Geometric Formulation of Unique Quantum Stress Fields

We present a derivation of the stress field for an interacting quantum system within the framework of local density functional theory. The formulation is geometric in nature and exploits the relationship between the strain tensor field and Riemannian metric tensor field. The resultant expression obtained for the stress field is gauge-invariant with respect to choice of energy density, and therefore provides a unique, well-defined quantity. To illustrate this formalism, we compute the pressure field for two phases of solid molecular hydrogen. 03.65.-w, 62.20.-x, 02.40.-k, 71.10.-w Typeset using REVTEX 1 The stress, or the energetic response to deformation or strain, plays an important role in linking the physical properties of a material (e.g. strength, toughness) with the behavior of its microstructure. In addition, the spatial distribution of stress is an invaluable tool for continuum modeling of the response of materials. The stress concept has been applied at atomistic scales as well. Over the last fifteen years, there has been a continuing trend toward understanding various structural and quantum-mechanical phenomena in materials in terms of their response to stress [1]. For example, the residual stress at equilibrium has been used to assess the structural stability of systems containing surfaces or strained interfaces. It has been demonstrated that the desire to minimize surface stress can give rise to reconstructions on high symmetry surfaces [2–8], and the stability of epitaxially grown bimetallic systems has been attributed to the formation of incommensurate overlayers, defects, and dislocations which minimize the stress near the metal-metal interface [9,10]. The stress can have significant effects on chemical reactivity as well. It has been shown that small molecule chemisorption energies and reaction barriers on certain strained metal and strained semiconductor surfaces are quite different from those on the unstrained surface [11,12]. Formally, studies of the above phenomena must include a quantum-mechanical description of the system’s electronic degrees of freedom. Therefore, one must consider how a stress is defined quantum mechanically. Methods for calculating the stress in quantum mechanical systems have been developed since the birth of quantum theory itself [13]. However, research in developing formalisms for determining the quantum stress in solid-state systems has recently been revitalized. This is mainly due to ever-increasing opportunities to perform accurate and efficient quantum-mechanical calculations on systems which exhibit stress mediated phenomena. The stress is a rank-two tensor quantity, usually taken to be symmetric and therefore torque-free. Two useful representations of the stress tensor are the volume-averaged or total stress, Tαβ , and the spatially varying stress field σαβ(x). The two representations are related since the total stress for a particular region in a system is the stress field integrated over 2 the volume. Nielsen and Martin developed a formalism for calculating the total quantum stress in periodic systems [14]. They define the total stress as the variation of the total ground-state energy with respect to a uniform scaling of the entire system. This uniform scaling corresponds to a homogeneous or averaged strain over the entire system. They further demonstrate that the total quantum stress is a unique and well-defined physical quantity. Their formulation has been successfully implemented to study a variety of solid state systems [2,3]. Other formalisms for determining the total quantum stress have been created as well [2,15,16]. Although these formalisms have provided important tools for studying quantum stress, the stress field is a more useful quantity that contains important information regarding the distribution of the stress throughout the system. A knowledge of the spatial dependence of the quantum stress is vital if one wishes to predict the spatial extent of structural modifications or understand phenomena at interfaces in complex heterogeneous systems. However, certain definitions of the quantum stress field suggest that it can only be specified up to a gauge. (This ambiguity manifests itself in classical atomistic models as well.) It has therefore been asserted that the quantum stress field is not a well-defined physical quantity, even though physical intuition may suggest otherwise. A traditional way to develop a quantum stress field formalism is to consider the stress field’s relationship with the force field. From this perspective, the stress field can be defined as any rank-two tensor field whose divergence is the force field of the system: F α = ∇βσ. (1) (Note that the Einstein summation convention for repeated indices is used throughout the Letter.) One can add to σ a gauge of the form ∂ ∂xγ A(x), (2) where A is any tensor field antisymmetric in β and γ, and recover the same force field, thereby demonstrating the non-uniqueness of this stress field definition. General formulations for computing non-gauge-invariant stress fields in quantum many-body systems 3 have been derived by Nielsen and Martin, Folland, Ziesche and co-workers, and Godfrey [14,17–19]. There have been several attempts to overcome this problem of non-uniqueness. For example, the stress field formalism of Chen and co-workers has been applied to numerous solid state systems to determine the local pressure around a region [20]. However, their method assumes that the potential is pair-wise only. Several ab-initio quantum stress field formulations have been developed, as well. Ramer and co-workers developed a method to calculate the resultant stress field from an induced homogeneous strain [21]. They incorporate the additional constraint that the field must be the smoothest fit to the ionic forces. This method cannot be used to calculate the residual stress field at equilibrium, nor can it determine the energy dependence on strains which do not have the periodicity of the unit cell. Filippetti and Fiorentini developed a formulation of the stress field based on the energy density formalism of Chetty and Martin [22,23]. Since this formulation is based explicitly on the energy density, which is not gauge-invariant, the resultant stress field is not unique. Mistura succeeded in developing a general gauge-invariant formalism for pressure tensor fields of inhomogeneous fluids within classical statistical models using a Riemannian geometric approach [24]. This Letter extends Mistura’s work, developing a Riemannian geometric formalism for computing gauge-invariant stress fields in quantum systems within the local density approximation (LDA) of density functional theory (DFT). We show that the response of the total ground-state energy of a quantum system to a local spatially varying strain is a unique and physically meaningful field quantity which can be determined at every point in the system. Using a procedure well known in continuum theory (see for example Ref. [25]), one can formally relate a Riemannian metric tensor field gαβ(x) with the strain field ǫαβ(x): gαβ = δαβ + 2ǫαβ , with ǫαβ ≡ 1 2 (δαγ∂βu γ + δγβ∂αu γ + δγκ∂αu ∂βu ), (3) where ∂α ≡ ∂/∂x [26]. Here the strain field is defined in terms of a vector displacement field u which maps coordinates x in the non-deformed system, to the coordinates x α = x+u 4 in the deformed system. The stress field σ(x) and strain field obey a virtual work theorem expressing the energy response to variations in the strain: δE = ∫ √ gσδǫαβd x, (4) where g(x) is the determinant of gαβ(x). It can be shown that the stress field is related to the functional derivative of the energy with respect to the metric field [24]: σ ≡ 1 √ g δE δǫαβ = 2 √ g δE δgαβ , σαβ = − 2 √ g δE δgαβ . (5) We now derive the quantum stress field of a many-electron system in the presence of a fixed set of classical positive charged ions using local density functional theory [27,28]. The ground state electronic charge density of the system is written as n(x) = ∑ i φ ∗ i (x)φi(x), where φi are single-particle orthonormal wavefunctions. For this derivation, we assume orbitals with fixed integer occupation numbers. The extension to metals with Fermi fillings is straightforward, simply necessitating use of the Mermin functional instead of the total energy [29]. The total charge density of the system can be written as a sum over all ionic charges and n: