## 1 Introduction  
Since the first half-metal (HM) NiMnSb compound was discovered in 1983 [1], the HM materials have attracted more and more scholars and researchers to investigate. The HM material has a very novel electronic structure, showing semiconducting property in on spin channel and metallic property in the other spin channel [2; 3]. This special electronic structure results in a perfect 100% spin polarization near the Fermi level, which makes HM a novel spintronic material. Until now, many different structures of materials have been found to be half-metallic, such as Heusler compounds, dilute magnetic semiconductors, and ferromagnetic metallic oxides, etc. [1; 2; 3; 4; 5; 6; 7; 8]. Among them, Heusler compounds are of great importance and present many unique physical properties, including the adjustable electronic structure and relative high Curie temperature [9; 10], etc.  
Heusler compounds, which have a general formula X\({}_{2}\)YZ, are a class of intermetallic compounds (also called full-Heusler compounds). In the formula X\({}_{2}\)YZ, X and Y atoms are usually transition metals, and Z is a main group element [11]. For full-Heusler compounds, there are two possible,different atomic arrangement fashions: the Cu\({}_{2}\)MnAl-type and Hg\({}_{2}\)CuTi-type structure [12]. Specially, when X and Y are the same atoms, the D0\({}_{3}\) structure is formed [13; 14]. And when one of the X atoms is substituted with a vacant, the Cl\({}_{\rm 1b}\) structure is formed [15]. When one of the X atoms is substituted by another different transition mental, the equiatomic quaternary Heusler compounds (EQHs) (LiMgPdSn structure) with the chemical formula of XMYZ are formed. Recently, several series of EQHs have been found and attracted wide attention. Compared with the binary and ternary Heusler HMs that have been mentioned above, the biggest advantage of the EQH HMs is their lower dissipation, which is caused by their lesser amount of atomic arrangement disorder in experiment [16].  
[PERSON] et al. have predicted several EQHs to be theoretical HM ferromagnets and have synthesized them experimentally, such as CoFeMnZ (Z = Al, Si, Ga, Ge), NiFeMnGa and NiCoMnGa [17; 18; 19; 20]. [PERSON] et al. have showed that NiFeMnGa and NiCoMnGa alloys are HM ferromagnets [14]. [PERSON] et al. Reported that CoFeCrAl and CoFeCrSi compounds are several excellent HM ferromagnets [21], [PERSON] et al. found that the NiFeTiP and NiFeTiSi alloys are true HM ferromagnets and that the NiFeTiGe has a near HM character [22]. All the above mentioned EQHs are those with 3d transition elements. Very recently, a series of new EQHs with a 4d transition element have been reported. Some of them are as follows: [PERSON] et al. have investigated the robust half-metallicity of the ZrCoTiZ (Z = Si, Ge Ga and In) [23]. The half-metallic and mechanic characters of ZrRhHfZ (Z = Al, Ga and In) [24] and FeCrRuSi [25] were investigated by [PERSON] et al. [PERSON] et al. have reported the excellent half-metallic property of ZrFeVZ (Z = Al, Ga and In) [26]. The new reports of EQHs with a 4d transition element expand the research scope of the spintronic materials. Thus, we can conclude that 4d-transition-elements-contained HMs are new potential spintronic materials worth researching.  
In our present work, the electronic, magnetic, half-metallic and mechanical properties of two newly designed EQH compounds, FeRhCrSi and FePdCrSi, were investigated by using the first-principles method. We also have discussed the effect of the hydrostatic strain and tetragonal distortion on their half-metallic properties. And the calculated results indicate that their half-metalicity can be kept in a wide lattice constant range and c/a ratio range. Moreover, the chemical stability and the mechanical properties of FeRhCrSi and FePdCrSi have also been studied in details.