Tian Functions: Pseudo‐Atom Method

• Tian Functions are defined as figure-of-merit measures that use pseudo structure factors to mimic density modification, improving phase determination accuracy.
• They overcome traditional metrics like RCF by penalizing nonphysical or trivial phase sets, especially when data resolution is limited.
• Empirical studies on systems such as GaN and complex organic molecules demonstrate the robust discrimination power of Tian Functions in phase retrieval.

The Tian function refers to the figure-of-merit $R_{\mathrm{Tian}}$, introduced within the Tian pseudo-atom method (TPAM) or pseudo atom method (PAM) for crystallographic phase determination. $R_{\mathrm{Tian}}$ replaces the conventional $R_{\mathrm{CF}}$ in the charge flipping (CF) structure solution algorithm, providing a robust criterion for evaluating phase sets, particularly when data resolution is limited. The fundamental innovation is the substitution of conventional observed structure factors with pseudo structure factors, which mimic the effects of density modification, thus overcoming specific limitations of standard metrics in low-resolution regimes (Li et al., 2017).

1. Formal Definition and Construction

$R_{\mathrm{Tian}}$ is defined as

$$R_{\mathrm{Tian}} = \sum_h W_h |F_h^{\mathrm{ps}} - F_{h,\mathrm{cal}}|$$

where $F_{h,\mathrm{cal}}$ is the structure factor amplitude from a cycle of Fourier transform (FT), density modification (DM, e.g., charge flipping), and inverse FT; $F_h^{\mathrm{ps}}$ is the pseudo structure factor amplitude. In practice, uniform weights $W_h = 1$ are used. This contrasts with the conventional charge flipping merit

$$R_{\mathrm{CF}} = \frac{\sum_h ||F_h^{\mathrm{obs}}| - |F_{h,\mathrm{cal}}||}{\sum_h |F_h^{\mathrm{obs}}|}$$

where $|F_h^{\mathrm{obs}}|$ are measured amplitudes. The Tian function substitutes $|F_h^{\mathrm{obs}}|$ with $F_h^{\mathrm{ps}}$, providing a modified residual tailored to density-modified amplitudes (Li et al., 2017).

2. Rationale and Relation to Conventional Metrics

The theoretical basis of $R_{\mathrm{Tian}}$ arises from the observation that $R_{\mathrm{CF}}$ is only effective as a global minimum indicator at sufficiently high resolution. At low resolution, incorrect or trivial phase sets (such as the all-zero solution) can yield deceptively low $R_{\mathrm{CF}}$ scores, failing to single out the correct solution. An ideal figure of merit, $R^*$, would directly compare trial amplitudes to those computed with the correct phases after density modification:

$$R^* = \frac{\sum_h |F_{h,\mathrm{cal}}^{\mathrm{corr}} - F_{h,\mathrm{cal}}|}{\sum_h F_{h,\mathrm{cal}}^{\mathrm{corr}}}$$

with $R^* = 0$ for the correct phase set and $R^* > 0$ elsewhere, irrespective of resolution. However, $F_{h,\mathrm{cal}}^{\mathrm{corr}}$ is inaccessible in practice. $R_{\mathrm{Tian}}$ employs $F_h^{\mathrm{ps}}$ as a surrogate, designed to approximate $F_{h,\mathrm{cal}}^{\mathrm{corr}}$ without requiring prior knowledge of the phases (Li et al., 2017).

3. Pseudo Structure Factor: Definition and Calculation

The pseudo structure factor, $F_h^{\mathrm{ps}}$, is constructed to emulate the effect of density modification on the true structure factors. Formally,

$$F(H) = \sum_{i=1}^N f_i(H) \exp(-2\pi i H \cdot R_i)$$

where $f_i(H)$ is the atomic scattering factor and $R_i$ the atomic position. The pseudo form,

$$F^{\mathrm{ps}}(H) = \sum_{i=1}^N f_i^{\mathrm{ps}}(H) \exp(-2\pi i H \cdot R_i)$$

uses pseudo-atom scattering factors $f_i^{\mathrm{ps}}(H)$, derived by applying the same density-modification operator (e.g., charge flipping) to an isolated atom and computing its Fourier transform. In structure determination, exact atomic positions are unknown; thus, the method estimates $|F_h^{\mathrm{ps}}|$ using Wilson statistics:

$$|F_h^{\mathrm{ps}}| \simeq |F_h^{\mathrm{obs}}| \sqrt{\frac{\sum_i (f_i^{\mathrm{ps}})^2}{\sum_i f_i^2}}$$

Tabulation of $f_i^{\mathrm{ps}}(H)$ for each atomic type, as a function of $s = \sin\theta/\lambda$, is carried out via algorithmic procedures: place an atom at the origin, compute its charge density, apply density modification, and invert the process to obtain $f_i^{\mathrm{ps}}(H)$ (Li et al., 2017).

4. Theoretical Properties and Advantages

$R_{\mathrm{Tian}}$ possesses several theoretical properties:

• Resolution-Independence: $R_{\mathrm{Tian}}$ vanishes only for the true phase set, even at coarse resolutions, provided the density modification on single atoms is a suitable proxy.
• Low-Resolution Robustness: Unlike $R_{\mathrm{CF}}$, $R_{\mathrm{Tian}}$ strongly penalizes non-physical and trivial phase sets, providing higher discriminative power when data are incomplete or of modest resolution.
• No Maximum-Entropy Requirement: $R_{\mathrm{Tian}}$ omits maximum-entropy calculations present in previous metrics (e.g., Tian2 function), yielding lower computational complexity.
• Ideal for Global Optimization: The function approaches zero if and only if the trial density reproduces the pseudo-modified density. A plausible implication is that $R_{\mathrm{Tian}}$ constitutes a near-ideal target function for global phase search strategies (Li et al., 2017).

5. Empirical Performance and Examples

Tests performed on GaN (at 1.0 Å resolution) and a large organic molecule (C$_{52}$H$_{326}$O$_{19}$, at 2.5 Å), demonstrate that $R_{\mathrm{Tian}}$ unambiguously reaches its minimum at the correct phases, while $R_{\mathrm{CF}}$ may incorrectly favor wrong or trivial phase sets. Key empirical results are summarized as follows:

Structure	Phase Set Type	$R_{\mathrm{CF}}$	$R_{\mathrm{Tian}}$
GaN	Correct	75.9	25.6
GaN	All-zero-phase	$\sim$lower than 75.9	64.7
GaN	"Wrong" (2 Ga)	75.1	29.3
C$_{52}$H$_{326}$O$_{19}$	Correct	14,858.40	7,097.39
C$_{52}$H$_{326}$O$_{19}$	All-zero	14,376.63	7,685.98
C$_{52}$H$_{326}$O$_{19}$	Wrong (lowest $R_{\mathrm{CF}}$)	11,127.59	11,437.14
C$_{52}$H$_{326}$O$_{19}$	Random-phase	19,302.53	11,484.14

In all tested cases, $R_{\mathrm{Tian}}$ attained a unique global minimum at the correct phase set, unlike $R_{\mathrm{CF}}$, which could be minimized by incorrect phase solutions. The ratio $$H_{\mathrm{hy}} = R_{\mathrm{Tian}} / R_{\mathrm{CF}}$$ may provide a compact indicator, though the principal focus remains on $R_{\mathrm{Tian}}$ itself (Li et al., 2017).

6. Limitations, Validity, and Extensions

Principal limitations of the Tian function include:

• Commutativity Assumption: The PAM assumes the density-modification operator and assembly of pseudo-atomic densities commute ($P_{\mathrm{cryst}}^* \simeq P_{\mathrm{cryst}}$). Strongly collective density modulations may violate this approximation.
• Wilson-Statistics Approximation: Wilson scaling neglects inter-reflection correlations. For low-multiplicity or highly anisotropic situations, $F_h^{\mathrm{ps}}$ may be systematically biased.
• Tabulation of Pseudo-Scattering Factors: Each element demands a tabulated $f^{\mathrm{ps}}(\sin\theta/\lambda)$. Heterogeneous occupancy or disorder introduces definitional ambiguity.
• Optimization Requirements: Unlike iterative CF/HIO approaches, PAM frames phase retrieval as a global optimization across phase space. Efficient global search algorithms may be necessary to exploit $R_{\mathrm{Tian}}$ effectively.
• Applicability Limits: In cases of extremely limited resolution or very weak scattering (e.g., nanocrystals, powder data), the validity of the pseudo-atom approximation requires further empirical validation (Li et al., 2017).

7. Summary and Context within Crystallographic Phase Determination

$R_{\mathrm{Tian}}$ provides a rigorously formulated residual measure for phase determination in crystallography, particularly efficacious under conditions that undermine conventional metrics. The pseudo structure factor framework enables density-modification-informed merit evaluation without direct phase knowledge, substantially improving discrimination against incorrect solutions at low resolution. Within the broader context of direct methods and global optimization in crystallographic phase retrieval, the Tian function represents a targeted advancement in the construction of phase-informative residuals that leverage density modification effects absent in traditional figures of merit (Li et al., 2017).