Over the past 30 years, coupled-cluster (CC) theory has developed as perhaps the most reliable computational method of electronic structure theory for the prediction of molecular properties of well-behaved systems. In 1966[1] and later in a review article in 1969[2] , Cízek laid down the foundations of the use of the method in electronic structure theory in formal terms. He used sophisticated techniques such as diagrams and second-quantization which, unfortunately, were beyond the experience of most quantum chemists at the time. Later, in the 1970s, CC theory began to gain the interest of a wider audience. Cízek and Paldus[3] derived expressions for coupled-cluster doubles (CCD) theory (at the time referred to as coupled-pair many-electron theory or CPMET ). This was actually an algebraic re-derivation of the diagrammatic equations presented by Cízek[1]. CCD was re-constructed from a different perspective by Hurley[4] . Around this same time, Monkhorst[5] presented a response theory for calculating molecular properties from general coupled-cluster wavefunctions. At the end of the 1970s, computer implementations of the theory began to appear for realistic systems. Pople , Krishnan, Schlegel and Binkley[6] and Bartlett and Purvis[7] presented spin-orbital CCD codes. In the early 1980s, a particularly important achievement was that of Purvis and Bartlett [8], who derived the coupled-cluster singles and doubles (CCSD) equations and implemented them in a practical computer code. At the end of the 1980s and beginning of the 1990s, the popularity of CC methods grew at a rapid pace, as more effort was expended in the construction of highly efficient CCSD energy codes[9, 10, 11, 12], spin-adaptation of open-shell wavefunctions[13, 14, 15, 16, 17], and inclusion of higher excitations in the CC wavefunction[18, 19, 20, 21, 22, 23, 24].
In these notes, we will present many of the fundamental concepts of the ever-growing field of coupled-cluster theory in a manner which will hopefully be accessible to students who are new to the subject. In the next section, we will discuss some of the basic ideas behind cluster functions and the exponential wave operator, as well as the phenomenon known as size-extensivity. In section 3, we take a more traditional approach as the fundamental equations of coupled-cluster theory are formally presented. In section 4, we will attempt to present some of the techniques frequently (and some not so frequently) used to produce programmable equations from formal ones. These techniques include both algebraic and diagrammatic approaches. In subsequent sections (yet to be written, obviously) we will discuss the relationship between CC methods and perturbation theory, the inclusion of higher excitation levels in CC, and perhaps even some of the difficulties of derivative methods.
Some important sources of general CC theory include the 1981 review by Bartlett[25] , which provides a very large number of references and gives a basic overview of CCD ; a 1989 article, also by Bartlett [26], which clearly presents the governing equations of CCSD and many of its approximants and discusses some of the future directions of the field; a very recent review by Taylor[27] , which provides an excellent discussion of some of the fundamentals from often unique and insightful perspectives; and a recent article by Lee and Scuseria[28] , which assesses the predictive abilities of CC theory for realistic chemical systems. Some of the analyses presented in these notes are implicitly based on these reviews and we gratefully acknowledge the authors' contributions.