylsilane (TMS) is a common reference in ¹³C and ¹H NMR. The C and H isotropic shielding constants of TMS calculated at the TZ2P/BLYP level by the T.A. are 178.6 ppm and 31.6 ppm respectively. Calculate the ¹³C and ¹H chemical shifts of ethene and methane. Use the TZ2P basis set and BLYP functional.

9.2.6 Spin-Spin Coupling

The spin of a nucleus will not only interact with the applied magnetic field but also the magnetic field created by the spin of nearby nuclei. In the NMR of organic compounds this is most often seen as the characteristic splitting of a ¹H NMR signal into a number of peaks depending on the number of viscinal hydrogen atoms.

The size of the splitting is characterized by a spin-spin coupling constant. This constant can be negative or positive though it is usually difficult to determine its sign experimentally.

It should be recalled that spin-spin coupling is mutual. If one nucleus creates a splitting of a certain size in the NMR signal of a second nucleus then the signal of the first nucleus will be split by the same amount by the second nucleus.

From a theoretical point of view, spin-spin coupling constants are usually described as being made up of four contribution called the Fermi-contact (FC), diamagnetic spin-orbit (DSO), paramagnetic spin-orbit (PSO) and spin-dipolar(SD) terms. The Fermi-contact contribution is usually the most important though not always. The PSO and particularly the SD terms can take a lot longer to calculate than the other two.

The other spectroscopic properties also can be analyzed in terms of contributions but we will not consider that here.

9.2.7 Spin-Spin Coupling Constants and ADF

Figure 9.5: Choose to calculate some spin-spin coupling constants.

Like shielding constants the main control of the calculation of spin-spin coupling constants is through the Properties: NMR menu option (figure 9.5). To choose to calculate spin-spin coupling constants, click the button next to "NMR spin-spin coupling constants". You now need to choose some perturbing and some responding nuclei. Spin-spin coupling arises from the interaction of two nuclear spins. Any nucleus in the list of perturbing nuclei will interact with all nuclei in the list of responding nuclei. Thus, if you have four nuclei, A, B, C and D and A and B are perturbing nuclei and C is a responding nucleus then you will get spin-spin coupling constants between A and C and between B and C but not between A and B or D and any other nucleus. This somewhat weird way of specifying a calculation is because of the way the spin-spin coupling constants are calculated. There are other technical reasons why it might be better to make a nucleus perturbing or responding but we won't go into them here.

By default, the calculation of spin-spin coupling constants will only include the contribution from the Fermi-contact term. To control which terms are included go to the Details: NMR menu option (figure 9.6). Here you can choose which contributions you want through the nice buttons.

Figure 9.6: Choosing the contributions to spin-spin coupling.

Much like the case with shielding constants, the results of a spin-spin coupling constant calculation are found at the end of the logfile and in the detailed output. The appropriate command for the OUTPUT program is the Other Properties: NMR spin-spin coupling option.

Two values are quoted for each constant, k and j. The value j is in MHz and represents the splitting that would be observed in a spectrum given the presence of the most common nuclear isotopes. The constant k is the splitting with the contribution due to the nuclear magnetic moment removed ie, independent of the isotope chosen.

Exercise 9.4

Use a DZP basis set and the BLYP functional for this exercise. Optimize the geometry of ethane. Calculate all possible spin-spin coupling constants of ethane. Assume that the methyl groups are not rotating rapidly. Use the minimum number of responding and perturbing nuclei. The symmetry of ethane means that many of the constants are the same by symmetry. Calculate the spin-spin coupling constants in three different calculations

With just the Fermi-contact contribution.
With the Fermi-contact, diamagnetic spin-orbit and paramagnetic spin-orbit contribution.
With all four contributions.

Compare the calculated spin-spin constants and computational time of the three calculations. Note: Since the spin-spin coupling constants are calculated by a seperate program when you are looking for the timings you must be sure to select CPL from the Include menu of the OUTPUT program or you will be looking at the timings of the ADF program.

9.3 EPR Spectroscopy

9.3.1 The g-factor

Just like nuclear spins, electron spins can interact with an applied magnetic field. Early in the 20th century it was noticed that electrons have intrinsic angular momentum (spin) and that this angular momentum is quantized to one of two possible values described by the quantum number m_s. The energy of an electron with quantum number m_s in a magnetic field was found to be

E_Z = m_B gm_s B

(9.2)

where m_B is the Bohr magnetion, B is the applied magnetic field and g is the electron's g-factor. This g-factor is usually approximately 2. For a free electron it is 2.0023192778. This particular value is usually labelled g_e. It turns out that if the unpaired electron is in a molecule then the observed g will be different from g_e. The difference is characteristic of the chemical environment that the electron is in. Thus, measuring g for a particular molecule can give insight into the properties of that molecule. The quantity that is usually measured in an experiment is the transition corresponding to Dm_s=1 so the observed signal is normally at an energy of m_B g B. For technical reasons the derivative of the intensity with respect to energy is usually what is plotted resulting in the characteristic spikey appearance of EPR spectra.

9.3.2 ADF and g-factors

Figure 9.7: Choosing to calculate the g-factor.

The calculation of EPR parameters with ADF is a little complicated because of some involved options that must be chosen. Be careful that all options are set correctly and beware of crashed calculations caused by incompatible choices.

The first thing to note about EPR spectroscopy is that a molecule will have an EPR spectrum only if it has one or more unpaired electrons. In the context of this lab this means that whenever we are calculating EPR spectra we will be running an unrestricted calculation and the spin polarization must be set.

We can choose to calculate a g-factor by selecting the Properties: ESR, EPR, EFG menu option (figure 9.7). Exactly how to proceed from here depends on how relativistic effects are dealt with. If relativity is not included in your calculation or relativity is included through the scalar Pauli approximation then a g-factor calculation is chosen through the first button (figure 9.7). It is not possible to calculate g-factors if you choose the scalar ZORA approximation. If a spin-orbit ZORA calculation is being run then the second from bottom button must be chosen.

Figure 9.8: Choosing not to take advantage of symmetry.

Both g-factor options require an extra option.

The g-factors with the first option cannot be calculated in combination with symmetry. Normally, ADF will detect any symmetry that the molecule has and make use of that symmetry to speed up the calculation. To turn this detection off select the Details: Symmetry option (figure 9.8). The top option in the symmetry section allows you to choose what the symmetry of the molecule should be (this must agree with the symmetry of the molecule that you have built). The default choice for this option is AUTO or automatic detection. To run an EPR calculation this option should be set to NOSYM or no symmetry.

Figure 9.9: Choosing the collinear approximation in order to run a ZORA spin-orbit g-factor calculation.

In the second choice, an extra option pertaining to the spin-orbit coupling must be chosen. The spin-orbit options can be accessed through the Details: Relativity menu option. The option that needs to be chosen is Collinear (figure 9.9). Noncollinear can be chosen also but we won't consider that here. Exactly what collinear means is beyond the scope of this course. If you are interested you can look at F. Wang, T. Ziegler J. Chem. Phys 122 074109 (2005).

Once a calculation is finished, you can find the results in the detailed output. Open the OUTPUT program. Depending on how the g-factor was calculated choose either the Other Properties: ESR g-tensor (ESR program) (nonrelativistic or Pauli) or Other Properties: ESR g-tensor (ADF program) (ZORA spin-orbit) option. This will move to a part of the output with lots of numbers but the important number is next to "isotropic g-factor". For example, if the g-factor is calculated by the first method we obtain what is displayed in figure 9.10.

Figure 9.10: The results of a g-factor calculation.

Exercise 9.5: Calculate the g-factor of H₂O⁺. Use a TZ2P basis and the PBE functional. The experimental value is 2.010. How does your result compare with this?

9.3.3 Magnetic Hyperfine Splitting

The spin of an unpaired electron can interact with nuclear spins just as was the case in NMR where nuclear spins interacted with each other. Also much like NMR the interaction with a nuclear spin splits the peak in an EPR spectrum into several peaks with characteristic intensities depending on the number and spin of the perturbing nuclei. The separation between these new peaks (the hyperfine splitting) is also characteristic of the molecular environment and is parameterized by the hyperfine coupling constant A_iso where the iso emphasizes that this is the isotropic value.

9.3.4 Hyperfine Coupling Constants and ADF

Figure 9.11: The results of an A_iso calculation.

To choose to calculate A_iso we also make use of the Properties: ESR, EPR, EFG menu option. The second to last button controls the calculation of hyperfine splitting (figure 9.7). This is the same button that was used to calculate the ZORA spin-orbit g-factor. It is not necessary to run a ZORA spin-orbit calculation to evaluate A_iso. Thus, if the second button is chosen what is calculated will depend on what relativistic theory is chosen. A_iso is calculated no matter what level of relativity is chosen but g is calculated only if a ZORA spin-orbit calculation is performed.

The results of your calculation are found in the detailed output with the command Other Properties: ESR A-tensor. The values are given in MHz (figure 9.11).

Exercise 9.6: Calculate A_iso due to N and O in NO₂. Use a TZ2P basis set and the PBE functional.

9.4 Projects

Do project 1 and project 2 or project 3.

9.4.1 Other Properties

ADF is capable of calculating many other properties beyond those that we have discussed in this lab. In this project you will consider one of these properties. Build a molecule of methanol. Optimize its geometry using LDA and a DZP basis set. From the Properties menu choose one of CD, Hyperpolarizability, Magnetizability, Optical Rotatory Dispersion, Polarizability, Raman and VanderWaals. Run a calculation using that option. Many of the properties have a number of extra options that can be chosen. Consult the keywords list of the online manual at http://www.scm.com/Doc/Doc2008.01/ADF/ADFUsersGuide/page1.html to find out exactly what you are calculating and to decide what options you want to choose. Note that the properties "Magnetizability" and Öptical Rotatory Dispersion" can be found under the keyword ÄORESPONSE"

Answer the following questions.

What have you calculated? Describe the property or physical effect.
What values have you calculated?
How long did the calculation take?

9.4.2 The NMR Spectra of 2-methylcyclopropanone

Predict the solution ¹³C and ¹H NMR spectra of 2-methylcyclopropanone. Recall that experimental ¹³C NMR spectra usually have the proton spin-spin coupling decoupled. We will assume that all three hydrogens on the methyl group are equivalent. Use the BLYP functional and a DZP basis set. To save optimizing the geometry of this molecule, you can import the XYZ coordinates from file /home/seth/575/methylcyc_lab9.xyz.

Do not include the spin-dipole contribution to the spin-spin coupling unless you want to wait an hour for your calculation to finish rather than 10 minutes.

9.4.3 The EPR Spectrum of Nb⁺₂

Predict the solution EPR spectrum of Nb⁺₂. Naturally occuring Nb is 100% the ⁹³Nb isotope which has a nuclear spin of 9/2. The ground state of Nb⁺₂ is a doublet. Use the PBE functional, a TZ2P basis set and a small core size.

Lab 9 NMR Spectra, EPR Spectra, Other Properties and Newer ADF