分子间的转移积分、电子迁移率、空穴迁移率的计算
粒子迁移率对于有机电子器件例如场致发射晶体管（OFET）、有机发光二极管、光伏电池非常关键。载流子从一个位置迁移到另一个位置，迁移率主要有转移积分决定。本功能仅适用于分子间的载流子迁移，不适用于分子或团簇内部的电荷转移。
理论公式
Marcus hopping rate：
k=V2/ћ * (π/λkBT)1/2 * e-λ/4kBT
ADF可以直接计算电子、空穴的V值（电子或空穴的转移耦合），也可以计算λ，上述公式中，其他均为常数，因此这样可以计算得到Marcus hopping rate。根据Marcus hopping rate再进行后续处理，一般有两种：
- 方法一：参考Kwiatkowski et al., Phys. Chem. Chem. Phys. 10, 1852-1858 (2008)，通过Monte Carlo得到迁移率
- 方法二：参考Wen et al., J. Phys. Chem. B 113, 8813-8819 (2009)，通过解析的方法计算扩散系数，进而得到Einstein关系
这里以萘之间的电子迁移为例演示计算V、λ值，以及Marcus hopping rate的过程。
建模
建模的操作，参考：AMS软件建模教程
建模完成之后，一般需要对结构进行优化（即，能量最小化，找到最稳定的结构）.这里我们跳过建模和优化，直接计算下面给定的结构的电子、空穴迁移率：
建模如下图：
并按照如何创建分区将分子分为两个片段，默认分别命名为Region_1、Region_2，如上图所示。
设置计算参数
查看结果
Marcus hopping rate公式中的V值
点击SCM LOGO > output，选择Properties > Charge transfer integrals，即看到如下所示的内容：
Charge transfer integrals relevant for hole or electron mobility calculations Electronic coupling V (also known as effective (generalized) transfer integrals J_eff) V = (J-S(e1+e2)/2)/(1-S^2) For electron mobility calculations the fragment LUMO are considered. For hole mobility calculations the fragment HOMO are considered. Warning: (near) degeneracies found in fragment HOMO/LUMO orbitals. Couplings are printed also for (near) degenerate fragment orbitals within 0.100 eV of fragment HOMO or LUMO. This threshold for treating fragment MOs as degenerate can be set by the TIDegeneracyThreshold key. e1(hole) Site energy HOMO-1 fragment 1: -6.39747 eV e2(hole) Site energy HOMO-1 fragment 2: -6.39742 eV J(hole) Charge transfer integral HOMO-1 fragment 1 - HOMO-1 fragment -0.18135 eV S(hole) Overlap integral HOMO-1 fragment 1 - HOMO-1 fragment 2: 0.00901 V for hole transfer: -0.12373 eV e1(hole) Site energy HOMO-1 fragment 1: -6.39747 eV e2(hole) Site energy HOMO fragment 2: -6.40353 eV J(hole) Charge transfer integral HOMO-1 fragment 1 - HOMO fragment 2: -0.07501 eV S(hole) Overlap integral HOMO-1 fragment 1 - HOMO fragment 2: 0.00564 V for hole transfer: -0.03893 eV e1(hole) Site energy HOMO fragment 1: -6.40361 eV e2(hole) Site energy HOMO-1 fragment 2: -6.39742 eV J(hole) Charge transfer integral HOMO fragment 1 - HOMO-1 fragment 2: -0.07500 eV S(hole) Overlap integral HOMO fragment 1 - HOMO-1 fragment 2: 0.00564 V for hole transfer: -0.03893 eV e1(hole) Site energy HOMO fragment 1: -6.40361 eV e2(hole) Site energy HOMO fragment 2: -6.40353 eV J(hole) Charge transfer integral HOMO fragment 1 - HOMO fragment 2: 1.10836 eV S(hole) Overlap integral HOMO fragment 1 - HOMO fragment 2: -0.08709 V for hole transfer: 0.55490 eV e1(electron) Site energy LUMO+1 fragment 1: -1.35037 eV e2(electron) Site energy LUMO+1 fragment 2: -1.35036 eV J(electron) Charge transfer integral LUMO+1 fragment 1 - LUMO+1 fragment -0.27885 eV S(electron) Overlap integral LUMO+1 fragment 1 - LUMO+1 fragment 2: 0.03415 V for electron transfer: -0.23301 eV e1(electron) Site energy LUMO+1 fragment 1: -1.35037 eV e2(electron) Site energy LUMO fragment 2: -1.31648 eV J(electron) Charge transfer integral LUMO+1 fragment 1 - LUMO fragment 2: -0.05813 eV S(electron) Overlap integral LUMO+1 fragment 1 - LUMO fragment 2: 0.00680 V for electron transfer: -0.04908 eV e1(electron) Site energy LUMO fragment 1: -1.31658 eV e2(electron) Site energy LUMO+1 fragment 2: -1.35036 eV J(electron) Charge transfer integral LUMO fragment 1 - LUMO+1 fragment 2: -0.05814 eV S(electron) Overlap integral LUMO fragment 1 - LUMO+1 fragment 2: 0.00680 V for electron transfer: -0.04908 eV e1(electron) Site energy LUMO fragment 1: -1.31658 eV e2(electron) Site energy LUMO fragment 2: -1.31648 eV J(electron) Charge transfer integral LUMO fragment 1 - LUMO fragment 2: 0.17239 eV S(electron) Overlap integral LUMO fragment 1 - LUMO fragment 2: -0.02920 V for electron transfer: 0.13407 eV e1 Site energy HOMO-1 fragment 1: -6.39747 eV e2 Site energy LUMO+1 fragment 2: -1.35036 eV S(Charge_Recombination) Overlap integral HOMO-1 fragment 1 - LUMO+1 fragment 2: -0.00592 J(Charge_Recombination) Charge transfer integral HOMO-1 fragment 1 - LUMO+1 fragment 0.05209 eV V for charge_recombination 1-2: 0.02917 eV e1 Site energy HOMO-1 fragment 1: -6.39747 eV e2 Site energy LUMO fragment 2: -1.31648 eV S(Charge_Recombination) Overlap integral HOMO-1 fragment 1 - LUMO fragment 2: 0.04109 J(Charge_Recombination) Charge transfer integral HOMO-1 fragment 1 - LUMO fragment 2: -0.39978 eV V for charge_recombination 1-2: -0.24171 eV e1 Site energy HOMO fragment 1: -6.40361 eV e2 Site energy LUMO+1 fragment 2: -1.35036 eV S(Charge_Recombination) Overlap integral HOMO fragment 1 - LUMO+1 fragment 2: -0.08832 J(Charge_Recombination) Charge transfer integral HOMO fragment 1 - LUMO+1 fragment 2: 0.91775 eV V for charge_recombination 1-2: 0.57986 eV e1 Site energy HOMO fragment 1: -6.40361 eV e2 Site energy LUMO fragment 2: -1.31648 eV S(Charge_Recombination) Overlap integral HOMO fragment 1 - LUMO fragment 2: -0.02649 J(Charge_Recombination) Charge transfer integral HOMO fragment 1 - LUMO fragment 2: 0.27771 eV V for charge_recombination 1-2: 0.17557 eV e1 Site energy LUMO+1 fragment 1: -1.35037 eV e2 Site energy HOMO-1 fragment 2: -6.39742 eV J(Charge_Recombination) Charge transfer integral LUMO+1 fragment 1 - HOMO-1 fragment -0.05207 eV S(Charge_Recombination) Overlap integral LUMO+1 fragment 1 - HOMO-1 fragment 2: 0.00592 V for charge_recombination 2-1: -0.02915 eV e1 Site energy LUMO+1 fragment 1: -1.35037 eV e2 Site energy HOMO fragment 2: -6.40353 eV J(Charge_Recombination) Charge transfer integral LUMO+1 fragment 1 - HOMO fragment 2: -0.91758 eV S(Charge_Recombination) Overlap integral LUMO+1 fragment 1 - HOMO fragment 2: 0.08830 V for charge_recombination 2-1: -0.57976 eV e1 Site energy LUMO fragment 1: -1.31658 eV e2 Site energy HOMO-1 fragment 2: -6.39742 eV J(Charge_Recombination) Charge transfer integral LUMO fragment 1 - HOMO-1 fragment 2: 0.39992 eV S(Charge_Recombination) Overlap integral LUMO fragment 1 - HOMO-1 fragment 2: -0.04110 V for charge_recombination 2-1: 0.24181 eV e1 Site energy LUMO fragment 1: -1.31658 eV e2 Site energy HOMO fragment 2: -6.40353 eV J(Charge_Recombination) Charge transfer integral LUMO fragment 1 - HOMO fragment 2: -0.27776 eV S(Charge_Recombination) Overlap integral LUMO fragment 1 - HOMO fragment 2: 0.02650 V for charge_recombination 2-1: -0.17560 eV Electronic coupling V (summed over printed couplings) Square root of (sum of squares of printed V's): ----------------------------------------------------- V for hole transfer: 0.57118 eV V for electron transfer: 0.27764 eV V for charge_recombination 1-2: 0.65295 eV V for charge_recombination 2-1: 0.65290 eV
- V for hole transfer：一般为负值，绝对值越大，转移越容易，一般绝对值与J的绝对值成正比
- V for electron transfer：一般为负值，绝对值越大，转移越容易，一般绝对值与J的绝对值成正比
- (hole) Site energy HOMO fragment N：外来一个空穴，放置到片段N（也就是放到片段N的HOMO上面）带来的能量变化量（不考虑空穴到达之后的弛豫过程）
- (electron) Site energy LUMO fragment N：外来一个电子，放置到片段N（也就是放到片段N的LOMO上面）带来的能量变化量（不考虑电子到达之后的弛豫过程）
- 对于分子HOMO、LUMO存在（近）简并的情况，分别进行了计算，最后以“方均根”（Square root of (sum of squares of printed V's)）的形式，综合考虑。近简并的阈值为0.1ev。如果需要考虑更多近简并轨道，可以通过如下所示TIDegeneracyThreshold关键词调节近简并阈值（单位为eV），即与HOMO差值不大于该值的占据轨道，均考虑在内：
Engine ADF
Basis
Type TZP
End
Fragments
Region_1 Region_1.rkf
Region_2 Region_2.rkf
End
XC
GGA PBE
DISPERSION GRIMME4
End
Relativity
Level None
End
Symmetry NOSYM
TransferIntegrals Yes
TIDegeneracyThreshold 0.2
EndEngine
这里我们得到了电子和空穴的V值0.57118 eV、0.27764 eV。这个值是考虑到单分子存在两个简并HOMO、LUMO的结果。
Marcus hopping rate公式中的λ值
根据前面的Marcus hopping rate公式，还需要另一个数值，也就是电子或空穴的重组能λ。重组能的计算公式：
- 电子重组能：λ = (Eanion(neutral geometry) - E(neutral)) + (Eneutral(anion geometry) - E(anion))
- 空穴重组能：λ = (Ecation(neutral geometry) - E(neutral)) + (Eneutral(cation geometry) - E(cation))
其中，
- E(neutral)，表示单个中性分子几何结构优化完成之后，得到的Bonding energy（在*.logfile尾部）
- Eanion(neutral geometry)，表示单个中性分子几何结构优化完成之后得到的结构，修改带电量为-1，计算single point得到的bonding energy
- E(anion)，表示单个分子，带电量为-1时，进行几何结构优化，得到的Bonding energy
- Eneutral(anion geometry)，表示单个分子，带电量为-1时，结构优化优化得到的结构，带电量改为0，计算single point得到的Bonding energy
- 如果是两种不同分子之间转移，参考：两种分子之间的λ值
Marcus hopping rate公式中的常数
Marcus hopping rate公式中用到的数据，统一能量单位到J，则各个常数的数值为:
- 1eV = 1.60×10-19 J
- h = 6.62606896×10-34 J·s
- ћ = h/2π=1.05457161×10-34 J·s
- kB = 1.38064881×10-23 J/K
知道V、λ值，以及上述常数，带入Marcus hopping rate公式，即可得到迁移率（单位为s-1，也就是每秒跳跃多少次）。
两个以上分子
本例说明的是2个Region之间的计算，实际上如果是多个片段也是一样，例如三个片段，会给出两两之间的数据。
Site energy (hole) HOMO fragment 1 (eV): -6.90814 Site energy (hole) HOMO fragment 2 (eV): -6.31622 Site energy (hole) HOMO fragment 3 (eV): -6.75325 Charge transfer integral (hole) HOMO fragment 1 - HOMO fragment 2 (eV): 0.10886E+01 Charge transfer integral (hole) HOMO fragment 1 - HOMO fragment 3 (eV): 0.16242E+01 Charge transfer integral (hole) HOMO fragment 2 - HOMO fragment 3 (eV): 0.18609E+00 Site energy (electron) LUMO fragment 1 (eV): -1.80417 Site energy (electron) LUMO fragment 2 (eV): -1.22862 Site energy (electron) LUMO fragment 3 (eV): -1.63222 Charge transfer integral (electron) LUMO fragment 1 - LUMO fragment 2 (eV): 0.16682 Charge transfer integral (electron) LUMO fragment 1 - LUMO fragment 3 (eV): 0.22993 Charge transfer integral (electron) LUMO fragment 2 - LUMO fragment 3 (eV): 0.17881 Charge transfer integral (Charge_Recombination) HOMO fragment 1 - LUMO fragment 2 (eV): 0.27072 Charge transfer integral (Charge_Recombination) HOMO fragment 1 - LUMO fragment 3 (eV): -0.25022 Charge transfer integral (Charge_Recombination) HOMO fragment 2 - LUMO fragment 3 (eV): -0.04055 Charge transfer integral (Charge_Recombination) LUMO fragment 1 - HOMO fragment 2 (eV): -0.27291 Charge transfer integral (Charge_Recombination) LUMO fragment 1 - HOMO fragment 3 (eV): 0.25191 Charge transfer integral (Charge_Recombination) LUMO fragment 2 - HOMO fragment 3 (eV): 0.05039 --------------------------------------------------------------------- Overlap integrals relevant for hole or electron mobility calculations --------------------------------------------------------------------- Overlap integral (hole) HOMO fragment 1 - HOMO fragment 2: -0.87092E-01 Overlap integral (hole) HOMO fragment 1 - HOMO fragment 3: -0.10910E+00 Overlap integral (hole) HOMO fragment 2 - HOMO fragment 3: -0.12533E-01 Overlap integral (electron) LUMO fragment 1 - LUMO fragment 2: -0.02920 Overlap integral (electron) LUMO fragment 1 - LUMO fragment 3: -0.02923 Overlap integral (electron) LUMO fragment 2 - LUMO fragment 3: -0.01751 Overlap integral HOMO (Charge_Recombination) fragment 1 - LUMO fragment 2: -0.02650 Overlap integral HOMO (Charge_Recombination) fragment 1 - LUMO fragment 3: 0.01887 Overlap integral HOMO (Charge_Recombination) fragment 2 - LUMO fragment 3: 0.00325 Overlap integral LUMO (Charge_Recombination) fragment 1 - HOMO fragment 2: 0.02650 Overlap integral LUMO (Charge_Recombination) fragment 1 - HOMO fragment 3: -0.01888 Overlap integral LUMO (Charge_Recombination) fragment 2 - HOMO fragment 3: -0.00323
需要用户根据公式V = (J-S(e1+e2)/2)/(1-S2)手动计算一下V值。
内层电子、空穴引起的电荷转移
如果电子、空穴的转移不是发生在HOMO、LUMO之间，而是发生在内层轨道，可以通过修改碎片的电子占据情况实现。
固体
块体材料的电子迁移率、Seebeck 系数、热导率、Hall系数以及Hall电导率张量，可以使用QuantumATK计算。