Research: Science and Education Analyzing and Interpreting NMR Spin–Spin Coupling W Constants Using Molecular Orbital Calculations Jochen Autschbach* and Boris Le Guennic Department of Chemistry, State University of New York at Buffalo, Buffalo, NY 14260–3000; *[email protected] The pedagogical value of computational chemistry in the the bonds and lone pairs (and core shells) of the Lewis for- chemistry curriculum, in particular the value of first-prin- mula of a molecule. Unfortunately, they are not prominently ciples calculations of NMR chemical shifts (1, 2), has been displayed in most textbooks. previously pointed out in this Journal. Emphasis has been To the best of our knowledge, NMR spin–spin coupling put on providing recipes for addressing well-defined prob- computations and orbital analyses based thereupon have not lems computationally and on comparing experimental with yet been discussed in this Journal. We believe that the mate- calculated data. These are important ingredients that can en- rial will be most useful in a course about NMR spectroscopy able students to support their future (usually experimental) or as a computational exercise as part of a computational graduate research projects with their own calculations, to get chemistry or molecular modeling course. It is now possible a feeling about the accuracy that can be achieved, and to be to compute NMR spin–spin coupling constants rather accu- able to assess the quality of published theoretical data. Quan- rately with inexpensive DFT methods. This makes a study tum chemistry is now a component of many graduate research of small molecules attractive for use in a classroom or for projects. Introducing students to the power and the limita- homework. We hope that this article will be beneficial to a tions of quantum chemical methods, and to the theoretical wider range of courses since the concepts used here can be background, is therefore an important aspect of a modern applied to analyze other properties as well (see Appendix A, chemistry education. Table 8, in the Supplemental MaterialW for a list of such prop- NMR spectroscopy is one of the most important experi- erties). A computational analysis of trends in J couplings might mental techniques since it is able to provide useful informa- also be suitable as a computational undergraduate or gradu- tion about the electronic and geometric structures of a wide ate research project (e.g., if applied to a different set of mol- variety of systems investigated in organic and inorganic chem- ecules). Course prerequisites depend on the level at which an istry. Additionally, theoretical methods have been used ex- instructor teaches a particular course. Introductory physics and tensively to compute chemical shifts and spin–spin coupling some basic knowledge of NMR spectroscopy as acquired in constants from first-principles theory. (See refs 3–10 for a undergraduate organic chemistry classes will be essential. A selection of available review articles.) A comprehensive over- qualitative explanation of J coupling at this level is given in view of computational methods for magnetic resonance pa- the first half of the second section. Basic quantum theory as rameters has also recently become available (11). With the taught in undergraduate physical chemistry courses will be help of calculations, it is possible to simulate NMR spectra, required to make use of the quantum theory of J coupling, propose data for yet unknown substances, get detailed in- which is outlined in the Appendix in the Supplemental sight into the mechanisms that determine the experimental Material.W This theory supports the arguments regarding the outcome, and so forth. s character of bonds presented in the second half of the sec- Indirect NMR spin–spin coupling constants, also known ond section. However, more qualitative explanations might as J coupling constants, provide a wealth of structural infor- be chosen instead. We do not think that prior experience with mation as well as detailed insight into the bonding of an atom computational chemistry is necessary for the calculations pre- to its neighbors. In this article, we show how density func- sented in the third section, though of course it would be help- tional theory (DFT) calculations of J couplings can help to ful. Based on these calculations, we apply various analysis clarify the relation between such properties and the electronic concepts to the J coupling in ethane, ethene, ethyne, and ben- structure and bonding in molecules. This is achieved by the zene in the third section. Here, we make use of molecular or- analysis of J coupling in terms of contributions from mo- bital plots. Some experience with qualitative molecular orbital lecular orbitals (MOs), localized molecular orbitals (LMOs), theory will be required to use these graphics. and fragment orbitals (FOs). We should point out that such analyses for organic (12–17) and inorganic systems (18–22) Origin and Mechanisms of J Coupling are also an active research topic. Here, we apply an orbital analysis to one-bond carbon–carbon coupling, J(C,C), in For further details regarding the material presented in ethane, ethene, and ethyne. The CϪC coupling constants this section we refer to refs 23 and 24. in the aromatic molecule benzene as well as CϪH couplings for all molecules are also discussed. We show how a graphi- Magnetic Dipole Moments cally-oriented approach, combined with “hard” numerical Moving charged particles (electrons, nuclei) represent an data, can lead to a better understanding of the nature of J electric current, which is the source of a magnetic field. In coupling and of the orbital description of the electronic struc- particular, electric ring currents are the source of a magnetic ture of molecules. In addition, we use the concept of local- field equivalent to that of a magnetic dipole (Figure 1). In ized molecular orbitals and apply it to rationalize the turn, a magnetic field will induce electric ring currents1 with magnitudes of the CϪC coupling constants. Unlike the an associated magnetic dipole moment in a system of charged “usual” MOs, these orbitals are the theoretical equivalent of particles (atoms, molecules). 156 Journal of Chemical Education • Vol. 84 No. 1 January 2007 • www.JCE.DivCHED.org Research: Science and Education When considering a single rotating charged particle, the magnetic moment m is proportional to its angular momen- tum L. The constant of proportionality is the magnetogyric ratio γ, that is, m = γL. For instance, for the orbital rota- ᎑ ͞ tional motion of an electron, we have γe = e (2me), where e is the electronic charge and me is the mass of the electron. Spin and Magnetic Moment The property “spin” is an angular momentum of an el- ementary particle. Spin is often incorrectly interpreted as the particle’s intrinsic rotation. However, according to the previ- ous paragraph this analogy from classical physics illustrates Figure 1. A rotating particle with a charge q has an angular mo- why there must be a magnetic moment associated with the mentum L. The charge rotation also represents a ring current that is spin angular momentum of a charged particle. Atomic nu- the source of a magnetic field and a magnetic dipole moment. The clei with an odd mass number or with an odd number of magnetic moment m is proportional to L. The magnetic field is sym- neutrons have a nuclear spin angular momentum (usually la- bolized here by field lines. beled I, not L). Thus, nuclei with nonzero spin are tiny mag- nets. The orientation of the spin vector and therefore the orientation of the magnetic moment with respect to a refer- cific quantized amount of energy. This forms the basis of ence axis is quantized and described by the spin quantum nuclear magnetic resonance (NMR). number I. For a nuclear spin quantum number I, there are 2I + 1 possible orientations. The length of the I vector is Nuclear Spin–Spin Coupling and NMR [I(I + 1)]᎑1/2ប. The magnetic moment of a nucleus A is pro- Consider two magnetic nuclei A and B in a molecule. portional to its spin angular momentum IA and is given by Since nucleus A is the source of a magnetic field, the other nucleus B will behave like a “quantum compass needle” in mIAAA= γ (1) this magnetic field. This results in a set of quantized orienta- tions with quantized energy levels for this nuclear spin. A The constant of proportionality, γ , is called the nuclear mag- A phenomenological energy expression that describes the en- netogyric ratio. For atomic nuclei it is determined from ex- ergy of interaction between the two nuclei is periment rather than calculated theoretically. Electrons also have a spin angular momentum S with 1 an electron spin quantum number of S = /2. The magneto- ∆E =,mKA ()AB+, D() AB) mB gyric ratio γS for the electron spin magnetic moment mS is approximately twice that for the orbital motion, that is, γ = S =,mKA,,iij()AB +,D ij()AB mB j (3) g γ , with g ≈ 2.0023 being the g factor of the electron. If an e e e i = j = atom or a molecule has all orbital levels doubly occupied there xyz,, x,,yz is no net electron spin magnetic moment present (other than the tiny induced magnetic moments due to the presence of The two 3 × 3 matrices K and D relate the interaction magnetic nuclei as will be discussed below). We will restrict of the x, y, z component of one nuclear spin magnetic mo- the discussion to such systems here. ment to the x, y, z component of the other nuclear spin mag- Interaction of Magnetic Dipole Moments and Magnetic netic moment. They mediate the strength of the nuclear Fields interaction; for example, they depend on the distance between the two nuclei, the relative orientation of their spins, and in The interaction energy of a magnetic dipole moment m the case of K on the details of the molecule’s electronic struc- with a magnetic field B is given as ture.