International Journal of Spectroscopy

Volume 2016, Article ID 2390109, 9 pages

http://dx.doi.org/10.1155/2016/2390109

## The Heteronuclear Multiple-Quantum Correlation Experiment: Perspective from Classical Vectors, Nonclassical Vectors, and Product Operators

^{1}Departamento de Farmacia, Instituto de Farmacia y Alimentos, Universidad de La Habana, Avenida 23, No. 21425 e/214 and 222, La Coronela, La Lisa, 13600 La Habana, Cuba^{2}Departamento de Química Física, Facultad de Química, Universidad de La Habana, Avenida Zapata y G, Vedado, 10400 La Habana, Cuba

Received 16 October 2015; Accepted 6 December 2015

Academic Editor: Guillermo Moyna

Copyright © 2016 Karen de la Vega-Hernández and Manuel Antuch. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

It is usually accepted that most 2D-NMR experiments cannot be approached using classical models. Instructors argue that Product Operators (PO) or density matrix formalisms are the only alternative to get insights into complex spin evolution for experiments involving Multiple-Quantum Coherence, such as the Heteronuclear Multiple-Quantum Correlation (HMQC) technique. Nevertheless, in recent years, several contributions have been published to provide vectorial descriptions for the HMQC taking PO formalism as the starting point. In this work we provide a graphical representation of the HMQC experiment, taking the basic elements of Bloch’s vector model as building blocks. This description bears an intuitive and comfortable understanding of spin evolution during the pulse sequence, for those who are novice in 2D-NMR. Finally, this classical vectorial depiction is tested against the PO formalism and nonclassical vectors, conveying the didactic advantage of shedding light on a single phenomenon from different perspectives. This comparative approach could be useful to introduce PO and nonclassical vectors for advanced upper-division undergraduate and graduate education.

#### 1. Introduction

Nuclear magnetic resonance (NMR) has become an indispensable technique in diverse fields such as chemistry [1–4], biochemistry [5–8], structural biology [9–12], materials science [13–15], and biomedicine [16–18]. However, the richness and complexity of NMR, joined to a vast literature, appear intimidating to novice users.

Concerning the educational literature, NMR is a recurrent topic. Many didactic papers are devoted to structure elucidation [19–22], to review media [23, 24], and other applications [7, 8, 25, 26]. However, the knowledge behind spin evolution in modern 2D-NMR is beyond the grasp of many users since it is not mandatory to interpret NMR spectra.

Most books dealing with 2D-NMR build on Product Operators (PO) or density matrix formalisms to explain spin evolution. Conversely, some other books present simplified treatments of some 2D-NMR experiments through vector representations. Unfortunately, there is a gap between the more elementary books, usually ignoring the most complex theoretical bases, and sophisticated books, treating rigorous methods as almost self-evident [27]. Consequently, an approach lying midway between simple and more elaborated explanations would be useful for didactic purposes.

For such an approach we selected the classical vector (CV) model [28] which permits visualizing some NMR experiments with special comfort. In nowadays (under)graduate teaching, it is widely accepted that most complex 2D-NMR experiments cannot be approached using the classical vector model, in particular those experiments involving Multiple-Quantum Coherence (MQC) such as the Heteronuclear Multiple-Quantum Correlation (HMQC). In those cases, instructors argue that PO or density matrix formalisms are the only alternative to get insights into complex spin evolution.

The use of PO for practical purposes is not so complicated since it consists in learning some established rules. However, the physical meaning of mathematical manipulations remains unclear in some cases [29]. This fact prompted some authors to develop graphical representations for PO [30, 31]. An interesting work describes what was named the nonclassical vector (NCV) model [32]. NCV take PO as the starting point, and its representations are images for each PO. Therefore, in order to comprehend such model, previous knowledge of PO is required, which is beyond the scope of undergraduate and some graduate courses. The great value of NCV resides in offering graphical representations for the equations in PO formalism. Consequently, NCV should be used to accompany PO, and not as an independent model in order to explain multiple-pulse NMR.

The main educational disadvantages of current visual representations of spin states for 2D-NMR experiments are the circumvention of graphical representations for MQC [33] or the use of PO and wave functions as the starting point [29, 32].

In a previous work [28], we provided a classical vector model for the sequence of events occurring during the Heteronuclear Single-Quantum Correlation (HSQC) experiment and the further comparison with PO. In this paper, we present a graphical representation of the HMQC experiment, following the same spirit as in our previous publication. In addition, we extend the comparative analysis to NCV and provide a more exhaustive and rigorous view of MQC evolution during the evolution period.

The classical representation of the HMQC allows an intuitive understanding as far as possible of the resulting spectrum appearance without the use of quantum mechanics. The further correspondence with PO and NCV allows for the comparison of the same phenomenon from the perspective of different models. Such comparison permits validating the proposed classical vector model as a pedagogical tool for introducing 2D-NMR [28].

#### 2. Some Initial Comments

The specialized literature offers many ways of presenting NMR. It is well recognized that NMR is a quantum phenomenon. However, classical mechanic approaches are often preferred because of its simpler nature and inherent intuitiveness [34]. Such methodology is consistent and in some cases presents excellent correspondence with more rigorous treatments [28].

Henceforward, only nuclei with shall be considered. It is well established that, for a single nucleus having , a measurement of gives only one of two possible orientations, namely, *α* and *β*, for the projection of along the field direction.

However, this is not the case in systems composed by many spins, in which the average orientation associated with in a magnetic field (i.e., the expectation value in quantum mechanics) gives bulk macroscopic magnetization that does not have quantized values [34, 35]. Deepening into the orientations of individual spins in the magnetic field is unnecessary in this context. Accurately speaking, the system is in a mixed state and there are innumerable microscopic configurations that would result in the same mixed state. Suffice it to say that, at equilibrium, the polarization of a population of spin-up versus spin-down nuclei is one possibility that gives the correct density matrix for the particular mixed state (Figure 1).