Chapter 1
Introduction

If you want to find the secrets of the universe, think in terms of energy, frequency and vibration.

Nicolas Tesla

This is a book about molecular properties, or to be more specific, molecular response properties. Response properties tell us about how molecules respond to electromagnetic fields. To understand these responses, we have to enter the microscopic world of atoms and molecules, governed by the laws of quantum mechanics. For that reason, the reader of this book can expect several intellectual challenges ranging from profound and conceptual cornerstones of quantum theory itself to trivial, yet mind-boggling, issues relating to the smallness of atomic sizes. Consider for instance the situation in which a collection of molecules are being exposed to the intense electric field of a laser, as illustrated in Figure 1.1. From a human perspective, the focal point of a laser is a dangerous place to be, but, from the atomic perspective, it is far less dramatic. In our example, there will be fewer photons than molecules, and, for instance, if the purpose is to protect the eye by efficient optical power limiting, only about every second molecule needs to absorb an incoming light quanta in order to reduce the energy in the transmitted light pulse to an eye-safe level. Furthermore, as strong as the electric field may appear to our eyes, to the individual electron it is several orders of magnitude smaller than the dominating forces exerted by the atomic nuclei and fellow electrons. To get an idea of magnitudes, one may note that the electric field below overhead power lines may reach and the maximum electric field strength possible in air without creating sparks is . In contrast, at the Bohr radius in the hydrogen atom, the electric field strength is . This is a key point, namely, that we can expose molecules to fields that are strong enough so that we can detect the responses of their charges (nuclei and electrons) while at the same time the fields are weak enough to act as probes, not significantly perturbing the electronic and nuclear structure of the molecule.

Figure 1.1 Liquid benzene in a small volume corresponding to the focal point of a laser operating at 532 nm and releasing pulses with an energy of 1 mJ.

Take a very simple example: What happens if a neutral atom (not even a molecule) is placed in a uniform electric field? An experimentalist will ask nature—that is, he or she may perform an experiment, where every macroscopic experiment relates to a very large number of probabilistic microscopic quantum events—by probing how the charge distribution of the atom is modified by the applied field. A theoretician will ask the wave function . The quantum-mechanical equivalent to the outcome of the experiment is the expectation value

where is the quantum-mechanical operator corresponding to the observable monitored by the experiment. Quantum mechanics is a probabilistic theory. The link between theory and experiment is made by considering a large number of systems prepared in the same state, prior to switching on the field. If we disregard measurement errors, then the possible outcomes of the individual quantum events are given by the eigenvalues of the operator , defined by the eigenvalue equation

Following the postulates of quantum mechanics, the operator is by necessity Hermitian, and the eigenvalues are thus real (corresponding to real-valued observables), and there is a probability for the outcome in each of the single quantum events, leading to an expectation value that is

For example, indirect information about the charge distribution of the atom can be obtained from measurements of the electric dipole moment since the two quantities are connected through an expectation value of the form

where denotes the number of electrons and is the elementary charge. However, the electronic charge density can in itself also be expressed as an expectation value

and it is possible to probe in for instance X-ray diffraction experiments.

Figure 1.2 Electronic charge density of neon expanded in orders of the applied electric field . Light and dark gray regions indicate positive and negative values, respectively.

If the external electric field is weak compared to the internal atomic fields, we can expand the induced electronic charge density in a Taylor series with respect to field strength. In Figure 1.2, such a perturbation expansion is illustrated to fifth order for a neon atom. The electric field of strength is applied along the vertical -axis (directed upward in the figure) and will tend to pull the positive charge along the field and the negative charge in the opposite direction, resulting in an electronic charge density that can be expanded as

The zeroth-order density refers to that of neon in isolation and integrates to . It follows from charge conservation that the higher-order densities all integrate to zero. The first-order density shows the charge separation of a dipole, and we then get more and more complicated structures with increasing order. It is also clear that the higher the order, the more diffuse the density becomes, and we can expect that an accurate description of higher-order responses put strong requirements on the wave function flexibility at large distances from the nucleus.

If we insert the expansion of the charge density into the expression for the dipole moment [Eq. (1.4)], even orders of the density will not contribute due to symmetry—this is a reflection of the fact that odd-order electric properties vanish in systems with a center of inversion. The resulting induced dipole moment, directed along the -axis, becomes

This expression defines a series of proportionality constants between the induced dipole moment and powers of the field. The linear and cubic coupling constants are known as the electric dipole polarizability and second-order hyperpolarizability, and they are conventionally denoted by Greek letters and , respectively.1 It is the focus of this book to understand how these and other molecular properties can be determined by means of quantum-chemical calculations.

Figure 1.3 Hierarchy of quantum-chemical methods.

When judging the quality of quantum-chemical calculations, one typically considers the choice of method and basis set. These two quantities combined constitute a theoretical model chemistry, that is, a certain approximation level reaching toward the exact solution of the electronic wave function equation. There exist hierarchical series of basis sets that allow for systematic convergence toward the complete one-particle basis set limit, as indicated in Figure 1.3. An increase in the cardinal number of the basis set, from double- to triple- and so forth, improves the description of the ground-state wave function, whereas levels of augmentation with diffuse functions in the basis set are particularly important for the description of the excited electronic states, and therefore also for many molecular properties. Likewise, in conventional wave function-based electronic structure theory, the configuration interaction (CI) and coupled cluster (CC) expansions provide systematic ways to reach the complete -particle limit. Increased complexity of the theoretical model chemistry comes, however, at a sometimes staggering computational cost. In general, the computational cost scales as , where the base represents the size of the one-particle basis set, and therefore implicitly scales with the system size, and the exponent is associated with a given electronic structure method. Starting from the Hartree–Fock (HF) method, which formally scales as , each excitation level treated variationally (perturbatively) increases the exponent by two units (one unit). Accordingly, CC and CI models that include single and double (SD) excitations, CCSD and CISD, respectively, scale as , but the CC expansion includes electron correlation in a more efficient manner than does CI and has other advantages such as size extensivity. Adding triple excitations perturbatively, as in CCSD(T), increases the exponent to seven. A great achievement of quantum chemistry has been to devise algorithms that significantly reduce these formal scalings.

Kohn–Sham density functional theory (KS-DFT) has become the most widely used method in quantum chemistry due to its efficient treatment of electron correlation at modest computational cost. It formally has the same scaling as HF theory as it also employs a single Slater determinant to describe the reference state of the fictitious noninteracting KS system, constrained to have the same electron density as the real interacting system. The similarity in the parametrization of the reference state has implications in the presentation of the time-dependent response approaches. Until the very final stages, we need not specify which of these two approaches we address, treating them instead in a uniform manner. We will present this formulation, common to time-dependent HF and KS-DFT theories, under the name self-consistent field (SCF) theory. At the end of the day, there is only one drawback that stands out as critical with the KS-DFT technique, and that is the lack of a systematic way to improve the exchange-correlation (XC) functional, which makes it impossible to provide a general ranking of DFT functionals....