The energy minimization involved in density functional calculations of electronic systems can be carried out using an exponential transformation that preserves the orthonormality of the orbitals. The energy of the system is then represented as a function of the elements of a skew-Hermitian matrix that can be optimized directly using unconstrained minimization methods. An implementation based on the limited memory Broyden-Fletcher-Goldfarb-Shanno approach with inexact line search and a preconditioner is presented and the performance compared with that of the commonly used self-consistent field approach. Results are presented for the G2 set of 148 molecules, liquid water configurations with up to 576 molecules and some insulating crystals. A general preconditioner is presented that is applicable to systems with fractional orbital occupation as is, for example, needed in the k-point sampling for periodic systems. This exponential transformation direct minimization approach is found to outperform the standard implementation of the self-consistent field approach in that all the calculations converge with the same set of parameter values and it requires less computational effort on average. The formulation of the exponential transformation and the gradients of the energy presented here are quite general and can be applied to energy functionals that are not unitary invariant such as self-interaction corrected functionals.

电子系统密度泛函计算中涉及的能量最小化可以使用保持轨道正交性的指数变换来执行。然后，系统的能量被表示为斜赫米特矩阵元素的函数，该矩阵可以使用无约束最小化方法直接优化。基于有限内存 Broyden-Fletcher-Goldfarb-Shanno 方法的实现，具有不精确的行搜索和预处理器并与常用的自洽场方法的性能进行了比较。结果显示为 G2 组 148 个分子、液态水配置（最多 576 个分子）和一些绝缘晶体。提出了一种通用的预处理器，它适用于具有分数轨道占用的系统，例如，在周期系统的 k 点采样中需要。发现这种指数变换直接最小化方法优于自洽场方法的标准实现，因为所有计算都收敛于相同的参数值集，并且平均需要更少的计算工作。